Clovis Lapointe
email: clovis.lapointe@univ-lorraine.fr

Thomas D Swinburne
email: thomas.swinburne@cnrs.fr

Laurent Proville

Charlotte S Becquart

Normand Mousseau

Mihai-Cosmin Marinica
email: mihai-cosmin.marinica@cea.fr

Machine learning surrogate models for strain-dependent vibrational properties and migration rates of point defects

Keywords: Harmonic approximation, defects, Machine Learning, vibrational entropy, attack frequency, migration rates, energy-entropy correlations, compensation law

Machine learning surrogate models employing atomic environment descriptors have found wide applicability in materials science. In our previous work, this approach yielded accurate and transferable predictions of the vibrational formation entropy of point defects for O(N ) computational cost. The present study investigates the limits of data driven surrogate models in accuracy and applicability for vibrational properties. We propose an improvement of the accuracy by extending the fitting capacity of the model by increasing the dimension of the descriptor space. This is achieved by using a non-linear relation between descriptors -target observables and when it is possible by including physical relevant information of the underlying energy landscape. The non-linear extension is used to learn the formation entropy of defects with or without applied strain whilst including physical information such as the minimum-saddle point sequences are employed for the migration of point defects, a key ingredient of transition state theory rate approximations. We find excellent predictive power after augmenting the dimensionality of the descriptor space, as demonstrated on large defect databases in α-iron and amorphous silicon based on semi-empirical force fields. The current linear surrogate models are used to investigate the correlation between migration entropy and energy. Our approaches reproduce the Meyer-Neldel compensation law observed from direct calculations in amorphous Si systems. Moreover, the same abstract descriptor space representation for entropy and energy is then used for the statistical correlation analysis. For linear surrogate models, we show that the energy-entropy statistical correlations can be reinterpreted in descriptor space. This provides a simple statistical criterion for marginal interpretation of the compensation law. More generally, the present work shows how linear surrogate models can accelerate high-throughput workflows, aid the construction of mesoscale material models and provide new avenues for correlation analysis.

I. INTRODUCTION

The thermodynamic and kinetic properties of defects drive the microstructural evolution of materials [1,[START_REF] Caillard | Thermally Activated Mechanisms in Crystal Plasticity[END_REF]. The appropriate thermodynamic potential of defects, such as free energy and enthalpy, gives the equilibrium density of defects, while the defects' kinetic properties steer the kinetic pathways of the microstructure towards equilibrium. Therefore, the study of solids at finite temperature requires the correct description of the free energy thermodynamic potential of defects. The defects' free energy accounts for the contribution of the system's microstates around a particular atomic configuration. In materials, the morphology of defects shows extraordinary variety and is related to the complexity of the underlying energy landscape.

Defects have a wide range of shapes and sizes: from localized 2D dislocation loops [3,4] to 3D clusters such as voids [START_REF] Loretto | Defect Analysis in Electron Microscopy[END_REF][START_REF] Hull | Introduction to Dislocations[END_REF], stacking fault tetrahedra [START_REF] Vítek | [END_REF]8], small interstitial cluster [9], insertions with particular crystallographic structure [10] etc. The complexity of the energetic landscape is amplified by the fact that those defects are embedded in a host matrix. The complexity does not come from just the arrangement of atoms within the defects, but also from the interface between the defects and the surrounding matrix. Very often, in order to characterise these complex energetic landscapes, the configurations of defects are indexed by the local minima of the energy landscape and the fluctuations associated with thermal vibrations are neglected [11][12][13][14][15][16][17][18]. Therefore, the entire domain of the free energy's phase space R 3N , of N atoms, has particular topology: it is sparse and sliced into basins of attraction, which can be indexed with discrete labels. This approach is similar to that used to describe atomic clusters in vacuum [19]. Any configuration q ∈ R 3N is then a member of some discrete states, which belong to the same basin of attraction [19]. Hence, the entire basin can be represented by the corresponding local minimum to which q converges to under local minimization of the internal energy U (q). Within this framework, probably the most common visual representation of energetic landscape is provided by the disconnectivity graph techniques [19][20][21]. For defects, it is very practical to account for the formation free energy, i.e. the difference between the free energy of the perfect and the defective solid containing the same number of N atoms. The formation free energy converges to a well-defined value in the thermodynamic limit N → ∞ [22]. However, the kinetic evolution of microstructure is driven by the pathways and the connections of the sequences minimum -saddle point -minimum [11,[START_REF] Dederichs | Point Defects in Metals II[END_REF][24][START_REF] Wales | Energy Landscapes[END_REF]. For these kinds of sequences, the quantification of the energy contribution has been widely studied and the community has well-established tools [12,24,[START_REF] Henkelman | [END_REF][START_REF] Henkelman | Theoretical Methods in Condensed Phase Chemistry[END_REF][START_REF] Henkelman | [END_REF]. The entropic contribution is still challenging to quantify because of the methodological and numerical complexity [1,29,30].

The present study focuses on the evaluation of the vibrational entropy S vib for minima and saddle points configurations of the energy landscape in the framework of the harmonic approximation. The vibrational entropy S vib (q) of some defect is directly related to the curvature of the phase space q ∈ R 3N in some particular point of the potential energy surface on which the defect is located. Many alternative methods to harmonic approximation have been designed and tested in the community [START_REF] Lelièvre | Free energy computations: A mathematical perspective[END_REF][START_REF] Torrie | [END_REF][33][34][35][36][37][38][39] to compute the free energy of defects, including even the non-harmonic contributions from energy and entropy in an indistinguishable manner. However, these methods remain computationally very demanding as they usually rely on sampling the phase space of the system through the construction of random or optimized trajectories. Furthermore, in most cases the main limitation is not the numerical efficiency and the poor scalability of computational methods. There is a conceptual problem: the community does not have the appropriate theoretical tools in order to handle in a systematic manner the anharmonic finite temperature vibrational entropy of a complex energy landscape. Currently, there are no existing general sampling methods to estimate the free energy barrier profile between two states of a complex energy landscape. The community proposes some promising methods for particular cases [START_REF] Lelièvre | Free energy computations: A mathematical perspective[END_REF][33][34][35][36][37][38][39][40] where is possible to build intuitively or automatically an ad hoc reaction coordinate.

In the harmonic approximation, the vibrational entropy can be computed from the knowledge of the frequency of normal modes, which itself requires the evaluation and the diagonalization of local Hessian matrix operations that have O(N 2 ) and O(N 3 ) complexity, respectively. This traditional procedure requires an interatomic interaction that can be treated in the framework of electronic structure calculations, such as ab initio calculations, or of empirical interatomic potentials. This workflow can be bypassed by the recent surrogate model proposed by Lapointe et al. [41], which proposes a linear correlation between the atomic descriptors of the local atomic environment and vibrational entropy. That model was applied for point defects in crystals and nanoparticles [41] and opens many perspectives for the fast eval-uation of vibrational properties around energy-minimum configurations. Moreover, the comparison of the surrogate model with direct calculations of formation vibrational entropies of defects reveals an excellent accuracy and predictive power. The direct evaluation of the Hessian (O(N 2 )) and its diagonalization (O(N 3 )) is replaced with O(N ) computational effort. The utility of such a model is huge: the numerical efficiency increases drastically. Consider the case of evaluating the formation entropy of a dislocation loop containing 200 self-interested atomsin α-Fe. To avoid finite size effects, traditional direct evaluation requires simulation cells with over 120,000 atoms. For this size, solely the diagonalization of the Hessian matrix requires 6 hours on 3000 modern CPUs, whilst the surrogate model provides the same observable, within 5 % error, in less than 10 minutes on one CPU i.e. more than 10 5 faster.

