Haoran Wang

Johann Guilleminot

Christian Soize

Modeling uncertainties in molecular dynamics simulations using a stochastic reduced-order basis

Keywords: Model Uncertainties, Molecular Dynamics, Stochastic Reduced-Order Modeling, Uncertainty Quantification

Modeling uncertainties in molecular dynamics simulations using a stochastic reduced-order basis

Introduction

Molecular Dynamics (MD) simulations are widely used in computational chemistry, materials science, and multiscale approaches. Applications include the prediction of material behavior under normal and extreme conditions, the interpretation of small-scale experimental observations and the design of new functionalized materials, to list a few. The construction and identification of relevant functional forms for interatomic potentials is a central aspect of all MD simulations. These intricate issues are usually undertaken with the aim to accurately estimate some problem-specific quantities of interest. As a result, several modeling solutions can coexist for a given system of (potentially coarse-grained) particles and can lead to very different predictions at the macroscopic level. Graphene is a relevant example for which many potentials were proposed, including the Morse, Tersoff, Airebo, and LCBOP potentials (see [START_REF] Belytschko | Atomistic simulations of nanotube fracture[END_REF], [START_REF] Tersoff | Modeling solid-state chemistry: Interatomic potentials for multicomponent systems[END_REF], [START_REF] Stuart | A reactive potential for hydrocarbons with intermolecular interactions[END_REF], and [START_REF] Los | Intrinsic long-range bond-order potential for carbon: Performance in monte carlo simulations of graphitization[END_REF] respectively). In addition, identified potential parameters often vary from one study to another, and nine versions of Tersoff-style potentials can be found for graphene [START_REF] Tersoff | Empirical interatomic potential for carbon, with applications to amorphous carbon[END_REF][START_REF] Tersoff | Modeling solid-state chemistry: Interatomic potentials for multicomponent systems[END_REF][START_REF] Tersoff | Carbon defects and defect reactions in silicon[END_REF][START_REF] Tersoff | Chemical order in amorphous silicon carbide[END_REF][START_REF] Tang | Atomistic simulation of thermomechanical properties of β-sic[END_REF][START_REF] Nordlund | Formation of ion irradiation induced small-scale defects on graphite surfaces[END_REF][START_REF] De Brito Mota | Structural properties of amorphous silicon nitride[END_REF][START_REF] Titantah | sp3/sp2 characterization of carbon materials from first-principles calculations: X-ray photoelectron versus high energy electron energy-loss spectroscopy techniques[END_REF][START_REF] Lindsay | Optimized tersoff and brenner empirical potential parameters for lattice dynamics and phonon thermal transport in carbon nanotubes and graphene[END_REF]. While the issue of selecting interatomic potentials can be in part circumvented by resorting to ab-initio MD simulations (for reasonably small systems), it remains a major bottleneck that can drastically affect the fidelity of the simulations.

In recent years, there has been a growing interest in the quantification and propagation of parametric uncertainties in MD simulations [START_REF] Angelikopoulos | Bayesian uncertainty quantification and propagation in molecular dynamics simulations: a high performance computing framework[END_REF][START_REF] Kim | Quantification of sampling uncertainty for molecular dynamics simulation: Time-dependent diffusion coefficient in simple fluids[END_REF], atomistic-to-continuum coupling strategies (see, e.g., [START_REF] Liu | Robustness of a new molecular dynamics-finite element coupling approach for soft matter systems analyzed by uncertainty quantification[END_REF][START_REF] Le | Stochastic continuum modeling of random interphases from atomistic simulations. Application to a polymer nanocomposite[END_REF]), and calibration and validation strategies in the presence of uncertainties (see [START_REF] Farrell | A bayesian framework for adaptive selection, calibration, and validation of coarsegrained models of atomistic systems[END_REF][START_REF] Farrell-Maupin | Adaptive selection and validation of models of complex systems in the presence of uncertainty[END_REF] and the references therein); see [START_REF] Ghanem | Handbook of Uncertainty Quantification[END_REF] for a general review about uncertainty quantification techniques. For instance, the effect of a stochastic Lennard Jones potential depth on the ionic conductance of Na + and Li + through a nanopore in SiO 2 was studied in [START_REF] Rizzi | Uncertainty quantification in MD simulations of concentration driven ionic flow through a silica nanopore. ii. uncertain potential parameters[END_REF], and the Bayesian solving of an inverse problem for the stochastic potential parameters of water molecules, based on density, enthalpy, and self-diffusion, was performed in [START_REF] Rizzi | Uncertainty quantification in MD simulations. part ii: Bayesian inference of force-field parameters[END_REF]. The case of model-form uncertainties, generated by the a priori selection of functional forms for interatomic potentials, has received very little attention to date. A methodology to estimate the sensitivity of MD predictions with respect to empiric potentials was proposed in [START_REF] Reeve | Error correction in multi-fidelity molecular dynamics simulations using functional uncertainty quantification[END_REF]. The proposed strategy provides a path towards the identification of relevant sources of modeling errors and allows errors in thermodynamical averages to be corrected between two models. The aim of this work is to provide a probabilistic framework to assess, model and propagate model-form uncertainties in MD simulations. The approach is based on the randomization of a reduced-order basis, using a random matrix approach [START_REF] Soize | A nonparametric probabilistic approach for quantifying uncertainties in low-dimensional and highdimensional nonlinear models[END_REF]. The stochastic reduced-order model thus constructed is specifically introduced as a mean to account for modeling errors in a nonparametric way, acting on the coefficients of the dynamics in the low-dimensional space. In contrast to the approach advanced in [START_REF] Reeve | Error correction in multi-fidelity molecular dynamics simulations using functional uncertainty quantification[END_REF], the methodology does not rely on an infinitesimal perturbation around a given potential, but rather consists of constructing a probabilistic model that is identified on a family of potentials that represent different functional forms and, possibly, different numerical parameterizations of the same potential. This strategy enables a complete probabilistic description and propagation of uncertainties, generating a family of random systems that includes random-system realizations that can be "far" from a nominal system, defined through a reference potential.

This paper is organized as follows. The molecular dynamics background is briefly reminded in Section 2. The construction of the stochastic reduced-order basis is then presented in Section 3. Applications are provided in Section 4.

List of notations.

d: dimension of the physical domain. n: reduced-order dimension. f: global force vector (physical space). f i : force vector acting on atom i (physical space). q: global coordinate vector (physical space). q i : position vector of atom i (physical space). N : total number of dofs before applying zero Dirichlet conditions. N a : total number of atoms in the physical space (standard MD). N CD : total number of dofs subjected to zero Dirichlet conditions. N : number of increments of engineering strain. N s : number of snapshots used to construct the reduced-order basis. M N,n : set of all the N × n real matrices. M n : set of all the n × n real matrices. M + n : set of all the symmetric positive-definite (SPD) n × n matrices. M +0 n : set of all the symmetric positive-semidefinite n × n matrices. M u n : set of all the n × n upper triangular real matrices. M +u n : set of all the n × n upper triangular real matrices with strictly positive diagonal. [I n ]: n × n identity matrix. [0 n ]: n × n zeros matrix.

[W ]: deterministic reduced-order basis (ROB).

[W]: stochastic reduced-order basis (SROB). E: mathematical expectation. Tr: trace operator. Var: variance.

Background in molecular dynamics

Basic equations

Consider a system composed of N a atoms, interacting in an open subset Ω ⊂ R d over a finite simulation time T . The dynamics of atom i is described by the Newton law

m i qi (t) = f i (q i (t), t) , q i (0) = q i 0 , q(i) (0) = qi 0 , (1)
where m i is the mass of the atom, qi is the acceleration vector corresponding to its position vector q i , f i is the force exerted on it, and (q i 0 , qi 0 ) are given initial conditions in R d × R d . The force vector can be written as

f i (q i (t), t) = f i int (q i (t)) + f i ext (t) , (2)
where f i int and f i ext represent interatomic contributions from surrounding atoms and external forces specified to prescribe, e.g., boundary conditions. Let q and f be the vectors of positions and forces for all atoms, gathered component-wise, and denote by [M ] the associated N ×N diagonal mass matrix (with N = d×N a ). The internal force is derived from a given interatomic potential V , defined on the configuration space, such that

f i int = -∇ q i V . (3)
The potential V typically incorporates several types of interactions, such as bonded and/or long-range interactions. The state of the system can be described in phase space, using the variables q and p, where p denotes the vector of momenta, p = [M ] q. The dynamics of the whole system can be expressed in a global matrix form as [M ]q(t) = f (q(t), t) , q(0) = q 0 , q(0) = q0 .

To simplify the presentation, the action of any control algorithm devised to impose target macroscopic properties (such as temperature, pressure, volume or energy) is omitted.

For later use, we introduce zero Dirichlet boundary conditions acting on a group of N CD atoms in the following matrix form:

[B] T (q(t) -q(0)

) = 0 NCD , t ∈ [0, T ] , (5)
where [B] is an N ×N CD matrix satisfying, by construction, the orthogonality property

[B] T [B] = [I NCD ].
In this work, MD simulations are conducted in the NVE ensemble, by using LAMMPS [START_REF] Plimpton | Fast parallel algorithms for short-range molecular dynamics[END_REF] for both standard and reduced-order MD simulations (note that the proposed SROM was implemented in LAMMPS).

Exploring the effect of potential selection: case of a graphene sheet

In order to qualitatively illustrate the impact of potential selection on the MD predictions, we consider the case of a 40.52 Å × 15.60 Å graphene sheet containing 272 carbon atoms, fixed on its left boundary and subjected to a sinusoidal load f i ext (t) = A sin(2πωt)e 2 for any atom i on its right boundary [START_REF] Lee | Proper orthogonal decomposition-based model order reduction via radial basis functions for molecular dynamics systems[END_REF], with A = 6 kcal.mol -1 . Å-1 and ω = 20 GHz; see Fig. 1. Figs. 2 and3 compare the e 1 -and e 2 -displacements of all atoms after 60,000 femiseconds (fs). These results are obtained by considering the two different initial configurations shown in Fig. 1 with different interatomic potentials. The impact of potential selection is clearly observed and turns out to be significant. In the case of an equilibrium initial configuration, the maximum magnitude of the e 1 -displacement for the atoms on the right end, predicted by the modified Morse potential, is twice as much as the one obtained with Tersoff potential. These differences are even more pronounced when the simulation is initialized with a non-equilibrium configuration. In this case, atoms are seen to move to completely opposite directions under the same loading conditions, depending on the interatomic potentials. Such differences at the atomistic scale can give rise to very different continuum characteristics of graphene. Indeed, it has been reported that the elastic moduli can differ by several hundred gigapascals when different potentials are used [START_REF] De Tomas | Graphitization of amorphous carbons: A comparative study of interatomic potentials[END_REF]. In the next section, we present a methodology that allows such model-type uncertainties to be incorporated in MD simulations. The model obtained with the full MD description or any reference model is referred to as the high-fidelity model (HFM) hereinafter.

Stochastic reduced-order modeling in MD simulations

This section is concerned with the construction of a stochastic reduced-order model for MD computations. To that end, a two-step approach is pursued. A reduced-order model is first introduced in Section 3.1. In a second step, the projection basis is subsequently randomized to account for uncertainties in a nonparametric approach, in Section 3.2. It should be noticed that the reduced-order model is not introduced to save computation time (which is not efficient here since evaluating the vector of forces necessitates mapping quantities back-and-forth), but rather to enable the proper randomization.

Reduced-order model for molecular dynamics

In this work, a reduced-order model (ROM) is constructed by using a proper orthogonal decomposition (POD) and the method of snapshots [START_REF] Sirovich | Turbulence and the dynamics of coherent structures. i. coherent structures[END_REF][START_REF] Kerschen | The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview[END_REF]. Let {V , = 1 . . . N V } be a set of candidates for the interatomic potential (see Eq. ( 3)). The dependence of the global force vector on the selected potential V is denoted by f (q(t), t; V ). In order to construct the reduced-order basis, for each 1 N V , we consider the state variable z (t) ∈ R N defined as

z (t) = q(t; V ) -q(0; V ) , ∀t ∈ [0, T ] . (6)
The function z satisfies

[M ]z (t) = f (t; V ) , (7)
supplemented with appropriate boundary conditions, with f (t; V ) = f (z (t) + q(0; V ), t; V ). In this work, two reduced-order bases are constructed: one basis is associated with a given potential V , while the other basis is aimed at representing the aforementioned set of potentials. These bases are denoted as {[W ], 0 N V }, where each element in {[W ], 1 N V } is the basis associated with each potential in the set {V , 1 N V }, and [W 0 ] is the basis adapted to all potentials. For 1, consider the discretization t 0 = 0 < t 1 < . . . < t Nt = T of time interval [0, T ] (taken independent of the potential), where t j = j∆t and ∆t is the constant time step. Let J = {j 1 , . . . , j Ns( ) } ⊂ {1, . . . , N t }, with 1 N s ( ) N t , not necessarily ordered and with distinct elements, and consider the sequence of MD snapshots {µ (k) } Ns( )

k=1 such that µ (k) = q(t j k ; ). For 1, let [X ] be the N × N s ( ) matrix defined as [X ] = [µ (1) . . . µ (Ns( )) ] , (8)
where µ (k) = µ (k) -q(0) for 1 k N s ( ). Upon introducing the singular value decomposition

[X ] = [U ][S ][V ] T , (9)
where the singular values are assumed decreasingly ordered, the reduced basis [W ] is classically obtained by retaining the n first columns (which are referred to as POD modes below) of [U ]. The number n of modes can be determined through a convergence analysis enabling a tradeoff between dimensionality reduction and projection error (see Section 4 for an example).

Similarly, let [X 0 ] = [[X 1 ] . . . [X N V ]],
where the number of snapshots N S ( ) is such that convergence is reached for all 1 N V . Consider next the singular value decomposition (with decreasingly ordered singular values)

[X 0 ] = [U 0 ][S 0 ][V 0 ] T . ( 10
)
The reduced-order basis [W 0 ] is then constructed by retaining the n 0 first columns of [U 0 ]. For 0 N V , we now introduce the linear mapping

z (t) = [W ]y (t) , ( 11
)
where y is the reduced variable with values in R n , n N . The matrix [W ] satisfies the orthogonality property

[W ] T [W ] = [I n ] (12)
and the boundary condition

[B] T [W ] = [0 NCD,n ] . (13)
It should be noticed that this construction slightly differs from the one followed in other studies, where snapshots are centered with respect to the mean value (estimated from the snapshots) and the basis is made of the dominant eigenvectors of the snapshot covariance matrix [START_REF] Lee | Proper orthogonal decomposition-based model order reduction via radial basis functions for molecular dynamics systems[END_REF][START_REF] Hoang | An hp-proper orthogonal decomposition-moving least squares approach for molecular dynamics simulation[END_REF]. The strategy proposed above was found to provide more accurate results in MD simulations, especially when N a becomes large. For 0 N V , the Galerkin projection of Eq. ( 7) expressed in terms of atom displacements (with physical variable z ) reads as

[M ]ÿ (t) = F (t) (14)
where

[M ] = [W ] T [M ][W ] (15)
is the projected mass matrix and F is the reduced force vector

F (t) = [W ] T f (t; V ) . (16)
In general, and as opposed to other frameworks involving nonlinear continuum mechanics formulations, solving for the reduced variable does not allow for a reduction of the computation time. This is due to the fact that most CPU time is spent on assembling f (t; V ) and computing F (t), which is still necessary to evaluate the right-hand side of Eq. ( 16). It should be noticed that surrogate models can be used to accelerate this intensive task. A discussion about this particular aspect is beyond the scope of this paper; see, e.g., [START_REF] Lee | Proper orthogonal decomposition-based model order reduction via radial basis functions for molecular dynamics systems[END_REF] and the references therein.

Stochastic reduced-order model for molecular dynamics

In order to account for model-type uncertainties in the MD formulation, for 0 N V , the deterministic reduced-order basis [W ] is replaced by a random matrix [W ], defined on a probability space (Θ, T , P) and with values in the subset

S N,n = {[Y ] ∈ M N,n | [Y ] T [Y ] = [I n ] , [B] T [Y ] = [0 NCD,n ]} (17)
of the compact Stiefel manifold

S N,n = {[Y ] ∈ M N,n | [Y ] T [Y ] = [I n ]} , (18)
where M N,n is the set of all N × n real matrices and

S N,n ⊂ S N,n ⊂ M N,n . It follows that [W ] satisfies [W ] T [W ] = [I n ] (19)
and

[B] T [W ] = [0 NCD,n ] (20)
almost surely and can thus be used as a stochastic counterpart for the ROM introduced in Section 3.1. In this work, the probabilistic model for the stochastic reduced-order basis [W ] constructed in [START_REF] Soize | A nonparametric probabilistic approach for quantifying uncertainties in low-dimensional and highdimensional nonlinear models[END_REF] is deployed.

In order to make the paper self-contained, some elements of this construction are provided in Section 3.2.1.

The identification of the hyperparameters defining this random matrix is discussed in Section 3.2.3.

Construction of a stochastic reduced-order basis.

In this section, the value of is fixed in {0, . . . , N V }. The probabilistic model for the SROB [W ] must be constructed such that (i) Eq. ( 19) and ( 20) hold almost surely and (ii) the number and dimensionality of model parameters (which are referred to as hyperparameters) remain small enough to make model identification tractable.

In essence, the modeling approach proposed in [START_REF] Soize | A nonparametric probabilistic approach for quantifying uncertainties in low-dimensional and highdimensional nonlinear models[END_REF] consists of (i) making the random matrix [W ] fluctuate around the deterministic ROB [W ], by using the tangent vector space T [W ] S N,n of S N,n at [W ] ∈ S N,n , and (ii) introducing an appropriate, smooth mapping between this tangent space and the compact Stiefel manifold to ensure that the SROB fulfills Eqs. [START_REF] Ghanem | Handbook of Uncertainty Quantification[END_REF][START_REF] Rizzi | Uncertainty quantification in MD simulations of concentration driven ionic flow through a silica nanopore. ii. uncertain potential parameters[END_REF] almost surely. Such a construction can be achieved through a sequence of algebraic decompositions, inferred from proper parameterizations of the involved matrix manifolds. These decomposition are detailed hereafter.

Let [U ] = [U ,1 . . . U ,n ] be a random matrix defined on the probability space (Θ, T , P), with values in M N,n , and introduce the random matrix

[Z ] = [A ] -[W ][D ] , (21)
where the random matrices [A ] and [D ] are given by

[A ] = [U ] -[B][B] T [U ] (22)
and

[D ] = 1 2 [W ] T [A ] + [A ] T [W ] . (23)
It can then be shown that [Z ] takes its values in T [W ] S N,n and that [A ] satisfies Eq. ( 20) almost surely.

A smooth mapping between T [W ] S N,n and S N,n is then introduced to define [W ] as

[W ] = ([W ] + s [Z ]) [I n ] + s 2 [Z ] T [Z ] -1/2 , ( 24
)
where 0 < s is a model parameter controlling the statistical fluctuations of [W ]. These fluctuations occur around [W ] when the random matrix [Z ] is centered, which in turn implies that [A ] (and thus, [U ]) must be centered as well:

E{[U ]} = [0 N,n ] . (25)
Notice, however, that the mean [W ] of [W ] does not coincide with [W ]. To provide some modeling flexibility while keeping the parameterization reasonably low-dimensional, the random matrix [U ] is written as

[U ] = [G ][σ ] , (26)
where [σ ] is any matrix in M u n and where [G ] is a random matrix defined on the probability space (Θ, T , P) (see below), with values in M N,n , that satisfies

E{[G ]} = [0 N,n ] (27)
and is constructed such that

E{[G ] jk [G ] j k } = [C N, ] jj δ kk . (28)
It can then be deduced that the fourth-order covariance tensor

E{[U ] ⊗ [U ]} of the random matrix [U ] reads as E{[U ] jk [U ] j k } = [C N, ] jj [c n, ] kk , (29)
where