The aim of this study is to explore the performance of surrogate model approaches for other physical observables in the field of materials science. Here, we apply the surrogate model for amorphous systems as well as defects in crystals under deformation. The deformed systems can be defects in minima configurations but equally first-order saddle points configurations. With this motivation, the previous surrogate model [41] is revisited by integrating specific physics, such as the metastable character of saddle point configurations. Moreover, in order to have a better representation in the descriptor space of the non-deformed minimum configurations, we introduce a non-linear extension of the machine learning surrogate model. The surrogate model formalism is then used to learn and predict the kinetical transition rates during defect migration.

Figure 1(A) and Figure 1(B) present the graphical summary of the traditional and current workflow, respectively, for the computation of vibrational harmonic entropies. The only inputs required for the present surrogate model are the optimized atomic coordinates of various defect configurations. Minimum and saddle point configurations are generated employing a 0 K method for systematic search in complex energetic landscapes, Activation-Relaxation Technique nouveau (ARTn). The collection of found defect configurations, in α-Fe and amorphous Si, defines the database of our machine learning surrogate model. The efficient ARTn 0 K method is used employing an inter-atomic force field based on a semi-empirical potential. Then, the system's Hessian matrix is computed and diagonalized to obtain the vibrational harmonic entropy S vib . Figure 1(B) emphasizes the machine learning (ML) sequence of the workflow. Descriptors of each configuration of the database are computed. Then, using a supervised regression models the machine learning model is trained in order to estimate S vib .

The paper is structured as follows. In Section II, we summarize the link between the vibrational entropy and the Green function formalism. The latter enables the total entropy of the system to be decomposed into local

(a) MD minimization ARTn Configuration q ∈ R 3N Hessian → O(N 2 ) Diagonalization → O(N 3 ) S vib (b) MD minimization ARTn Configuration q ∈ R 3N Descriptors → O(N) Regression S vib

Data saddle

Data minimum

November 21, 2022 1 / 4

FIG. 1: Graphical abstract of the vibrational entropy calculations using the traditional (A) and the present workflows of vibrational harmonic entropies. Atomic configurations are generated and optimized using the ARTn method and semi-empirical potentials for α-Fe and amorphous Si. Then, in the traditional approach (A), the system's Hessian matrix is computed and diagonalized in order to obtain vibrational harmonic entropy S vib . In the present machine learning workflow (B), to estimate S vib we calculate the atomic descriptors of the same atomic configurations. Then the surrogate regression model is trained using the corresponding database of minimum and saddle point configurations. atomic contributions. In the same section, we describe the descriptor space used in order to obtain an appropriate representation of the atomic environments. The construction of this space is based on the encoding, using atomic descriptors, of the local atomic neighborhood of each atom by preserving all the geometrical invariances. In Section III, we explore the limits of the present non-linear extension of the surrogate model in order to increase the accuracy of vibrational entropy predictions when the system is constrained to a small deformation ϵ. The efficacy of our approach is demonstrated on strained defects in α-Fe. In Section IV, we use our approach to build a surrogate model for attack frequencies, in particular for the logarithm of the attack frequency, which is proportional to a vibrational entropy difference between the initial minimum atomic configuration and the saddle point configuration. The model is applied for the case of amorphous silicon. In Section V, the same surrogate model architecture is employed to predict the transition energy barrier, giving a complete surrogate model for harmonic transition state theory rate calculations. We use linear surrogates models to investigate the correlation between migration energy and entropy. Within the present framework we give a statistical insight of the compensation Meyer-Neldel law from the perspective of the energetic and entropic surrogate models and descriptor space.

II. THE HARMONIC VIBRATIONAL ENTROPY AND GREEN FUNCTIONS

The normal modes of a system with N atoms are obtained from the spectrum of the dynamical matrix D:

( D -ω 2 ν I) • êν = 0, (1)
where I is the identity matrix. When the dynamical matrix includes only the phonons from the center of Brillouin zone, it reduces to the mass normalized Hessian operator of the system:

M -1 2 .H.M -1 2
, where M is a diagonal matrix which return the mass of each atom. For unary systems of mass m, M = mI. The Hessian is built from the second derivatives of the potential energy U (q) of the system: H iαjβ = ∂ 2 U /∂q iα ∂q jβ . Vibrational mode (or Γ point phonons) frequencies ω 2 ν and displacements êν are the eigenvalues and eigenvectors of the above secular equation (1). At high temperature T (above the Debye temperature in cristalline solids) in the so-called harmonic approximation, the classical vibrational entropy of the system becomes:

S vib (T, N ) = k B ν ln k B T ℏω ν + 1 , (2)
where k B and ℏ are the Boltzmann and Planck constants, respectively.

A. Green function formalism for vibrational entropy calculations

Within the harmonic approximation, vibrational entropy estimation requires the full spectrum of the secular equation (1). The Green function formalism provides an iterative solution to the mentioned eigen-problem. The total density of states of vibrational modes Ω(ω) is extracted from the imaginary part of the trace of Green function G ∈ C 3N ×3N [START_REF] Dederichs | Point Defects in Metals II[END_REF]:

Ω(ω) = 2ω π ℑ (Tr {G(ω)}) , (3)
G(ω) = ν êν ⊗ êν ω 2 ν -ω 2 = D -ω 2 I -1 , (4)
and the above total density of state (DOS) of Γ-point phonons, Ω(ω), verifies the following constraint with respect to the degrees of freedom of the system under 3D periodic boundary conditions:

∞ 0 Ω(ω)dω = 3N -3. ( 5
)
We can express the classical vibrational entropy of the system at a given temperature T , from the total density of vibrational modes states:

S vib = -k B ∞ 0 ln ℏω k B T -1 Ω(ω)dω. ( 6
)
Very often, the eigenmode (ê ν , ω ν ) is delocalized over several atoms. For example, low energy and large wavelength Rayleigh phonons induce concerted atomic motions across the entire system. However, the local contribution to S vib i can be deduced [START_REF] Dederichs | Point Defects in Metals II[END_REF] by transforming from the delocalized basis êν of the Green function in equation (4) to a localized basis êiα of atomic sites, which will yield the local density of states Ω i (ω). This transformation is not unique, and, in the next section, we will choose the standard local projection.

B. Local basis for densities of states of vibrational modes

The orthonormal bases of normal modes êν and atomic sites êiα (where êiα are unit vectors for displacement of atom i along direction α) are both complete over the configurational phase space, and are therefore related by a unitary transformation:

êν = i α ξ iα (ν) êiα , (7)
where the square of the rotation matrix elements, |ξ iα (ν)| 2 , can be seen as the probability of the mode êν to be localized on atom i and along the α direction. By analogy with equation ( 4), the local density of state of Γ-point phonons reads:

Tr(G(ω)) = i,α êiα • G(ω) • êiα , (8)
Ω i (ω) = α 2ω π ℑ (ê iα • G(ω) • êiα ) , (9)
where ℑ(•) selects the imaginary component, giving a local vibrational entropy

S vib i = -k B ∞ 0 ln ℏω k B T -1 Ω i (ω)dω. ( 10
)
We note that the vibrational entropy of the system is completely defined by projected normal modes on each atom. Furthermore, in this local basis the total entropy can be exactly decomposed into local entropies. Regression of vibrational entropy can thus be achieved by a local approach depending on the atomic neighborhood. This local problem needs an accurate and systematic representation of the local atomic environment as well as the fact that this projection mixes the eigenvectors.