[C N, ] ∈ M + N and [c n, ] = [σ ] T [σ ] ∈ M +0
n are covariance-like matrices (note that this construction includes the particular case where [σ ] is chosen as the Cholesky factorization of [c n, ]). It is seen from Eq. ( 29) that the matrix [C N, ] controls the correlation along the components of the random vector U ,k , 1 k n , while [c n, ] governs the correlation between any pair of random vectors U ,k and U ,k , 1 k, k N . From the previous derivations, it is seen that the stochastic model can, indeed, be constructed on the random matrix [G ]: provided that Eqs. ( 27) and ( 28) hold, the use of Eqs. ( 21), ( 22), ( 23) and [START_REF] De Tomas | Graphitization of amorphous carbons: A comparative study of interatomic potentials[END_REF] in Eq. ( 24) makes the SROB [W ] a mathematically consistent randomization of the deterministic ROB [W ]. It now remains to define the random matrix [G ]. As discussed in [START_REF] Soize | A nonparametric probabilistic approach for quantifying uncertainties in low-dimensional and highdimensional nonlinear models[END_REF], the definition of the probability law of [G ] is somewhat arbitrary in the present context, since the framework is not focused on modeling parametric uncertainties. In this work, the formulation derived in Appendix D of [START_REF] Soize | A nonparametric probabilistic approach for quantifying uncertainties in low-dimensional and highdimensional nonlinear models[END_REF] is used and [G ] is defined and generated by discretizing independent copies of a second-order, stationary and centered Gaussian random field (see the above reference for details). The covariance function of the latter is parameterized by a scalar parameter β > 0 controlling the rate of decay of the correlation along the rows of [G ] (note that the columns of [G ] are uncorrelated by construction; see Eq. ( 28)), and

[C N, ] is thus an implicit function of β , [C N, ] = [C N, (β )].
The stochastic model for the SROB is hence parameterized by a maximum of 2 + n (n + 1)/2 hyperparameters (namely s > 0, β > 0 and the nonzero entries (with a maximum of n (n + 1)/2 terms) of the upper triangular matrix [σ ]), and a minimum of 2 + n hyperparameters. A strategy to identify these parameters is sketched in the next section. The approach will be exemplified and purposely adapted in Section 4.

Definition of the SROM [W ]

For 0 N V , the SROM is written as

Z (t) = [W ]Y (t) (30)
and

[M ] Ÿ (t) = F (t) , (31)
where

[M ] = [W ] T [M ][W ] (32)
and

F (t) = [W ] T f (t; V ) . (33)
In the equation above, V is the selected interatomic potential when 1 N V . For = 0, the potential V 0 represents a reference potential in the family of potentials under consideration, denoted by V * , with 1 * N V .

Identification of the SROM [W ]

Throughout this section, the value of is fixed in {0, . . . , N V } and the dependence of the hyperparameters on is omitted to simplify the presentation. In general, the definition of a strategy to identify the parameters of the SROB is problem specific and typically based on the selection of a quantity of interest (QoI). Regardless of the latter, it is reasonable to assume that the (potentially time-dependent) mean and variance of the QoI constitute target properties. This allows us to derive the following prototypical form for the cost function:

J(α) = wJ m (α) + (1 -w)J v (α) , (34)
where α symbolically gathers all hyperparameters (α = (s, β, [σ])), J m and J v are normalized relative errors in mean and variability (which can be expressed in terms of variance or confidence interval) between some reference dataset and results predicted with the SROM (see Section 4 for examples), and w ∈ [0, 1] is a weight allowing the two contributions to be balanced. The optimal solution is then defined as

α opt = arg min α ∈ Cα J(α) , (35)
where the admissible set is taken, in this paper, as

C α = R >0 × R >0 × M +u n (see Section 3.2.1
). The choice of the reference dataset is goal-oriented. For instance, if this reference is taken as the prediction of the QoI obtained through the HFM (which is the non-reduced molecular dynamics simulation in this study), then the SROM aims at accounting for the modeling error generated by the reduction (see Section 4.2 for an application). If some variability is introduced in the HFM using different interatomic potentials, the SROM is intended to incorporate both the reduction error and model-type uncertainties. This case will be specifically addressed in Section 4.3. Finally, the reference dataset can be composed of experimental data, in which case the SROM integrates model reduction error, model-form uncertainties and possible parametric uncertainties.

The optimization problem given by Eq. ( 35) is nonconvex and generally high-dimensional when [σ] ∈ M +u n . It should be noticed that the dimension of this problem can be reduced if [σ] is sought in M u n (see [START_REF] Soize | Probabilistic learning for model-form uncertainties in nonlinear computational mechanics[END_REF]). In order to solve this problem at a reasonable computational cost, a multi-step methodology is pursued as follows [START_REF] Soize | A nonparametric probabilistic approach for quantifying uncertainties in low-dimensional and highdimensional nonlinear models[END_REF]. In the description below, variables that are not active at a given step are assigned the values determined from the previous step.

Step 1. Active variables: (s, β, σ), where σ is such that

[σ] = σ[I n ].
An optimal point that minimizes the cost function J over a tensor product grid on the parameter space is sought. The following support restrictions are applied: 0 < s s s u , 0 < β β β u and 0 < σ σ σ u . The solution is denoted by

α 1 = (s 1 , β 1 , [σ 1 ]), with [σ 1 ] = σ 1 [I n ].
Step 2. Active variables: (s, β, [σ]), where [σ] is a diagonal matrix.

The interior point method is used to solve Eq. ( 35), considering the inequality constraints from Step 1 and starting from α 1 . The solution is denoted by α 2 = (s 2 , β 2 , [σ 2 ]).

Step 3. Active variable: [σ * ], where [σ * ] is an upper triangular matrix with null diagonal such that

[σ] = [σ 2 ] + [σ * ].
The interior point method is used to solve Eq. [START_REF] Bernal | The structure of graphite[END_REF], where no constraint is imposed on the entries of [σ * ]. The solution is denoted by

α 3 = (s 2 , β 2 , [σ 3 ]), with [σ 3 ] = [σ 2 ] + [σ * 3 ].
Step 4. Active variable: β.

The optimal value β opt of β is found by minimizing the cost function J over a uniform discretrization of [β , β u ]. The solution is denoted by α 4 = (s 2 , β opt , [σ 3 ]).

Step 5. Active variables: (s, [

σ d ]), where [σ d ] is a diagonal matrix such that [σ] = [σ d ] + [σ * 3 ].
The interior point method is used to solve Eq. ( 35) and find the optimal values s opt and [σ opt d ]. The optimal solution is then defined as

α opt = (s opt , β opt , [σ opt ]), with [σ opt ] = [σ opt d ] + [σ * 3 ].
Note that the lower and upper bounds in the above steps are problem specific.

Applications

Overview

In this section, we assess the relevance of the stochastic reduced-order model to represent, and eventually to propagate, model-form uncertainties in MD simulations. A graphene sheet system is selected as a prototypical application, and the following set of potentials (available in LAMMPS) will be considered:

• V 1 : modified Morse [25]; • V 2 : Airebo [3]; • V 3 : BOP [31]; • V 4 : LCBOP [4];
• V 5 : REBOOII [START_REF] Brenner | A second-generation reactive empirical bond order (rebo) potential energy expression for hydrocarbons[END_REF];

• V 6 : Tersoff-2010 [START_REF] Lindsay | Optimized tersoff and brenner empirical potential parameters for lattice dynamics and phonon thermal transport in carbon nanotubes and graphene[END_REF] .

The values of the parameters used for each potential can be found in the references above.

The case of uncertainties raised by model reduction is first addressed in Section 4.2 for N V = 1, with V 1 corresponding to a modified Morse potential (application 1). The objective is to identify the associated SROB [W 1 ] modeling the uncertainties generated by the use of the ROM instead of the HFM.

Model uncertainties generated by the selection of potentials is then discussed in Section 4.3, including the effect of the truncation error (application 2). In this case, we consider the six different interatomic potentials listed above, N V = 6. A SROB [W 1 ] is constructed by randomizing the ROB [W 1 ], associated with a modified Morse potential V 1 , and the SROM is identified to account for the variability induced by the use of the family of potentials {V 1 , . . . , V 6 }.

In the last application, discussed in Section 4.4, we investigate the use of a SROB [W 4 ], associated with the LCBOP potential V 4 (application 3-a), and a SROB [W 0 ] constructed by considering snapshots associated with all potentials {V 1 , . . . , V 6 } (application 3-b). In both cases, the SROM is identified to represent the model error between molecular dynamics computations and a continuum-mechanics model calibrated with experimental data.

All MD simulations reported in this section are performed in LAMMPS with a time step of 1 fs. As previously indicated, the SROMs were also implemented in LAMMPS.

Application 1: quantification of model uncertainties induced by order reduction

In this first application, the 272-atom graphene sheet system introduced in Section 2.2 is considered (with A = 6 kcal.mol -1 . Å-1 and ω = 20 GHz), given the high sensitivity of the results to interatomic potentials (see the parametric analysis in the aforementioned section). A modified Morse potential [START_REF] Belytschko | Atomistic simulations of nanotube fracture[END_REF] is used for the construction of the deterministic reduced-order basis.

Construction of the deterministic ROB

In order to construct the deterministic reduced-order basis [W 1 ], the graphene sheet is subjected to a 300K thermo-equilibrium and snapshots are collected every 1, 000 fs. The number of snapshots necessary to construct the ROB is determined by analyzing the convergence of the eigenvalues of the empirical estimate of the second-order tensor moment of µ, using the function N s (1) → Tr([S(N s (1))] 2 )/N s (1) (see Eq. ( 10)) and the value N s (1) = 1, 000 is selected (with snapshots uniformly sampled over a period of the excitation). Three snapshots are shown in Fig. 4.

in which N s (1) was determined from the convergence analysis (N s (1) = 1, 000), and selecting n 1 < N s (1) such that ε(n 1 ) ε. The graph of the function n 1 → ε(n 1 ) is shown in Fig. 5. In this application, the value ε = 10 -4 is chosen, which corresponds to n 1 = 10, and computations are initialized with a non-stress-free configuration. A comparison between the HFM (standard molecular dynamics) and ROB-based predictions for the e 2 -displacement is displayed in Fig. 6. The discrepancies observed between the HFM and ROB-based results indicate that the ROM cannot fully capture the nonlinear response of the system, despite the use of a small value for the error parameter ε. In the next section, the SROM is used to account for this reduction-induced error.

Stochastic reduced-order molecular dynamics

In order to deploy the SROM, the hyperparameters defining the stochastic reduced-order basis [W 1 ] must be identified on a reference dataset. As discussed earlier, the definition of this dataset and of the cost function J is goal-oriented. Following the results in Section 4.2.1 (see Fig. 6), the e 2 -displacement at time t * = 60, 000 fs is selected as the QoI, and the cost function is constructed to impose a match between the HFM result and the mean of the SROM-based predictions. Fluctuations in the stochastic model are calibrated to account for the fluctuations of the HFM outcomes around the mean response of the SROM.

Let y HFM and y ROM be the vectors in R Na (with N a = 272) gathering the e 2 -displacements of all atoms at time t * , obtained with the HFM and the ROM, respectively. Denote by Y SROM (α; V 1 ) the random vector defined in (Θ, T , P), with values in R Na , corresponding to the predictions (expressed in physical space) at 34)) is then defined as

J m (α) = y HFM -E{Y SROM (α; V 1 )} 2 / y HFM 2 , ( 37
)
while the contribution related to fluctuations is taken as

J v (α) = v HFM-ROM -v SROM (α; V 1 )} 2 / v HFM-ROM 2 , ( 38
)
where v HFM-ROM is a γ-weighted relative error between the HFM and ROM results,

v HFM-ROM i = γ|y HFM i - y ROM i | for 1 i N a (with γ 0), and the components of v SROM (α; V 1 ) are v SROM i (α; V 1 ) = Var{Y SROM i (α; V 1 )} for 1 i N a .
The hyperparameters defining the SROM are identified following the steps presented in Section 3.2.3, with w = 0.99 (notice that this value allows J m and J v to have different contributions in the global cost function). The e 2 -displacement for all atoms at time t * is shown in Fig. 7 for the HFM, the (deterministic) ROM and the SROM. The confidence interval is estimated with 1, 000 independent realizations of the SROM.

It is seen that the mean value of the SROM computation closely replicates the result obtained with the HDM, regardless of the value of γ. The effect of the latter on the spread of the confidence interval can also be observed. In particular, the variability exhibited by the QoI decreases with γ, with a plateau observed for some minimal value of the parameter. In the present case, γ = 0.5 yields a reasonable approximation in the sense that (i ) the mean obtained with the SROM is close of the HFM result, and (ii ) the HFM prediction is included in the confidence region associated with the identified SROM. The effect on the total energy is next illustrated in Fig. 8. In this figure, ROM-based results are presented for several values of n. Note that the construction of SROMs for all these values of n was not pursued, since the number of hyperparameters raises up significantly. In addition, the Gaussian kernel density estimation and confidence region for the interatomic bond length, obtained with the HFM, ROM, and SROM, are shown in Fig. 9. The discrepancies between the HFM and the ROM, observed for both the energy and the probability density function of the bond length, are due to the value of n, which is not sufficiently large. It should be noticed that n could have been chosen greater (at the expense of additional computational cost): the selected value is, however, sufficient to construct the SROM, which still delivers a good approximation for the energy (see Fig. 8). Moreover, it is seen that the probability density function estimated for one realization of the SROM is close to the probability density function obtained with the HFM, and that the latter is, indeed, included in the 98% confidence region of the SROM. This demonstrates the ability of the SROM to generate a coherent confidence region, even when the ROM has not converged to the HFM.

Application 2: quantification of model uncertainties due to interatomic potential selection

We now turn to the assessment of model uncertainties induced by interatomic potentials. To that end, the graphene sheet model is still considered and six widely used potentials (namely, the Airebo, BOP, LCBOP, modified-Morse, REBOII, and Tersoff-2010 potentials) are considered to generate a spread in the HFM results. The predicted e 2 -displacement at a given time instance is shown in Fig. 10. It is seen that the choice of the potential has a significant impact on the predictions, with a maximum difference over all atoms equal to 59.7% (note that this relative error is normalized by the average between the maximum and minimum values). More specifically, it is observed that the BOP potential predicts the most compliant response, while the Tersoff-2010 potential yields the most rigid response (under the same boundary conditions).

In the present application, the modified Morse potential is used to construct the ROB [W 1 ]. In order to identify the parameters in the SROB [W 1 ], the definition of the cost function J is adapted such that the minimum and maximum envelopes of the response taken over all potentials are contained within the 98% confidence region of the SROM. This constitutes an alternative to the choice given by Eqs. (37) and (38). We adopt the same notations as in Section 4.2.2 for the quantity of interest (that is, the e 2 -displacement of all atoms at t * ), and let y HFM and y HFM + be the vectors of lower and upper envelopes over the potentials, (y HFM -

) i = min({y i (V j )} N V j=1 ) and (y HFM + ) i = max({y i (V j )} N V j=1 )
, where N V is the number of potentials considered in the HFM (N V = 6 in the present application). In addition, we introduce the two vectors q SROM -(V 1 ) and q SROM + (V 1 ) in R Na , the components of which correspond to the lower and upper values of the 98% confidence region of the components of

Y SROM (α; V 1 ): (q SROM - (V 1 )) i = P - 98% (Y SROM i (α; V 1 )) and (q SROM + (V 1 )) i = P + 98% (Y SROM i (α; V 1 )).
We then take J(α) = J v (α), with The SROM is calibrated pursuing the multi-step approach (see Section 3.2.3), and the comparison between the HFM outcomes (for several potentials) and the SROM is shown in Figs. 11 and12. It is seen that the SROM allows the variability generated with the considered set of potentials to be accurately reproduced for all atoms of the graphene sheet.

J v (α) = ( y HFM + -q SROM + (V 1 ) 2 + y HFM - -q SROM - (V 1 ) 2 )/ y HFM + -y HFM - 2 . ( 39

Application 3: quantification of model uncertainties based on a continuum model

In this final application, HFM predictions obtained for different potentials are gauged against results obtained by a hyperelastic continuum model. The latter was specifically constructed and identified in [START_REF] Cadelano | Nonlinear elasticity of monolayer graphene[END_REF], showing a good agreement with the AFM experiments conducted in [START_REF] Lee | Measurement of the elastic properties and intrinsic strength of monolayer graphene[END_REF]. It is used below to generate a reference dataset under uniaxial loading conditions, referred to as the continuum model. The SROM is then deployed to represent the variability induced by the choice of an interatomic potential.

Molecular dynamics setup

A 41.74 Å × 42.54 Å graphene sheet is built with periodic boundary conditions as shown in Fig. 13. The graphene model includes 680 carbon atoms, which is large enough to make the response of the system size independent. An interatomic bond distance of 1.418 Å , widely accepted in the literature, is adopted [START_REF] Bernal | The structure of graphite[END_REF].

In order to construct an appropriate ROB, two MD simulations are performed: the graphene model is subjected to uniaxial-strain deformation (Dirichlet boundary conditions), first along the zigzag direction, and then along the armchair direction (see the right panel in Fig. 13). Periodic Dirichlet boundary conditions are then applied, and a classical relaxation is imposed after each increment of the boundary displacement. The boundary displacement is imposed through a stepping scheme with increments imposed every 100 fs (leading to N increments of the engineering strain), and the sheet is stretched up to a strain of 0.3, for T = 30, 000 fs. A time step of 1 fs is used and all the MD simulations are performed under the NVE ensemble. Five interatomic potentials are considered in our study, namely the Airebo [START_REF] Stuart | A reactive potential for hydrocarbons with intermolecular interactions[END_REF], BOP [START_REF] Zhou | An analytical bond-order potential for carbon[END_REF], LCBOP [START_REF] Los | Intrinsic long-range bond-order potential for carbon: Performance in monte carlo simulations of graphitization[END_REF], REBOII [START_REF] Brenner | A second-generation reactive empirical bond order (rebo) potential energy expression for hydrocarbons[END_REF] and Tersoff-2010 [START_REF] Lindsay | Optimized tersoff and brenner empirical potential parameters for lattice dynamics and phonon thermal transport in carbon nanotubes and graphene[END_REF].

Following [START_REF] Cadelano | Nonlinear elasticity of monolayer graphene[END_REF], the strain energy under uniaxial tension along zigzag and armchair directions (respectively denoted with underscripts zz and ac) can be expressed as

u zz ( ) = 1 2 E 1 -v 2 2 + 1 6 C 111 3 (40)
and

u ac ( ) = 1 2 E 1 -v 2 2 + 1 6 C 222 3 , (41)
where E = 312N m -1 is the Young's modulus, v = 0.31 is the Poisson ratio, C 111 = -1689.2N m -1 and C 222 = -1487.7N m -1 are two elastic constants and is the engineering strain. Fig. 14 shows the evolution of the strain energy as predicted by either the continuum model above (which matches the experimental results) or by molecular dynamics simulations, parameterized by the different potentials. The model-type uncertainties can clearly be visualized, and it is seen that none of the interatomic potentials captures the nonlinear behavior of the graphene system predicted by the continuum model. The ROB is constructed with the concatenation of 1, 000 snapshots in each direction (hence, a total of 2, 000 snapshots), obtained with the LCBOP potential V 4 . Following an analysis of convergence, a reduced dimension n 4 = 10 is used. Let u CM be the vector in R 2N gathering the values of the potential energy (strain energy) at all engineering strain increments along the two (armchair and zigzag) directions, obtained with the experimentally-consistent continuum model (denoted as CM). The hyperparameters are identified by using the cost function defined in Eq. (34), J(α) = wJ m (α) + (1 -w)J v (α), in which

J m (α) = u CM -E{U SROM (α; V 4 )} 2 / u CM 2 , (42)
and where U SROM (α; V 4 ) is the concatenated random vector with values in R 2N , corresponding to the two MD predictions (along the two directions) of the potential energy obtained with the SROB [W 4 ] for all strain increments, the components of v CM-ROM are defined as

J v (α) = v CM-ROM -v SROM (α; V 4 )} 2 / v CM-ROM 2 , (43)
v CM-ROM i = γ|u CM i -u ROM i | for 1 i 2N (with γ = 0.5), and the components of v SROM (α; V 4 ) are given by v SROM i (α; V 4 ) = Var{U SROM i (α; V 4 )} for 1 i 2N .
Results obtained with the identified model are shown in Figs. SROM allows the mean model to approach the continuum-mechanics target with a reasonably good accuracy, regardless of the loading direction. For the specific case of the armchair direction, the use of the SROB [W 4 ], constructed with the snapshots associated with V 4 only, leads to an increase of the variability. This suggests that the use of a SROB [W 0 ], constructed with the concatenation of snapshots obtained for the family of potentials, would decrease the level of uncertainties, that is, the width of the confidence region: this point will be discussed in detail in Section 4.4.3. In terms of quality assessment, it should be noted at this stage that while the MD results associated with all potentials are not considered in the identification of the SROM, the latter is still able to accurately reproduce the uncertainties generated by the considered family of potentials.