C. Local atomic environment encoded into local atomic descriptors

At the heart of the current approach is the relationship between the local density of states and the local atomic environment. Atomic descriptors are numerical tools developed to describe quantitatively the local atomic environments [42][43][44][START_REF] Bartók | Gaussian Approximation Potential : an interatomic potential derived from first principles Quantum Mechanics[END_REF][START_REF] Bartók | [END_REF]. A large number of the methods and studies were presented in the literature to build regression models between local physical observable and local atomic descriptors [START_REF] Bartók | [END_REF][47][48][49][50][51][52]. In the next section, we will introduce simple models which link the local density of states and the local atomic environment. Even if the choice of local basis is not unique we will numerically show that the above local basis is a good choice to state the proportionality of descriptors and the local environment. The relation between the two quantities is established by the use of local atomic descriptors that provide the non-linear encoding of the geometrical neighborhood of each atom. These atomic descriptors project the atomic configurations into descriptor space. This non-linear function maps the entire neighborhood of a central atom into a space with fixed dimension, hence, we assume the existence of a set of functions {D i } 0≤i≤N such that

D i : R 3N → R D q → D i (q),
where D i is the local atomic descriptor for the i th atom and q is the vector of coordinates of the entire system. Moreover, D is independent of the number of atoms in the system. Descriptor functions should respect the symmetries of the system, i.e. applying symmetry operations (e.g. permutations, translations and rotations) on the input coordinate vector q should not change the value of D i .

The notion of atomic descriptors in material science was introduced by Behler and Parrinello [42][43][44]. They proposed the G 2 descriptor, which is sensitive to the radial distribution of neighboring atoms, weighted by a Gaussian. Since then, many descriptors have been developed by (i) introducing the explicit angular description, as the G 3 descriptor [42], (ii) using the spectral decomposition in 3D or 4D spherical functions of the atomic density [START_REF] Bartók | Gaussian Approximation Potential : an interatomic potential derived from first principles Quantum Mechanics[END_REF][START_REF] Bartók | [END_REF] (iii) decomposition of the total energy in manybody contributions that are expanded in tensorial decomposition in particular basis [53][54][55][56][57] (iv) particular design for a given system [58][59][60][61][62][63], (v) or using machine / deep learning methods in order to find the appropriate descriptors [49,50,64,65], (vi) hybrid descriptors that can mix all other classes mentioned above [51]. The dimension of the descriptor space, D, ranges from few tens to few thousands. The dimension is flexible and is often used to control the level of accuracy necessary to represent the local atomic environment in the descriptor space. The numerical cost is also proportional to D.

In this paper we choose to work with bi-spectrum SO(4) (bSO(4)) descriptor [START_REF] Bartók | Gaussian Approximation Potential : an interatomic potential derived from first principles Quantum Mechanics[END_REF][START_REF] Bartók | [END_REF]. This descriptor is based on the 4D hyper-spherical harmonics decomposition onto the unit sphere of R 4 . With this projection any function returns values in R 3 as described in [START_REF] Bartók | Gaussian Approximation Potential : an interatomic potential derived from first principles Quantum Mechanics[END_REF]. The local environment of the i th atom is described as a density ρ i (q), and can be decomposed on the basis of 4D spherical harmonics

ρ i (q) = k∈Ri w k δ(q -q k + q i ), (11)
= k∈Ri ∞ j=0 j m,m ′ =-j c m,m ′ i,j U m,m ′ j , ( 12
)
w k is species-dependent weight, the c m,m ′ i,j
are the result of the scalar product between the density centered on atom i and the hyper-spherical harmonic U m,m ′ j . R i is the cut-off radius for atom i, q k and q i are the coordinates of atom k and i, respectively. The components of bSO( 4) are defined by the following equation:

B i jj1j2 = (c m,m ′ i,j ) † H j1j2 (c m1,m ′ 1 i,j1 ⊗ c m2,m ′ 2 i,j2
), (13) where j ≤ j max , |j 1 -j 2 | ≤ j ≤ j 1 + j 2 and H j1j2 is related with the Clebsch-Gordan coefficient of SO( 4) group. In this study we use j max = 4 and select only the diagonal components j 1 = j 2 [START_REF] Bartók | Gaussian Approximation Potential : an interatomic potential derived from first principles Quantum Mechanics[END_REF][START_REF] Bartók | [END_REF][START_REF] Kakarala | The bispectrum as a source of phase-sensitive invariants for Fourier descriptors: a group-theoretic approach[END_REF] yielding the total number of components to 35. Otherwise stated, the cut-off distance is set to 5 Å . The bSO(4) descriptor is an over-complete basis of representation of the SO(3) group. In the present study, we extend our previous formalism [41] to a quadratic machine learning model. The main improvements of the regression model for formation entropy or attack frequency are achieved by increasing the dimension of the descriptor space: from 2D (for attack frequencies) to D 2 (for formation entropy of systems under deformations), where D is the initial dimension of the descriptor space. The higher the dimension of fit, the more parameters are required. This high-dimensional regression increases the risk of overfitting and decreases the predictive power of the model. An accurate description of the local atomic environment can be obtained using the bSO(4) descriptor with a relatively low number (35) of components [START_REF] Bartók | Gaussian Approximation Potential : an interatomic potential derived from first principles Quantum Mechanics[END_REF][START_REF] Bartók | [END_REF]51] Therefore, descriptor bSO(4) provides the right balance between the low dimensional descriptor space (critical for the ML quadratic model) and the precision of the representation. This low dimensional descriptor space helps to prevent the risk of overfitting.

III. EXTENSION OF SURROGATE MODEL FOR VIBRATIONAL ENTROPY AT HIGHER ORDERS

In previous work [41] we have shown that the local entropy S vib i is proportional to the local atomic descriptors:

S vib i = w • D i , ( 14
)
where w is the weight vector that parametrizes the surrogate model.

D i = {B i jj1j2 } 0≤j≤jmax such as |j 1 -j 2 | ≤ j ≤ j 1 + j 2 is
the local atomic descriptor of the atom i given by Eq. ( 13). For j max = 4.0, the number of components is equal to 35. Moreover, this proportionality expressed in [41] is reinforced by the definition of harmonic vibrational entropy given by Dederichs et al. [START_REF] Dederichs | Point Defects in Metals II[END_REF], S vib as a sum of local terms on each atom :

S vib = i S vib i . (15)
The above equation ( 15) is exact when summing over all atoms in the system. In this study, we explore how the mere proportionality of Eq. 14 may be enhanced beyond linearity, e.g., by considering a quadratic model. Before introducing the quadratic model, we introduce the following vector and matrix notations for the rest of the study:

D = i D i ∈ R D , D = i D i • [D i ] ⊤ ∈ R D×D . ( 16
)
By using the above notations our quadratic model for the vibrational entropy reads

S vib = w • D + Tr {W • D} , ( 17
)
where w is the weight vector of the linear model Eq. ( 14) and W ∈ R D×D is a learnable matrix. We note that D/N and D/N -DD ⊤ /N 2 give the descriptor means and sample covariances, respectively. An equivalent way of writing this quadratic model is given by:

S vib = i [w • D i + W : D i ⊗ D i + o ∥D i ⊗ D i ⊗ D i ∥ ], (18)
where : denotes the double contraction operator. The second term reads W :

D i ⊗ D i = D k=1 D l=1 W k,l D i k D i l .
Finally, the third term accounts to

D i ⊗ D i ⊗ D i = D k=1 D l=1 D m=1 D i k D i l D i m e k ⊗ e
l ⊗ e m (where {e i } 1≤i≤D are the canonical basis of the descriptor space) and corresponds to 3 rd order (i.e. cubic) term. In the present study, we consider only the quadratic expansion. Increasing the order of expansion increases the number of model parameters and thus the fitting capacity of the model, but also increases the risk of overfitting. We show below that a well chosen quadratic model form gives better accuracy than the linear model while maintaning excellent transferability.