The confidence interval on the bond length is finally shown in Fig. 17, in both the armchair and zigzag directions. It is seen that the estimated confidence interval is very sensitive to the direction under which loading is applied. This property is due to the very structure of the graphene sheet, depicted in Fig. 13 (note, in particular, that the mechanical response is not invariant with respect to π/2 rotations). In addition, the variability exhibited by the probability density functions for the bond length is rather large, which is due to the spread observed with the identified [W 4 ] SROM (see Fig. 15). The impact of the latter can also be noticed on the mean bond length, estimated as a function of the applied engineering strain; see Fig. The SROM defined with the SROB [W 0 ], considering V 4 as the reference potential ( * = 4), is finally deployed. The results obtained in this case are shown in Fig. 19 and must be compared with those presented in Fig. 15. It is seen that the SROM with [W 0 ] allows us to predict the target (which is the experimentallyconsistent continuum-mechanics model) with a level of uncertainties that is significantly smaller than the one for the with [W 4 ]. More precisely, the level of variation at the largest strain (ε = 0.3) is decreased from 51.33% to 6.51% in the armchair direction, and from 24.17% to 10% in the zigzag direction. Quite remarkably, the SROM can predict the target with a confidence region that does not include the ROM response. This is an important difference with the results presented for the SROM defined with [W 4 ], in which the ROM results are included in the confidence region for both the armchair and zigzag directions (see Fig. 15) -at the expense of a larger confidence region.

Confidence intervals on the bond length are shown in Fig. 20 for the two loading directions. By comparing this figure with Fig. 17, where the confidence intervals were estimated with the SROB [W 4 ], it is seen that the use of the SROB [W 0 ] allows the experimentally-consistent continuum model to be approximated with much less variability. This conclusion can also be drawn from Fig. 21, where the mean bound length is represented as a function of the applied strain. The lower and upper bounds of the confidence interval for the mean bound length are almost indistinguishable, which is in contrast with the result obtained with the SROB [W 4 ] (see Fig. 18). These results demonstrate the robustness of the SROM, especially when constructed with the whole family of potentials.

How to use the proposed methodology to analyze uncertainties in molecular dynamics?

The proposed methodology can be summarized as follows.

1. Define a family of N V appropriate interatomic potentials that are adapted to a given quantity of interest. This family contains a reference potential V * that will be subsequently used by the SROM.

2. Perform a set of molecular dynamics simulations, potentially with different loading conditions, for all the N V selected potentials. Concatenate all the snapshots thus obtained in a deterministic matrix [X 0 ] (see above Eq. ( 10)).

3. Construct the ROB [W 0 ], using a singular value decomposition of [X 0 ]. Perform a convergence analyze to determine the number of mode n 0 , for a selected value of ε.

4. Define an observable and a suitable cost function. Identify the hyperparameters of the SROB [W 0 ], using V * and following the methodology detailed in Section 3.2.3.

For the SROM identified in

Step 4, with [W 0 ] and V * , perform the propagation of uncertainties and conduct a statistical analysis for any quantity of interest (mean, variance, confidence region, probability distribution).

It should be noticed that the first step is fundamental, since the selected family of potentials generates the level of uncertainties the SROM aims to model.

Conclusion

A methodology that allows for the integration of model-form uncertainties in molecular dynamics simulations was proposed. The strategy relies on the proper randomization of a reduced-order basis, obtained by the method of snapshots in the configuration space. The relevance of the approach was finally assessed on a variety of situations, including the representation of uncertainties raised by model reduction and interatomic potential selection. The presented framework can be used, in particular, to perform robust MD-based design optimization of materials.

It should be noticed that the proposed approach allows for a complete probabilistic modeling of uncertainties and their propagation for any relevant quantity of interest. This is in contrast with an interval method where only minimum and maximum deterministic bounds, associated with two potentials in the selected family, would be considered. In addition to model-form uncertainties, the SROM also enables the robust, simultaneous treatment of parametric uncertainties on a set of potentials.

Figure 1 :

 1 Figure 1: Graphene sheet initial configurations: equilibrium stress-free configuration (left panel) and non-equilibrium configuration (right panel). Red-circled atoms on the left edge satisfy a zero displacement condition, while atoms on the right edge are subjected to force f .

Figure 2 :Figure 3 :

 23 Figure 2: Graph of the e 1 -displacement for all atoms at time t = 60, 000 (fs), starting from the equilibrium stress-free configuration (left panel) and the non-equilibrium configuration (right panel) shown in Fig. 1. The results associated with four potentials are displayed: Modified Morse (blue), Tersoff (red), Airebo (orange), and LCBOP(purple). Atoms are numbered from bottom to top and left to right.

Figure 4 :

 4 Figure 4: Examples of snapshots used to construct the reduced-order basis [W 1 ]. The dimension n 1 of the ROB can be determined in a standard manner, by characterizing the convergence of the function ε(n 1 ) = 1 -

Figure 5 :Figure 6 :

 56 Figure 5: Plot of the error function n 1 → ε(n 1 ).

Figure 7 :

 7 Figure 7: Vertical displacement for all atoms at time t * : HFM (blue line), ROM-based (red line) and SROM-based (orange line: mean; yellow region: 98% confidence interval) results. The left and right panels correspond to γ = 1 and γ = 0.5, respectively.

Figure 8 :

 8 Figure 8: Evolution of the total energy for the HFM, the ROM with n ∈ {10, 50, 250}, and the identified SROM.

Figure 9 :

 9 Figure 9: Gaussian kernel density estimator of the probability density function for the bond length for the HFM, ROM and SROM (left), and confidence interval for the probability density function estimated with the SROB [W 1 ] (right).

Figure 10 :

 10 Figure 10: Vertical displacement at t * = 60, 000 fs, predicted by the HFM with different interatomic potentials.

Figure 11 :

 11 Figure 11: Vertical displacement of all atoms at t * = 60, 000 fs, predicted by the HFMs, and associated 98% confidence interval estimated with the SROM (from 1, 000 independent samples).

4. 4 . 2 .

 42 Application 3-a: stochastic reduced-order modeling with SROB [W 4 ]

Figure 12 :

 12 Figure12: Vertical displacement of a group of atoms at t * = 60, 000 fs, predicted by the HFMs, and associated 98% confidence interval estimated with the SROM (from 1, 000 independent samples).

Figure 13 :

 13 Figure 13: Left panel: graphene sheet model with periodic boundary conditions (gray atoms are periodic images). Right panel: definition of the armchair and zigzag directions.

Figure 14 :

 14 Figure 14: Evolution of the strain energy predicted by the continuum-mechanics model and different molecular dynamics systems. The left and right panels correspond to the armchair and zigzag directions, respectively.

15 and 16 .Figure 15 :

 1615 Figure 15: Confidence region and evolution of the potential energy function in the armchair and zigzag directions: continuummechanics model and [W 4 ] SROM predictions.

Figure 16 :

 16 Figure 16: Confidence region and evolution of the potential energy function in the armchair and zigzag directions: continuummechanics model, molecular dynamics results for all potentials and SROM predictions.

Figure 17 :

 17 Figure 17: Confidence interval for the probability density function of the bond length, estimated with the SROB [W 4 ] in the armchair (left) and zigzag (right) directions.

18

 18

  .

4. 4 . 3 .

 43 Application 3-b: stochastic reduced-order modeling with SROB [W 0 ]

Figure 18 :Figure 19 :

 1819 Figure 18: Evolution of the confidence interval for the mean bond length (with SROB [W 4 ]), as a function of applied engineering strain (armchair direction).

Figure 20 :

 20 Figure 20: Confidence interval for the probability density function of the bond length, estimated with the SROB [W 0 ] in the armchair (left) and zigzag (right) directions..

Figure 21 :

 21 Figure 21: Evolution of the confidence interval for the mean bond length (with SROB [W 0 ]), as a function of applied engineering strain (armchair direction).


Olivier Capra
email: olivier.capra@etu.univ-lille1.fr

Florent Berthaut
email: florent.berthaut@univ-lille1.fr

Laurent Grisoni
email: laurent.grisoni@univ-lille1.fr

Toward Augmented Familiarity of the Audience with Digital Musical Instruments

Keywords: digital musical instrument, performance, familiarity, causality, augmentations, correspondences

The diversity and complexity of Digital Musical Instruments often lead to a reduced appreciation of live performances by the audience. This can be linked to the lack of familiarity they have with the instruments. We propose to increase this familiarity thanks to a transdisciplinary approach in which signals from both the musician and the audience are extracted, familiarity analyzed, and augmentations dynamically added to the instruments. We introduce a new decomposition of familiarity and the concept of correspondences between musical gestures and results. This paper is both a review of research that paves the way for the realization of a pipeline for augmented familiarity, and a call for future research on the identified challenges that remain before it can be implemented.

Introduction

Whether as a musician or as a spectator, the experience of live music has very particular characteristics. It is often immersive, as intimate as collective and implies different modalities with both low and high level cognitive engagements. Moreover, a musical experience is often a corporeal, aesthetic as well as emotional commitment and is therefore difficult to define. Despite their great diversity and the parallel treatments they require, we assimilate all these aspects without any apparent effort. We naturally build this intense feeling that we all have experienced by attending a concert. Beyond the unfathomable subjective part of this musical experience (i.e. tastes and colors), we can identify objective characteristics that influence how we perceive live music, and among these, the way musicians interact with their instrument and how we perceive and integrate these interactions.

On the one hand, in the production of music with acoustic instruments, gestures and sounds are intrinsically linked by the laws of physics or at least by intuitive connections that do not need any prior explanation. Beside the music they produce, every interaction is visible. Thereby the expressive intentions, as well as the intensity of emotions, are particularly vivid and underline the role of a multimodal integration [START_REF] Vines | Music to my eyes: Cross-modal interactions in the perception of emotions in musical performance[END_REF]. Along this line, the degree of perceived control, influenced by the additional information usually extracted from the musician's interactions, participates in the emergence of the liveness of the performance.

On the other hand, the experience of live music produced by digital musical instruments (DMIs) may suffer from a deterioration of this intuitive link between the behavior of the musician, the inner mechanisms of the instrument and the sounds actually produced. When what we hear is not directly linked to what we see or what we may infer from a given gestural behavior, when music is not the direct and causal consequence of the specific gestures of the musician, then we may loose the multimodal integration that contributes to the immersive experience of live music.

These considerations have led to an increasing amount of recent researches on measuring the audience experience [START_REF] Barbosa | Considering audience's view towards an evaluation methodology for digital musical instruments[END_REF][START_REF] Lai | Audience experience in sound performance[END_REF][START_REF] Fyans | Where did it all go wrong ? a model of error from the spectator's perspective[END_REF][START_REF] Fyans | Examining the spectator experience[END_REF][START_REF] Fyans | Perceptions of skill in performances with acoustic and electronic instruments[END_REF][START_REF] Bin | In-the-moment and beyond: Combining post-hoc and real-time data for the study of audience perception of electronic music performance[END_REF][START_REF] Baytas | The perception of live-sequenced electronic music via hearing and sight[END_REF][START_REF] Bin | Hands where we can see them! investigating the impact of gesture size on audience perception[END_REF], sometimes used as a way to evaluate the instruments themselves [START_REF] Wu | Evaluating the audience's perception of real-time gestural control and mapping mechanisms in electroacoustic vocal performance[END_REF]. The less familiar we are with an instrument, the less we are able to perceive the fine relationship musicians build with their instrument. This lack of coherence in multimodal information might downgrade the attributed agency [START_REF] Berthaut | Liveness Through the Lens of Agency and Causality[END_REF], that is the perception of how much the musician is controlling their performance. Thus the knowledge we have about a DMI or its obvious behavior are crucial features to understand and integrate the interactions and thus fully experience live music. How do we ensure familiarity with DMIs ?

Improving familiarity with Digital Musical Instruments

Familiarity is the feeling of knowing about the behavior and the possibilities of an instrument. It can be associated with other concepts in the literature as mental model [START_REF] Fyans | Where did it all go wrong ? a model of error from the spectator's perspective[END_REF] or as an extension of transparency [START_REF] Fels | Mapping transparency through metaphor: towards more expressive musical instruments[END_REF]. We believe that familiarity is a strong component of the live music experience as it provides the spectator the ability to detect the intentions and the virtuosity of the performance of the musicians as well as their errors. Several attempts have been made at improving the familiarity. In this section, we propose a quick review of these solutions and their limitations.

Building a repertoire Quite naturally, a first way to increase the familiarity of people with a particular instrument is to promote its use and dissemination by building a repertoire of compositions around it. Once a majority of spectators have seen the instrument played by many musicians in different contexts, or even practiced it themselves, they are aware of its potential for musical expression and of its behavior. Thereby, the familiarity issue does not exist anymore. This kind of "natural familiarity" is definitely effective but requires a large amount of time and energy to be achieved. Moreover, this method is not compatible with the very idea behind DMIs and the exponential creativity they embed. Whether a musician wants to evolve its instrument or let other musicians modify it, as soon as the instrument changes, the whole process of natural familiarity needs to start all over again.

Explaining the instruments. When the problem of familiarity lies partially in the understanding of the operations of an instrument, demonstrating the behavior of the instrument can be a valuable solution. Building the familiarity with a pedagogical method is a simple way to make an audience understand what is going on on stage. Before, after and even in breaks during the performance, the musicians can explain how their instrument works. Prior hands-on demos, where the audience can actually play the instrument, are also a good way to increase the familiarity. However, both these methods trigger some reservations as the technical understanding of an instrument is not necessarily linked to a better appreciation of the performance. For example, Bin et al. [START_REF] Bin | Skip the pre-concert demo: How technical familiarity and musical style affect audience response[END_REF] demonstrate that explanations before the performance do improve the understanding of the instruments and its mechanisms but do not increase either the appreciation or interest. Besides these results, the audience may forget or unconsciously misreport important details from the performance [START_REF] Loftus | Reconstruction of automobile destruction: An example of the interaction between language and memory[END_REF] and the demos may not be possible when dealing with a large audience.

Designing for transparency. Another strategy to increase the experience of spectators watching a musician playing an instrument they have never seen before lies in the design of the instrument itself. The idea is to expose a clear link between the gestures and the sound modifications operated by these gestures, what is called the mapping of an instrument. The fluency of perception of these mappings is often called "transparency", and is defined by Fels et al. [START_REF] Fels | Mapping transparency through metaphor: towards more expressive musical instruments[END_REF] as follows :

For the lay audience, this understanding is derived from cultural knowledge, including percepts of physical causality relationships.

Aiming for transparency from the very first steps of the design of a musical device can lead to "easier to perceive" instruments [START_REF] Murray-Browne | The medium is the message: Composing instruments and performing mappings[END_REF]. Following that lead, an interesting way to increase the transparency is the use of metaphors [START_REF] Fels | Mapping transparency through metaphor: towards more expressive musical instruments[END_REF]. In that case, the common background of inexperienced audience, their general knowledge, are used as a mold to grow new knowledge about an instrument mapping. For instance, the timbre of a sound can be modified as the shape of an associated graphical representation that is getting sculpted. However, the instruments designed according to these specifications tend to maintain the familiarity we have with acoustic instruments and physical laws. Thus, despite the gain in transparency, this method may narrow the design possibilities usually available for DMIs.

Visually augmenting the instruments. Berthaut et al. [START_REF] Berthaut | Liveness Through the Lens of Agency and Causality[END_REF][START_REF] Berthaut | Rouages: Revealing the Mechanisms of Digital Musical Instruments to the Audience[END_REF] proposed to increase the familiarity by recreating a link of causality through visual augmentations of the instruments. Based on the attribution of causality by spectators (attributed agency) to a music performance, the visual augmentations provide insights on the musician's interactions with its instrument. Animated 3D objects overlapping the device reveal the relationship between the gestures and the musical result to the audience. These augmentations are based on Wegner's criteria of apparent mental causation [START_REF] Wegner | Apparent mental causation: Sources of the experience of will[END_REF] (See 2.1). By exposing the details of the interactions, the setup contributes to re-link the gestures of the artist to their intentions and expressiveness. Besides, the augmentations enable the distinction between automated part and actual live music production. Such a discrimination between live and pre-recorded music appears to be crucial in the experience of live music precisely because it is supposed to be a live performance. However, these augmentations do not take into account the audience reaction or their expertise. Indeed, the familiarity with an instrument is a personal characteristic that cannot be generalized. A concert in front of specialists or naive people does not imply the same requirement of explanations or augmentations. Toward a new method. These different methods offer a rather effective way to increase the understanding of the behavior of DMIs. However they don't fully cover a major aspect of the music experience which is the real time multimodal integration of visual and auditory cues. Furthermore, while the visual augmentations gave good results they do not take into account the audience expertise that may appeal for specific levels of details in the augmentations. Moreover, in some cases, non-visual augmentations may be more relevant. Thereby, we propose new insights to improve the familiarity in a more adaptable and reliable way.

Contribution

Our contribution is two-fold. First, we propose a conceptual pipeline to improve the familiarity of the audience with DMIs in real-time by analyzing spectators' reactions, musicians' actions, musical outputs and by augmenting the instrument. To do so, we introduce the concept of correspondences. Second, for each module of this pipeline, we briefly review relevant knowledge in computer and cognitive sciences. We also identify interdisciplinary challenges that need to be solved to reach a functional software implementation of the pipeline.

Toward augmented familiarity

In this section we propose a novel approach for improving familiarity, formalized by a software pipeline. While it remains conceptual, it was designed so that it could be implemented by addressing the challenges described in Section 2.2.

Here is a scenario that we envision with our pipeline : Patricia attends an electronic music concert. At the entrance, she is given a small device equipped with physiological sensors (a choice of either a bracelet or a special glass that she holds). During the concert, she has trouble understanding what is happening, in particular what the musician's action on the sound is. The device senses a change in a set of physiological signals, that corresponds to a loss in familiarity, and sends the data to a server. Patricia may also directly indicate her loss in familiarity with a graphical slider on an app on her smartphone. Simultaneously, this server has been analyzing the musician's gestures, the flow of data inside the instrument and the musical output. When it receives Patricia's familiarity signals, the server, with settings defined by the musician, selects the adequate augmentations to be displayed. They aim at compensating for the familiarity disruption caused by the musical interaction context. Consequently, visual augmentations are displayed around the musician either for Patricia alone when she watches the performance through her smartphone (e.g. with video augmentedreality), or for the group of people around her using a mixed-reality display. They provide information that improve her degree of familiarity, allowing her for example to perceive the link between the musician's gesture and the resulting sound, and to enjoy the performance to a larger extent.

General approach

To improve the liveness of music performances with DMIs, we suggest increasing the familiarity thanks to a trans-disciplinary approach in which both human and technological signals are analyzed. To achieve this goal, we first clarify the notion of familiarity. We then propose the concept of correspondences, which describes musical interactions and the way they are perceived. Finally we present a potential pipeline that would extract and process signals from the musician and from the audience to compose augmentations.

Familiarity dimensions.

As other high-level cognitive abilities, familiarity is not a unitary notion. It can be decomposed into components that differ by their nature, their inner rules as well as the type of information they convey. Besides, familiarity relates to expertise or attention and thus impacts on brain and motor system activities. We will describe in Section 2.2 how previous research show that these activities can be detected in more detail. Familiarity may also relates to various parts of the musical interaction that is perceived. Spectators can be more or less familiar with : musical gestures, relations between gestures and sound, instruments capabilities or musical genres. All these factors contribute to the general familiarity one has with an instrument. Following research by Fyans et al. [START_REF] Fyans | Where did it all go wrong ? a model of error from the spectator's perspective[END_REF][START_REF] Fyans | Examining the spectator experience[END_REF] and Belloti et al. [START_REF] Bellotti | Making sense of sensing systems: five questions for designers and researchers[END_REF], Barbosa et al. [START_REF] Barbosa | Considering audience's view towards an evaluation methodology for digital musical instruments[END_REF] evaluate the audience experience through their comprehension of cause, effect, error, intention and mapping. However, measuring the familiarity might not be sufficient to inform the design of instruments or to augment performances if the origin of this familiarity is not understood correctly. For example, the perception of expertise, skill, error or intention of the performer, are consequences of the familiarity one has with a musical interaction rather than dimensions of the familiarity itself. For instance, the more familiar we are with an instrument, with the way the sound is produced and with the context of the performance, the better we can assess the quality of the performance and perceive the errors and the intention of the musician. Therefore while they can help measure the familiarity, to which they are correlated, these features do not provide insights on the reasons for this familiarity. One must therefore investigate the features which, when changed, have an impact on the overall familiarity, and in turn on the perception of expertise, skill, error or intention.