We will call as "Extended Quadratic" Machine Learning (EQML) this second order formulation. The current EQML model preserves the entropy extensiveness, is local and enables direct comparison with the linear model Eq. (14). The performances of EQML model (17) are tested on the database of small defects under strain in bcc iron built in our previous work [41].

A. The database: small defects in BCC iron

A point defect can be created in a perfect crystalline system of N b atoms by adding or removing N d atoms. We are going to denote with C the resultant defective structure. The formation vibrational entropy S f,C at temperature T reads:

S f,C (T, N d ) = S d,C (T, N b ± N d ) - N b ± N d N b S b (T, N b ), (19)
By definition, the vibrational entropies of the defected structure S d,C (T, N b ± N d ) and of the perfect bulk structure S b (T, N b ) are computed at the same volume V . Under the harmonic approximation, the formation entropy will thus depend only on the energies ℏω ν of normal modes. For high temperatures, larger than the crystal Debye temperature, such that max ℏων

k B T ≪ 1, the ex- pression becomes simply S f,C (N d ) = k B ln   ν b (ℏω ν b ) N b ±N d N b ν d ℏω ν d   , (20)
where ω ν b and ω ν d are the frequencies of the perfect bulk and defect configurations, respectively. This classical formulation of the formation vibrational entropy is useful in the present study because it is possible to assign to each defect configuration a single value for vibrational entropy, instead of a function that depends on temperature (as would be the case in the limit of quantified phonons). This formulation is not a limitation of the present study, but a choice to demonstrate the feasibility of current surrogate models.

To train our machine learning model we used a database of 4506 configurations of bcc iron, including various defect morphologies such as 2-4 self interstitials and 4 vacancy clusters in supercells of (8a 0 ) 3 (N b = 1024). All these instances of the defects are minima of the energy landscape and have been generated using the ARTn method. [24,[START_REF] Barkema | [END_REF]. The Activation-Relaxation Technique nouveau [24,[START_REF] Barkema | [END_REF][68][69] is a powerful method for searching saddle points and transition pathways of a given potential energy surface. The ARTn method is designed to explore the energy landscape of the system using only the lowest eigenvalue and the associated eigenvector of the Hessian. The ARTn method is composed of two main steps, the activation step and the relaxation step. The activation step consists in moving the system from a local minimum to a saddle point. The relaxation step consists in relaxing the system, from the computed saddle point, to another local minimum. At the end the ARTn method provides 0 K minimum -saddle point -minimum sequences. All the 4506 minima configurations are different. In order to obtain a more complete dataset, we perform data augmentation. On the ARTn minimum configurations of the dataset, we have applied a homogeneous and isotropic strain of -1% to 3%, resulting in a total of 22530 configurations. Reference harmonic vibrational entropies are computed by using molecular dynamics simulations with LAMMPS [70] and the PHONDY package [71][72][73] from the direct evaluation of vibrational spectrum. PHONDY package enables the evaluation of the phonon spectrum of crys-talline systems by direct diagonalization of the dynamical matrix of the system [1,[71][72][73] and allows the computation of vibrational properties in the framework of the harmonic approximation. We use the modified embedded atom potential (MEAM) developed by Alireza and Asadi [74]. Descriptors of each configuration have been computed by using the MILADY package [51,52,75].

B. Train/test procedure

The linear model is parametrized with the same procedure described in [41] using Eq. ( 14). The parametrization of the EQML model is sequential, in two steps procedure, being preconditioned by the linear model and then adjusting the quadratic part. Firstly, we set the values of the linear model, w in Eq. ( 17) by a linear fit using Bayesian ridge regression. Secondly, the target property becomes the differences ∆S k ≡ S k -Ŝlin k for each configuration k, where Ŝlin k is the estimation of S k with the linear model. The values of the elements of the tensor W , in Eq. ( 17), are parametrized using ∆S k values using the same Bayesian ridge regression. All the fitting procedure is performed with ridge Bayesian Regression by using scikit-learn package [76] (the initial value of σ w Bayesian prior has been set by default).

The robustness and the transferability of the surrogate EQML model is checked by performing a train/test procedure. Here, we define two statistical quantities in order to evaluate the quality of the surrogate model:

1 M M k Ŝvib k -S vib k 2 (RMSE), (21) 1 M
M k | Ŝvib k -S vib k | (MAE), (22)
where S vib k and Ŝvib k are the formation entropy and the predicted formation entropy of the k th configuration, respectively. RMSE and MAE are the Root Mean Square Error and the Mean Absolute Error, respectively. The database configurations were randomly divided into two sets at a certain ratio of p. The percentage of the training configurations is set at (1 -p) (from the total database) and the rest of the database, with a ratio of p, is reserved for tests. The surrogate model is fitted on the training set and the predictions are evaluated for the test set. RMSE and MAE are calculated for both sets. In order to reduce bias of the random procedure selection, we iterate the procedure a hundred times for a given ratio of p and we average the value of RMSE and MAE for the training and test set.

C. The performance of the EQML surrogate model

Here, we compare the two regression models presented in the previous sections: (i) the linear model given by equation ( 14) and (ii) the EQML model given by equation (17). The comparison is made using Figure 2: linear machine learning approach from [41] is presented in the Figure 2 2 it is possible to deduce that the linear model is transferable even for a very small fraction 1 -p. EQML remains stable even for large test / small train database's partition. EQML learning curves emphasize small over-fitting behavior for the ratio p > 0.7. Even under these conditions, the RMSE of EQML model is inferior to the linear model. It is important to note that the additional quadratic term preserves the good transferability capacity of the linear model. The strong linear preconditions of the quadratic parametrization keep the characteristics of the linear fit. Moreover, as we pointed out in [41], the information provided by the local vibrational modes is sufficient to quantitatively recover vibrational modes properties, even if they are delocalized over the cutoff distance of descriptors.