Taking this into account, our approach is to decompose familiarity into dimensions that we can both evaluate and improve. Based on results in HCI, NIME, and cognitive sciences that have explored various aspects of familiarity, we propose to decompose it into five components : Causality, Instrumentality, Instrument expertise, Musical culture and Musical genre expertise.

These in turn contain dimensions that can be evaluated independently : Causality relates to the apparent mental causation, e.g. the judgment of causality of one's action, defined by Wegner et al [START_REF] Wegner | Apparent mental causation: Sources of the experience of will[END_REF], where each of the three following dimensions are required to establish a judgment of causality of one's action:

-Priority -The thought should precede the action at a proper interval -Consistency -The thought should be compatible with the action -Exclusivity -The thought should be the only apparent cause of action The consistency aspect is also closely related to the notion of naive physics, that is a set of innate or common knowledge about the physical world like gravity or friction that can also be used in digital interfaces as physical metaphors [START_REF] Jacob | Reality-based interaction: A framework for post-wimp interfaces[END_REF].

As proposed in [START_REF] Berthaut | Liveness Through the Lens of Agency and Causality[END_REF], we transpose this model from the perception of one's own intentions/actions to the perception of others' actions. By doing so, we extend the concept of agency to the rather recent concept of attributed agency.

Instrumentality relates to common knowledge that allows one to predict the range of sound possibilities of a musical instrument from its appearance. It is composed of two dimensions : the composition, i.e. shape and material, and behavior, i.e. mechanisms and degree of autonomy from the musician's actions.

Instrument expertise relates to the exposure the spectator has had to the instrument, from a first-time observer to an expert player.

Musical culture corresponds to a basic knowledge of musical theory, that can for example be used to represent pitch as vertical position of a graphical element, i.e. mimicking a staff.

Musical genre expertise is composed of dimensions that correspond to the familiarity with the specifics of a genre, such as of structure and constraints. The effect of this category is obvious in the study by Bin et al. [START_REF] Bin | Skip the pre-concert demo: How technical familiarity and musical style affect audience response[END_REF], where the same instrument played in two genres has a different impact on familiarity.

Notice that each component has a specific weight on the overall process. Furthermore, this model is consistent with the partial results that previous strategies have achieved. Culture and expertise for the "building a repertoire" and the "prior explanations" strategies, Instrumentality and Culture for "the design for transparency" and Causality for the augmentation strategy. This decomposition might evolve according to findings from the implementation phase. Some dimensions could prove harder to evaluate or others may emerge.

Finally, the complete decomposition of familiarity that we propose includes the following dimensions :

F amiliarity = C exclusivity + C priority + C consistency + I composition + I behavior + E expertise + M culture + G structure + G rules (1)
Correspondences. In order to handle the heterogeneous data (physiological, behavioral, musical, visual, mechanical) associated with the dimensions of familiarity, we introduce the notion of correspondence: a conceptual object that stands as a digital multidimensional representation of a musical interaction. This object associates several properties : musical output (audio), gesture data (time series of 3D coordinates), visual data (video of the movement), control data (time series of sensor values) and the source of the interaction. This last property specifies the author of the interaction. Indeed, interactions are not only produced by one musician but can be triggered and mastered by another one in a collaborative performance or, more usually, by autonomous prepared processes, e.g. automations or playlists. Each property is subdivided in three elements : a raw format element, a semantic element, and a classification element.

The raw format element is a pointer to a collection of related files. For instance, the raw element of the visual property of a clap correspondence is a pointer to a collection of short videos showing a clap from different angles and velocities and the raw element of the audio property a pointer to a collection of short audio recordings of a clap.

The semantic element is composed of annotated descriptors of the property. For our clap example, the semantic element of the visual property could list obvious descriptors as "hands", "clap", "applause", "brief" but also more precise descriptors relative to specific taxonomies developed in the different analysis of gesture and sound.

The descriptors can be freely provided by the audience or taken from refined existing models such as [START_REF] Cadoz | Gesture-music[END_REF] or [START_REF] Jensenius | Musical gestures. Musical gestures: Sound, movement, and meaning[END_REF]. For instance, "excitation gesture" or "effective gesture" could be used to populate the semantic element of the gesture property of a correspondence. Interestingly, by holding descriptors for each property of a unique interaction (aka correspondence), the semantic element offers the possibility to analyze the links between descriptors from different properties by the analysis of their co-occurrence. Finally, looking forward to the implementation, the processing of meaningful descriptors can rely on the strong foundation of web semantics [START_REF] Berners-Lee | The semantic web[END_REF].

The classification element is dedicated to the classification of the raw elements through machine learning methods and thus allows for the pairing of close correspondences with respect to their different properties. To compute such a distance between correspondences, multidimensional vectors can be used alongside more specific techniques as neural networks, informed by an efficient semantic analysis thanks to the semantic element of a given property.

In addition, correspondences hold a score for each of the familiarity dimensions described earlier (causality, instrumentality, ...). For example, two correspondences with the same musical result, e.g. the fade-in of an audio loop, can have different gesture and control properties depending on the mapping chosen for the DMI. While a continuous gesture on a fader would have a high score (meaning, the most natural way to fade) for the consistency dimension, a discrete tap on a pad would have a low one, since the effect would no be consistent with the cause for a spectator, e.g. discrete input and continuous output. This score of familiarity held by each correspondence plays a central role in our approach for an increase of familiarity as it allows for sorting correspondences from obvious to abstruse for a given audience.

Finally, as an interaction may be more complex and decomposed in a sequence of more basic interactions, a correspondence can also be linked to other, simpler correspondences.

Pipeline. Our envisioned pipeline (

C

Via a phone application the audience requires augmentations to increase their familiarity. During the performance, an average level of required familarity can be computed and allow more precise adjustments.

Fig. 2. The proposed pipeline with its main modules and data flows based on a database of correspondences (DB C ). The pipeline is used at three different moments : before, during and after a performance. Before the performance, correspondences, coming from a shared online database, or recorded specifically for the instrument, are saved in the DB C module. 1 During the performance, IA receives musical interaction data extracted by EXT M . This data consists of both dynamic values such as gestural parameters, audio features and control values which will be used directly by IA, and of physical (position of gestures and sensors), logical (tracks, effects, synthesizers, ...) and structural information about the musical interaction, which will be used by AUG. 2 IA builds and manages a set of ongoing correspondences from these signals. 3 Simultaneously, IA receives the familiarity evaluation (with the identification of the sensed individual or group) from EXT A. If the familiarity evaluation is low, 4 IA finds in DB C correspondences similar to the ongoing one, and select the familiarity dimensions that need to be compensated depending on their scores in these. IA then sends AUG the data required for the augmentation : live correspondences, associated signals from EXT M with dynamic and structural data and identification of the source of the familiarity evaluation (in order to display the augmentations only to the correct person or group). 5 AUG creates (or selects if it already exists) the augmentation that matches the received structural data, for example a visual augmented-reality link between the physical position of a sensor and a virtual representation of an audio track. 6 Augmentations are then connected to EXT M and listen to the signals required to update the augmentation, e.g. control values for the sensor, loudness of the track. After the performance, the familiarity extracted from the audience can be reused to refine the scores for the familiarity dimensions in each detected correspondence of DB C .

Modules

Database of correspondences (DB C ). DB C manages all pre-existing correspondences. These can be generic, or specific to an instrument or performance. It receives queries from IA to select correspondences matching the ones detected during the performance. A very promising approach in dealing with heterogeneous data is the use of databases. Even if the efficiency of machine learning and analysis tools is still evolving, a numerous amount of initiatives, especially in analysis of emotion (DEAP [START_REF] Koelstra | Deap: A database for emotion analysis; using physiological signals[END_REF], RECOLA [START_REF] Ringeval | Introducing the recola multimodal corpus of remote collaborative and affective interactions[END_REF], EATMINT [START_REF] Chanel | Assessment of computer-supported collaborative processes using interpersonal physiological and eye-movement coupling[END_REF]) but also in music-related actions [START_REF] Godøy | Classifying music-related actions[END_REF], contribute to shared databases that compile multimodal and synchronized experimental data. The main goal of these databases is to predict complex and abstract states, e.g. the emotional state of an individual, thanks to the analysis of their physiological and behavioral signals such as face expression, electrodermal activity or heart rate variation. Most of the existing databases are composed of 15 to 30 entries referencing data of diversified nature. To allow the gathering of a more relevant amount of cross data, we first need to facilitate the indexing thanks to the more and more intuitive and effective frontend technologies of the web. We propose to develop a web interface that could provide the specific tools to aggregate the data required to constitute the properties of a correspondence. A typical correspondence would require a short video footage of a gesture, the motion capture of this gesture, sensor value and audio output of the instrument. Tags could also be manually added for each of these properties. The interface could be accessible from an open web platform where artists as well as researchers could populate the database, to constitute their own correspondences and therefore optimize their pipeline with more personal choices of gestures, mappings or sound processes. The same online platform can then be used for crowdsourced online evaluation sessions in which correspondences are exposed to participants with different levels of expertise. Their task is to indicate their understanding, using a questionnaire along the familiarity dimensions, and tag the correspondences.

Extraction of the musician's interactions (EXT M ). As explained in Section 2.1, EXT M extracts data from the instrument and musician's gesture which is then sent to IA in order to be aggregated into live correspondences. At the instrument level, EXT M extracts sensors states, mapping values as well as musical result of the musician interactions. At the musician level, EXT M extracts the control gestures, the body movements and physiological signals. In addition to these signals, structural information needs to be provided for further use in the augmentations, such as the position of the physical sensors of the DMI and the position of the musician's hands, the list of tracks, effects or other sound processes with their names, or the mappings between sensors and sound parameters. While some of these obviously need to be defined by the musician manually, or sensed by devices external to the instrument, others can be extracted through a trans-disciplinary approach. Regarding the instrument input, research on gesture recognition, especially concerning hand gestures as demonstrated by Rautaray & Agrawal [START_REF] Rautaray | Vision based hand gesture recognition for human computer interaction: a survey[END_REF], can be used to identify the performed musical gestures. Regarding the instrument output, research in music information retrieval (MIR) provides tools for segmenting music from the audio signal only [START_REF] Paulus | State of the art report: Audio-based music structure analysis[END_REF] using spectral, tonal, rhythmical descriptors and methods. Finally, the extraction can be facilitated by a multimodal approach, similarly to techniques developed for video indexing [START_REF] Snoek | Multimodal video indexing: A review of the stateof-the-art[END_REF], as some events might appear clearer on one channel than on others. For example, control data might inform on the temporal boundaries of a change in the sound that can then be analysed

We identify two main challenges for the implementation of this module. The first is the access to the data from the DMIs. In fact, while extraction from the audio signal provides many features that can be used to detect correspondences, it might not be enough for precise analysis. In order to access pre-mappings and post-mappings data, to differentiate between manual and automated changes and to analyze the output of individual tracks or other sound processes, one can not rely solely on the DMIs inputs and outputs, i.e. additional software components will need to be integrated. In most DMIs, plug-ins can be added at various stages of the instrument. However, the API might not provide enough information on the instrument to a single plugin. For example, one plugin per track might be needed to access and send the audio output features for each separately. The integration of EXT M will be simpler if DMIs are built using patching environments such as PureData or Max/MSP, where the musician has more control over the architecture of the instruments. The second challenge is to combine detailed but costly and slow analysis of features for the detection of correspondences with maximum accuracy, and fast but less accurate analysis of features for the update of augmentations in AUG.

Extraction of the audience familiarity (EXT A). The role of this module is to extract the audience subjective and objective information in order to inform IA.

It is now commonly accepted that the live music experience engages multiple complementary processes of low level perception, en-action and embodiment ( [START_REF] Fyans | Perceptions of skill in performances with acoustic and electronic instruments[END_REF], see Leman & Maes for a review [START_REF] Leman | The role of embodiment in the perception of music[END_REF]), processing of hierarchical and sequential information [START_REF] Molnar-Szakacs | Music and mirror neurons: from motion to 'e'motion[END_REF][START_REF] Kohler | Hearing sounds, understanding actions: action representation in mirror neurons[END_REF], as well as strong affective and social aspects. This perspective of complementarity is also included in recent studies relative to music produced with DMIs [START_REF] Schacher | Skill development and stabilisation of expertise for electronic music performance[END_REF] The evaluation of such a rich experience triggers multiple methodological difficulties. Subjective assessment is therefore a common method that can be conducted through post-performance questionnaires [START_REF] Lai | Audience experience in sound performance[END_REF][START_REF] Wu | Evaluating the audience's perception of real-time gestural control and mapping mechanisms in electroacoustic vocal performance[END_REF][START_REF] Bin | In-the-moment and beyond: Combining post-hoc and real-time data for the study of audience perception of electronic music performance[END_REF], emotion rating during the concerts [START_REF] Schubert | The six emotion-face clock as a tool for continuously rating discrete emotional responses to music[END_REF][START_REF] Bin | In-the-moment and beyond: Combining post-hoc and real-time data for the study of audience perception of electronic music performance[END_REF] or when viewing an audio-video recording of a performance [START_REF] Baytas | The perception of live-sequenced electronic music via hearing and sight[END_REF].

Post-performance questionnaires are a good source of information but may be less precise than live subjective reports. In addition, there are initiatives that do not require the participants to directly communicate their introspective evaluation. These more objective measurements require very specific equipment as eye trackers [START_REF] Baytas | The perception of live-sequenced electronic music via hearing and sight[END_REF] and, to date, suffer from a lack of out-of-the-lab physiological measurements. Familiarity is one of the key aspects of the experience of live music and, as its other dimensions, relies on multiple underlying mechanisms. Therefore, as the mentioned studies above, and to initiate ecological (i.e. out of the lab, in "real life") measurements of physiological signals, we propose a dual methodology to extract the familiarity of the audience. First, we base the subjective assessment on a familiarity application for mobile phone. Its main purpose is to supply IA with data about the ongoing familiarity of the audience, from a graphical familiarity slider that spectators activate.

As other authors [START_REF] Baytas | The perception of live-sequenced electronic music via hearing and sight[END_REF][12], we believe this continuous survey could be a good methodological answer to the reservations we exposed about questionnaires after the performance. The second part of the extraction relies on physiological signals. Neuroscience studies show that the expertise, a key component of familiarity, influences the perception of action [START_REF] Calvo-Merino | Action observation and acquired motor skills: an fmri study with expert dancers[END_REF]. Those findings, applied to music expertise, may lead to a better understanding of its role in the live music experience. By measuring the peripheral signals, we aim at discovering potential patterns that could correlate with the subjective data we extract. Widely used in the emotion studies, and rather suitable to extract in natural condition, four signals are particularly interesting in our musical context : Heart rate variability [START_REF] Sammler | Music and emotion: electrophysiological correlates of the processing of pleasant and unpleasant music[END_REF], electrodermal activity [START_REF] Sequeira | Electrical autonomic correlates of emotion[END_REF] and oculometry (eye tracking + pupillometry). The complementary analysis of extracted features of these signals already gave interesting result in the emotion classification by machine learning algorithms [START_REF] Kim | Emotion recognition based on physiological changes in music listening[END_REF] and need to be further extended in music experience studies. Moreover, these signals can already be acquired by wearable devices and the quantified self movement [START_REF] Swan | The quantified self: Fundamental disruption in big data science and biological discovery[END_REF] will surely provide more accurate and affordable devices in the near future.

Integration and Analysis (IA). IA is the central hub that connects to all the other modules. Its role is to : 1) compute the live correspondences with the data extracted from the musician and their instrument; 2) match the computed live correspondence with a correspondence from the database to calculate the familiarity dimensions that need to be augmented; 3) supply the AUG module with the information needed for the relevant augmentation selection. To fulfill these tasks, the module can rely on the classification element of each property of correspondences. Alongside the raw and the semantic parts, the classification element is a machine learning model dedicated to the classification of raw data. This model is pre-computed with the raw files registered in the correspondence. Its goal is to discriminate new stimuli and detect those who match with the recorded one. Considering the heterogeneity of the modalities, each property might require a specific machine learning model and specific extracted features. For example, MIR descriptors for a sound element and a deep convolutional network for picture classification. The main idea is to use the set of models as a global digital representation of the correspondence that can either be projected, depending of the context, on a single and more easy to handle property or be represented as a multidimensional vector that allows similarity comparison of whole correspondences (the matching process). Considering the variety of data and processes it has to handle, this module needs to be regularly updated with recent findings in signal processing, machine learning or movement models. Without a strongly modular structure of the available tools that the pipeline has at its disposal, the framework may not be able to evolve and thereby join a long list of deprecated initiatives.

Augmentation (AUG). This module manages both a database of available augmentations and a set of active ones. When a correspondence needs to be augmented, AUG receives the data required to create a new or select an existing augmentation, such as the physical position of the gesture and sensors and logical components of the instruments, the familiarity dimensions that need to be compensated, the destination of the augmentation (individual or group) and the data from EXT M that the augmentation should listen to. The augmentations are selected from a database of augmentations designed to compensate the various dimensions of familiarity.

A number of research have shown the opportunities opened by augmentations to provide information on DMIs to the audience. For example, Perrotin et al. [START_REF] Perrotin | Visualizing gestures in the control of a digital musical instrument[END_REF] used visualizations, projected on a screen behind an ensemble of DMIs, to help the audience understand better the contribution of each musician. In previous work, we proposed an augmented-reality (AR) approach where the visualizations are perceptually consistent with the physical instrument, e.g. visual links attached to the physical sensors. We designed a display where multiple spectators can reveal the augmentations [START_REF] Berthaut | Reflets: Combining and Revealing Spaces for Musical Performances[END_REF] and all perceive them consistently. Finally, we proved the effect of 3D visual augmentations on the causality aspect of familiarity [START_REF] Berthaut | Liveness Through the Lens of Agency and Causality[END_REF], i.e. the degree of control perceived by the audience. On the haptic side Armitage [START_REF] Armitage | Revealing timelines: Live coding and its gestures[END_REF] has experimented with using haptic feedback to provide information on musician's interaction during a live-coding performance. We believe that the first challenge is the creation of a framework that allows one to design augmentations according to the specific dimensions of familiarity that they compensate. Rules will need to be defined so that one can adapt an augmentation to the artistic specificity of a particular performance. A second challenge is the design of augmentations that provide just enough content to fill the multimodal gap without distracting the audience from the musical performance because of a too heavy cognitive load.

Conclusion

In this paper, we presented a novel approach for augmenting familiarity of the audience with Digital Musical Instruments and reviewed associated research results and challenges.

Among the perspectives, our first future work will be the construction of the initial database of correspondences that will help refining our familiarity decomposition.

The first ones will therefore be generated through a systematical process, i.e. by recording unitary interactions (e.g. one gesture one change in the sound) with variations on each dimension perceived by the audience, e.g. gesture type and amplitude, mapping, sound parameter, point of view, and so on.

In addition, we plan on investigating the impact of familiarity variations on electrophysiological signals from the audience, with the aim of finding realtime measurements that could be used to assess familiarity during performances and guide the choice of augmentations. This transdisciplinary approach could reveal significant research avenues especially with the democratization of wearable devices. We hope that this paper will trigger exciting new research in NIMEs, and both cognitive and computer sciences.

Fig. 1 .

 1 Fig. 1. Visual augmentations of DMIS for the increase of familiarity. The inner mechanisms of the instrument are dynamically revealed to the audience. (From Berthaut et al. 2013 [8])

Fig 2 )

 2 is composed of five modules that handle the extraction of the data from the musician and their instrument (EXT M ) , the extraction of physiological and subjective data from the audience (EXT A), the processing of the data (IA) and the selection of fitting augmentations (AUG) the augmentation is identified, a mapping is set between the ongoing sensors and the port of the augmentation