IV. APPLICATION OF LINEAR MODELS FOR HARMONIC TRANSITION STATE THEORY (HTST)

The kinetic pathways of the microstructural evolution of the system are driven by the migration energies / free energies landscape. Sequences of minima and saddle points drive complex phenomena such as the agglomeration of defects in a larger structure, e.g. a dislocation loop [77] or the mobility of the dislocations [1,78]. In the following, we will call a sequence of connected minimum -saddle point -minimum as an event E. The transition rate is the probability of realisation of that event in a specific order. The migration coefficients can be computed from the transition rates between the relevant minima of the energy landscape [13,15,[79][80][81][82]. As such, the transition rates are observable of paramount importance in the implementation of multiscale simulations such as Kinetic Monte Carlo (KMC), whatever the variant: objects [83][84][85][86][START_REF] Domain | Handbook of Materials Modeling[END_REF][START_REF] Jansson | [END_REF][89], events [90][91][92][93], or off-lattice framework [11,94]. Each variation of the KMC method mentioned above has a special recipe for defining the physical reality corresponding to the transition rate. In object KMC, for example, each rate is associated to an atomistic transition rate of a particular defect which is the main concept of the present study. The event and off-lattice KMC deals with rates, which correspond to some physical phenomenons such as collective migration and reaction of defects.

Let's consider an event E, which is defined by two instances: the initial minimum state E, m and an associated saddle point state E, s. We define the rate from state E, m to state E, s as R E,m→s . The overall sequence of the event contributes to the transition rate [95,96]. However, within the framework of the most used approximation in computational materials science for the transition rate, namely the Transition State Theory (TST) [95,97], only the initial minimum and the saddle point of the events account for the transition probability. In the harmonic approximation, the transition rates gives [95,97]:

R E,m→s = ν * E,ms e -β∆E E,m→s , (23)
where β = (k B T ) -1 , ∆E E,m→s is the energy difference between the saddle point and the minimum and ν * E,ms is the attack frequency defined in the framework of harmonic TST as:

ν * E,ms = ν m ′ ∈S(E,m) ν m ′ ν s ′ ∈S(E,s) ν s ′ , ( 24
)
where the numerator is the product of frequencies at the minimum S(E, m) and the denominator is the product of frequencies at the saddle point S(E, s). By S(E, m) and S(E, s), we denote the ensemble of the Hessian spectrum at the minimum (E, m) and saddle point (E, s) configuration of event E, respectively. Obviously, the unstable modes such as the three zero modes due to the periodic boundary conditions as well as the imaginary mode of the saddle point are not included in ensemble S. In practice, under harmonic approximation, ν * E,ms is obtained by diagonalizing the dynamical matrix of the system's minimum and saddle point. The numerical complexity of this procedure is O(N 3 ) and limits the size of the routinely studied systems to a few tens of thousands of atoms. Due to this complexity, the majority of multiscale studies use phenomenological laws of unique value for the attack frequencies (such as the Debye frequency for crystalline materials). Here, we propose to extend the linear machine learning approach in the descriptor space to compute and predict the attack frequencies for a collection of events {E i }. In this way, as in the case of formation vibrational entropy, we avoid a direct diagonalization of the dynamical matrix which is replaced by the O(N ) operation of computing descriptors.

A. Reformulation of the attack frequency

For the attack frequency defined in Eq. ( 24), it is better to handle the logarithm of frequencies for ensemble S, which are positively definite. which can be related to the density of states Ω j (ω) for the j th state of the system:

ln ν * E,ms = ν m ′ ∈S(E,m) ln (ν m ′ ) - ν s ′ ∈S(E,s) ln (ν s ′ ) , (25)
ν j ′ ∈S(E,j) ln (ν j ′ ) = +∞ 0 ln ω 2π Ω j (ω)dω. (26)
Similarly, the logarithm of the attack frequency can be decomposed into local contributions. The associated surrogate model can be set up using a linear model in descriptor space. The regression model attack frequencydescriptors now has the following formulation:

ln ν * E,ms = w 1 • D E,m ⊕ D E,s , (27)
where D E,m/s = d∈E,m/s D d ∈ R D is the total descriptor vector for E, m or E, s. We denote by • ⊕ • the direct concatenation operator for two descriptor vectors. This extended descriptor space is sufficiently general to capture the fact that the evaluation of reaction rate R E,m→s of the event E requires information about the minimum configuration m and the saddle point configuration s.

The direct sum of descriptor vectors for states E, m and E, s gives an extended descriptor vector of dimension 2D.

B. The database for transition rates: amorphous Si

The performance of the surrogate model for attack frequencies was tested on amorphous Si. This system is a challenging test case where each minimum of the energy landscape is connected to a large number of saddle points. This system was widely studied in the past using the ARTn method [15,94,98,99]. The amorphous system is contained within a cubic cell of 4096 atoms at constant volume. The spectrum of partial Hessian is estimated by using the Lanczos method described by Marinica et al. [9], and we fix the admissible error on Lanczos eigenvalues at 1×10 -6 (1.018049×10 -2 ps) -2 . The amorphous Si system is explored through inherent states. The inherent states are representative states of disordered materials corresponding to attraction basins (local minimum surrounded by many other minima and saddle points). Inherent states give access to a potentially astronomical number of different configurations and activation barriers. The database of events E is generated from 20 independent inherent states of amorphous silicon, simulated using a modified version of the Stillinger-Weber potential [100]. To obtain inherent states, we run canonical molecular dynamics simulations [70], with 4096 Si atoms, a fixed density of 2.192 g.cm -3 , and using a timestep of 1 fs. Random configurations are first equilibrated at 2300 K for 20 ns, and then directly relaxed (i.e. without any intermediate quench) at 700 K during 100 ns. Finally, the system energy is minimized using the FIRE algorithm [101], until all components of the force vector are lower than 10 -9 eV. Å-1 . Once initial minima are prepared, we sample saddle points with ARTn (converging towards them until all force vector components are lower than 10 -7 eV. Å-1 ). The connectivity of saddle points to initial minima is systematically checked using the steepest descent method, and duplicates are removed by comparing saddle points energies and displacements of the most displaced atom [30]. At the end, the amorphous Si database contains an average of 420 saddle points per minimum (a total of 10502 distinct activated events).

C. Surrogate model for attack frequencies

The surrogate model was trained using the bSO(4) with j max = 4.0. For consistency reasons, the descriptor was computed using a cut-off equal to that of the empirical potential used to perform the MD simulations, r cut = 3.77 Å [100]. The direct sum descriptor spans a descriptor space with 35 + 35 components. The performances of the linear model are presented in figure 3 for logarithm (a) and plain values (b) of attack frequencies. The spectrum of attack frequencies is wide over many orders of magnitude.The linear model is capable of predicting magnitude changes in a fairly accurate manner. The value of the log(ν/ν 0 ) (ν 0 is fixed at 1 THz) has the RMSE of 0.65 for a range of 10 log(ν/ν 0 ). The results of the train/test procedure, presented in the inset of the figure 3.(a), indicate that the model is stable even for a small train fraction of the database. This behavior of the present linear model is similar to that of the previous surrogate model for vibrational formation entropies [41].

The results of plain attack frequencies are shown in Figure 3.(b), the RMSE is about 1400 THz for a range of values from 1×10 -1 to 1×10 5 THz. The logarithm of the stochastic noise for the direct attack frequency model, log(ν/ν) where ν is the predicted frequency, follows a normal distribution presented in the inset of Figure 3.(b) with a mean µ = 0 and a standard deviation σ = 0.653. We can estimate that 98 % of the predictions ν verify the following ratio 0.27 ≈ e -2σ ≤ ν/ν ≤ e 2σ ≈ 3.7. Further analysis of the distribution of errors will be carried out in the context of the statistical formulation of the compensation law, in the next section.

In conclusion, under the linear formalism in descriptor space, the attack frequencies can be inferred and predicted. This approach is purely geometric and based on the local decomposition of the density of states of vibrational modes, bypassing the direct calculations and diagonalization of the dynamical matrix. This type of approach scales like O(N ) and has the same order of magnitude as the calculation of descriptors and could be used in any multi-scale atomistic methodology that uses transition rates.

V. THE COMPENSATION MEYER-NELDEL LAW IN DESCRIPTORS SPACE

It is frequently found in thermally activated processes -i.e. following an Arhenius-like law given in Eq. ( 23)that when the activation energy increases within a family of processes, the prefactor also increases. Thus, observed first in chemistry [102] and then in physics by Meyer and Neldel [103], the increase of a rate prefactor somewhat "compensates" for the decrease in the Arrhenius exponential term governing the dependence on temperature. This experimental observation can be expressed as a simple correlation between the observed prefactor (ν exp ) and the slope (∆E exp ) of the measured Arrhenius law: ln ν exp = a + b∆E α exp (where α is an exponent, which usually is taken as 1). Evidence of a direct link between the experimental measurements and numerical simulations is difficult to obtain [30,[START_REF] Philibert | Diffusion in Solids -Past, Present and Future, Defect and Diffusion Forum[END_REF][START_REF] Yelon | [END_REF]. The main complexity is that in experimental measured Arrhenius law there is a contribution of many kinetic pathways for which it is difficult to account theoretically / numerically the huge number of thermally activated sequences of events. Evidence of a direct link between experimental measurements and numerical simulations is difficult to obtain [30,[START_REF] Philibert | Diffusion in Solids -Past, Present and Future, Defect and Diffusion Forum[END_REF][START_REF] Yelon | [END_REF]. The main complexity is that in experimental measurements of the Arrhenius law, there is a contribution from many kinetic pathways which are difficult to theoretically and/or numerically account for the huge number of thermally activated sequences of events. However, in the case of simple energetic landscapes, a direct association can be established between the migration mechanism and experimental observation. This is the case for self-diffusion in crystalline solids or migration of particular defects, e.g. diffusion of ad-atoms on metallic surfaces [106,107] or thermal activated dislocation movements [108]. This compensation was even demonstrated analytically in the case of well identified thermal overcoming of energetic barriers, such as in the correlation between the magnitude of the gap in semiconductors and the thermally activated conductivity [109]. This compensation is explained by some authors with the concept of multi-excitation entropy and its consequences. When a fluctuation involving a large number of excitations occurs, for example when a large activation barrier is overcome, there must be a large entropy associated with this fluctuation [START_REF] Yelon | [END_REF]. Otherwise, when the kinetic pathways are difficult to identify, such as the migration in disordered solids, some studies [30,[START_REF] Philibert | Diffusion in Solids -Past, Present and Future, Defect and Diffusion Forum[END_REF][START_REF] Yelon | [END_REF] describe a compensation effect for large set of events between the average value of the pre-exponential factor logarithm ln ν * E,m→s , in a given bin / window of activation energy, and the energy barrier. This correlation is called enthalpy-entropy compensation law and results from the averaging over a large number of kinetic pathways [30]. Moreover, in the general framework of thermally activated diffusion of kinetic processes, beyond an array of interpretations all along the last 100 years, Gelin et. al [30] recently proposed (based on numerical simulations) a general interpretation for which the compensation is a statistical law associated with the deformation of the vibrational spectrum caused by the local deformations of the atomic lattice and that can be calculated within the harmonic TST.

The goal of this section is two fold: (i) we first examine if the present surrogate model for attack frequencies is able to recover enthalpy-entropy compensation law that was previously identified by the direct calculations. (ii) Then, in the framework of the linear model in descriptors space, we try to give a statistical insight of the compensation law by the simple fact that the present surrogate FIG. 3: Illustration of the linear model to adjust the (log) attack frequencies of the amorphous Si database, where ν 0 = 1 THz. The color gradient represents the data distribution, with yellow corresponding to dense data zones. model for the activation entropy and the activation energy share the same descriptor space. First, let's define the proportionality between the value of energy barrier ∆E E,m→s and the logarithm of attempt frequency log ν * E,ms . We define two possible statistical correlations: (i) the direct correlation of activation energy-entropy (DCAEE) for a particular event, and (ii) the enthalpy-entropy compensation (EEC) law based on averaging many activated events. In the case of DCAEE the correlation can be expressed in simple terms for event i:

log(ν i /ν 0 ) = γ∆E i + log(ν γ /ν 0 ), (28)
where γ in eV -1 and log(ν γ /ν 0 ) are parameters that define DCAEE correlation. As we have mentioned before, this law is observed in some simple thermally activated events such as metal conductivity [110], diffusion of adatoms on metallic surfaces [106,107], dislocation glide in Zr [108] etc. The other type of correlation is a marginal proportionality. This average EEC relation can be expressed as:

E [log(ν(∆E)/ν 0 )|∆E] = γ * ∆E + log(ν * γ /ν 0 ), (29)
where γ * in eV -1 and log(ν * γ /ν 0 ) are parameters. We denote by E [log(ν(∆E)/ν 0 )|∆E] the average value of log(ν(∆E)/ν 0 ) for each configuration whose associated energy barrier is between ∆E and ∆E + δϵ, where δϵ is the width of the energy bin. This expression has been used by Gelin et al. [30] for EEC in thermally activated systems in materials science. We note that the DCAEE law implies its marginal EEC definition. These correlations have practical consequences being a "physical" surrogate models in order to estimate the prefactor of some transitions without doing any computations or experiments. In the following, we give a new statistical insight of the DCAEE compensation law by representing the energy barriers and attack frequencies in the same descriptor space and using the same surrogate model with just different parameterisations. FIG. 4: The correlation between the predicted and computed energy barrier ∆E using the linear regression between descriptors and energy barriers on the amorphous Si database. The relative accuracy of the linear regression model is better for barriers than for the log of prefactors (a RMSE of 0.09 eV over a range of 6 eV).

Using the formalism of the current surrogate model, we built a linear regression model for energy barriers, using the relation:

∆E E,m→s = w 2 • D E,m ⊕ D E,s , (30)
where D E,m/s = d∈E,m/s D d ∈ R D is the total descriptor vector for E, m or E, s. The weight vector w 2 ∈ R 1×2D is parametrized using the same database of amorphous Si with the particularity that the target property is now the barrier energy of the event. The corresponding energy barriers have been calculated using the same ARTn exploration. For the mapping of the atomic configuration into the descriptor space we have used the same descriptor bSO(4) with the same parameters i.e. j max = 4.0 and the cut-off radius r cut = 3.77 Å . Regression results are presented in Figure 4. The surrogate energy model given by Eq. ( 30) provides a RMSE of 0.09 eV. Interestingly, the regression is more accurate for the energy barrier than for the log attack frequency. In [41], it is shown that the accuracy performances can be explained by the nature of the force fields. This source of errors for surrogate models for the migration energy or the attack frequency has exactly the same origins. With these two surrogate models for the energy and the attack frequency we have the appropriate tools to investigate the DCAEE and EEC "compensation laws".

Firstly, we investigate the ability of the present surrogate model to recover the direct EEC law from [30]. We emphasize in Figure 5 the correlation between the value of energy barrier ∆E and the corresponding attack frequency, in terms of log(ν/ν 0 ), using direct and machine learning surrogate approaches. We observe a clear linear relation between ∆E and log(ν/ν 0 ). In Figure 5.(a) the blue spots emphasize the direct atomistic calculations and in Fig 5 .(b) the red spots are the predictions of the current surrogate model. Regression models for both datasets are given in the inset in figure 5. The correlations between ∆E and log(ν/ν 0 ) for the direct method and the surrogate model are quantitatively very close. The current correlation coefficient is defined by the following equation:

r(log(ν/ν 0 ), ∆E) = C[log(ν/ν 0 ), ∆E] V[log(ν/ν 0 )]V[∆E] ( 31
)
where C[log(ν/ν 0 ), ∆E] is the covariance between log(ν/ν 0 ) and ∆E. V is the variance of the corresponding observables. The correlation coefficient is a quantitative measurement of whether there is a linear relation between two quantities. If the DCAEE law is exact, it implies this r(log(ν/ν 0 ), ∆E) = 1. The correlation coefficient is 0.61 for the database and 0.65 for the data predicted by the surrogate model. The current surrogate machine learning approach is accurate enough in order to reconstruct the correlation between the two observables.

Here, we go further. Are there particular conditions of realisation of DCAEE law that can be characterized in the descriptor space? Let's consider two distinct events E 1 and E 2 and the associated attempt frequencies ν * E 1 ,m→s and ν * E 2 ,m→s such as ln ν * E 1 ,m→s = α ln ν * E 2 ,m→s , Eq. ( 27) implies:

w 1 • (D E 1 ,m ⊕ D E 1 ,s -αD E 2 ,m ⊕ D E 2 ,s ) = 0, (32)
where D E 1 ,m , D E 1 ,s and D E 2 ,m , D E 2 ,s are the descriptors of minimum and saddle point for the two events E 1 and E 2 , respectively. w 1 is the weight vector introduced by Eq. ( 27). If we set ∆E i -∆E 0 = γ log(ν i /ν 0 ) and ∆E j -∆E 0 = γ log(ν j /ν 0 ), so we can deduce from equation ( 28) log(ν i /ν 0 ) = ∆Ei-∆E0 ∆Ej -∆E0 log(ν j /ν 0 ). If there exists a DCAEE relation then we can build the following relation based on Eq. ( 32) such as:

w 1 • D E i ,m ⊕ D E i ,s - ∆E i -∆E 0 ∆E j -∆E 0 D E j ,m ⊕ D E j ,s = 0, (33)
where E i , D E i ,m ⊕ D E i ,s are the events and the associated descriptors for each event E i of the Si amorphous database. w 1 is the weight vector defined by Eq. ( 27). This orthogonality relation is valid if the DCAEE law defined by equation ( 28) is exact. As shown in Figure 5 the DCAEE relation ( 28) is qualitatively true but implies an imperfect correlation i.e. 0.61 instead of 1.0 for the perfect theoretical correlation. In order to quantify the notion of orthogonality in the descriptor space, given by Eq. ( 33), for the realistic case of imperfect correlations of DCAEE law, we introduce the following vectorial quantity:

D ij = D E i ,m ⊕ D E i ,s -∆Ei-∆E0 ∆Ej -∆E0 D E j ,m ⊕ D E j ,s .
In the end, we focus on the following ratio:

κ ij = w 1 • D ij w 1 • D E i ,m ⊕ D E i ,s + D E j ,m ⊕ D E j ,s . (34)
If DCAEE law is valid, the following ratio κ ij should be as small as much as possible. For the amorphous Si dataset we can compute the average ratio:

⟨κ⟩ ≡ 1 M 2 M,M i,j κ ij ≪ 1, ( 35
)
where M is the number of events. The average value of κ over the entire database is equal to ⟨κ⟩ = 0.11, which is low value. This suggests a weak DCAEE law for this particular database. Furthermore, for the same amorphous Si database we investigate the variance of the marginal correlation of the EEC law. In order to compute the numerical value of bin average of the E [log(ν/ν 0 )|∆E], given by Eq. ( 29), the barrier energies domain ∆E is split into 50 uniform bins, each of them having the width δϵ = 0.144 eV. Consequently, E [log(ν/ν 0 )|∆E] is computed with the average of log(ν/ν 0 ) in each bin of the energy domain. The analysis is performed, over the Si database, for brute and predicted data from our surrogate model. For both we compute the marginal variance V [log(ν/ν 0 )|∆E]. The marginal variance represents the width of the intrinsic stochastic noise over log(ν/ν 0 ) for a bin in [∆E, ∆E +δϵ]. A constant marginal variance implies constant intrinsic noise for all energy bins. Consequently, constant variance over the entire database indicates that correlation between log(ν/ν 0 ) and ∆E is blurred by a noise with the same origin e.g. induced by the same physical phenomena over the entire range of the energy barriers domain ∆E. Moreover, marginal variance V [log(ν/ν 0 )|∆E] is a quantitative indicator of the DCAEE law's validity. From this perspective, the DCAEE is a particular case of marginal law with V [log(ν/ν 0 )|∆E] = 0. The less is the variance, the better is the accuracy of DCAEE to predict log(ν/ν 0 ) from ∆E.

Results of this analysis are given in Figures 5. If we suppose that all events present in the Si database are independent, we can give a simple stochastic reformulation of the DCAEE law, Eq. ( 28), by assuming that the stochastic noise follows a normal distribution:

log(ν/ν 0 ) = γ∆E + log(ν γ /ν 0 ) + N (σ 2 [log(ν/ν 0 )|∆E]), ( 36
) where N (σ 2 [log(ν/ν 0 )|∆E]) is a centered normal distri- bution of standard deviation equal to σ[log(ν/ν 0 )|∆E].
In the limit σ[log(ν/ν 0 )|∆E] → 0, we find the DCAEE relation. Combining the two perspectives of the compensation law, direct correlation and marginal, allows us to give a more general formulation of the enthalpy-entropy compensation. This formulation is valid for simple (e.g. small point defects in crystalline lattice) and disordered systems (such as the present amorphous system).

The present statistical analysis within the descriptors formulation of the enthalphy-entropy correlations gives two important information. Firstly, by using two surrogate models -one for barrier energy and one for attack frequency -which underlay the same descriptor space, we are able to recover the statistical correlations of the EEC law given by direct calculation of Gelin et al. [30]. The results, provided by our linear surrogate models over an amorphous Si database, emphasize the same correlations with the direct HA TST calculations and are solely based on geometrical considerations. Our statistical analysis of the compensation law underlines the ability of the present linear models to capture the complex information about the potential surface of the energy landscape. Secondly, the geometrical information that feed the surrogate models is local, i.e. the descriptors on which is based the linear models encode the geometric structure of atoms within a cut-off distance around a central atom. It is interesting to note that our surrogate model is able to reconstruct well the enthalpy-entropy compensation law solely from this local geometric information. This means that the current surrogate models enable the reconstruction of harmonic vibration quantities, which are diffuse by definition. The ability of the local surrogate model to reconstruct non-local components of vibrational quantities implies a particular structuration of the data in the descriptor space. In fact, data in the descriptor space seems to present a simple topology that allows for new formulations, in terms of elementary geometry in high dimensional space, of non-trivial physical laws in the configuration space, such as the present Meyer-Neldel correlation law.

VI. CONCLUSIONS AND PERSPECTIVES

The current study explores the limits of machine learning surrogate models for the vibrational properties of solids that make a direct link between atomic geometry and vibrational observables of interest. This current approach is developed using the traditional harmonic approximation, which allows vibrational properties to be exactly decomposed into local contributions around each atom. This feature allowed the model to be easily integrated within machine learning frameworks. Traditional methods based on harmonic approximation use for the evaluation of the vibrational entropy the derivatives of the interatomic potential energy surface in particular points of the potential energy surface i.e. minima or saddle points. Consequently, here we treat only the information from particular instances of the phase space such as minima or saddle points of the first-order.

Within the framework of harmonic approximation and transition state theory we proposed surrogate models for the vibrational entropy that account for (i) variations under applied hydrostatic strains (ii) attack frequencies for thermally activated events. The traditional evaluation of the system's Hessian and its diagonalization is bypassed; the only need is to provide reliable atomic positions for the minimum / saddle point configurations. In the present study we have used the ARTn method in order to provide those configurations.

For the present surrogate model for formation vibrational entropy, with or without strain, compared to the previous work, we have extended the dimension of atomic descriptors through quadratic coupling between various components of the original atomic descriptor. We have deduced the correlation between the new atomic descriptors and the vibrational observables within the framework of the quadratic surrogate model (EQML). This EQML model is more accurate than previous linear models and it is not too much less transferable. We keep the balance between accuracy and transferability of the fit by the parametrization procedure of the quadratic fit, which is preconditioned by the linear fit. Furthermore, the EQML model has very good transferability in the sense of train/test procedure. Over a database containing various point defects in α-Fe we have numerically demonstrated that this approach has better accuracy for homogeneous, isotropic deformations from -1% to 3% strain than the previous linear model introduced in [41].

We have replaced the local atomic descriptors with the appropriate descriptor of the activated event, i.e., the sequence minimum -saddle point, for estimating the attack frequency of thermally activated processes. For this purpose, the surrogate model uses structural information, through descriptor projection, of the geometrical configurations of the minimum and of the saddle configuration. The relation descriptors -target local observables is based on the linear model Eq. ( 14). However, this choice is not restrictive and the model framework can be extended to higher orders, such as quadratic EQML. This new surrogate model has been tested on a database of activated events in amorphous Si [30] and provides good accuracy over many orders of magnitude (6 orders of magnitude) of the attack frequency. This model has a good transferability with very stable train / test learning curves.

Our study shows that it is possible to adjust the formation entropy of defects and logarithm of the attack frequency of their activated events only with O(N ) numerical estimation. The present workflow avoids the time-consuming evaluation of system's dynamical matrix (O(N 2 )) and its spectrum (O(N 3 )). The current efficient solution opens many avenues in the field of on-the-fly exploration of complex energetic landscapes, for example using semi-automatic atomistic frameworks such as lattice or off-lattice and relaxed Kinetic Monte Carlo [11,[83][84][85][86][START_REF] Domain | Handbook of Materials Modeling[END_REF][START_REF] Jansson | [END_REF][89]94].

Finally, employing the framework of machine learning surrogate models, we have proposed an insight into the statistical analysis of enthalpy-entropy compensation law -a non-trivial conjecture observed in several materials [START_REF] Philibert | Diffusion in Solids -Past, Present and Future, Defect and Diffusion Forum[END_REF]111]. This law states that, for a given transition event, the link between the magnitude of the potential energy function (the value of ∆E) and its curvature (the value of log(ν/ν 0 )). Here we give a statistical formulation of the compensation energy -frequency law. Sharp statistical analysis emphasizes that we are able to reproduce the same correlation as the direct calculations by using two linear surrogate models for the barrier energy and the attack frequency of activated events. Moreover, by tackling the DCAEE law in the descriptor space, we have provided a geometrical insight into the conditions of realisation of this conjecture, which are quantified by an orthogonality relation. The present formulation requires more investigation in other systems and opens many perspectives for further investigation.

VII. CODES AND DATA

The ARTn package and databases of amorphous silicon events are available upon reasonable request to Normand Mousseau (normand.mousseau@umontreal.ca). The Milady package is open source software under ASL license and can be downloaded at https://ai-atoms.github.io/milady/.

ACKNOWLEDGMENTS

The authors sincerely thank S. Gelin for providing the amorphous Si database and for his valuable suggestions concerning the manuscript. This work has been carried out within the framework of the EUROfusion Consor-

  (a) while EQML model is presented in Figure 2(b). Both models, linear and quadratic, use the same parametrization for the bSO(4) descriptor i.e. j max = 4.0 and the cut-off radius is set to 5 Å . The number of dimensions for linear and quadratic fit are D = 35 and D + D(D + 1)/2 = 665, respectively. EQML model provides better RMSE/MAE fit errors than the simple linear model. The non-linearity of EQML model introduces coupling between descriptor components giving larger fit capacity. For each class of deformations, EQML model has lower RMSE (0.34k B ) compared to linear model, 0.82 k B . From the learning curves presented in the insets of Figure

FIG. 2 :

 2 FIG. 2: Illustration of the performance of linear and EQML surrogate models using deformed supercells of I 2-4 /V 4 clusters and by using bSO(4) 4 descriptor. The initial configurations have a (8a 0 ) 3 volume and have been deformed by applying an homogeneous and isotropic dilatation of the supercell. The deformation rates range from -1% to 3%. Figures illustrate the performances of linear (a) and EQML (b) of surrogate models.

Figure 3 .

 3 FIG. 3: Illustration of the linear model to adjust the (log) attack frequencies of the amorphous Si database, where ν 0 = 1 THz. The color gradient represents the data distribution, with yellow corresponding to dense data zones. Figure 3.(a) shows the regression of the logarithm of attack frequencies; the results of the train/test validation procedure are presented in the inset. The values of statistical indicators remain stable, even for a large proportion of validation sets. Figure 3.(b) emphasizes the results of attack frequencies. The logarithm of noise log(ν/ν) (ν is the frequency estimation using the surrogate model of the direct computed frequency ν from ML model) is presented as inset of Figure 3.(b) and follows a normal distribution with mean µ = 0 and standard deviation σ = 0.653.

  (

  a) and (b) emphasize the correlation between log(ν/ν 0 ) and ∆E for the Si amorphous database employing direct and surrogate model, respectively. Grey points indicate the marginal observable E [log(ν/ν 0 )|∆E] and his estimated standard deviation. The ajusted marginal EEC from equation (29) is given as an inset for each figures. Average EEC models are very similar for direct and surrogate models.

Figure 5 .

 5 (c) and Figure 5.(d) draw the marginal variance V [log(ν/ν 0 )|∆E] for direct and predicted data, respectively. For these fig-ures we also give the estimated value of standard deviation variance 2σ [V [log(ν/ν 0 )|∆E]]. This indicator allows to quantify the uncertainty of the marginal variance V [log(ν/ν 0 )|∆E]. Low deviation of marginal variance implies that DCAEE law could be extended depending only on V [log(ν/ν 0 )|∆E]. From Figure 5.(c) and Figure 5.(d) it can be noted that the marginal variance of log(ν/ν 0 ) remains almost constant for every bin of barrier domain ∆E. Moreover, the marginal variance σ[log(ν/ν 0 )|∆E] 2 = V [log(ν/ν 0 )|∆E] is quite similar between direct and predicted data. 2σ [V [log(ν/ν 0 )|∆E]] slightly depends on the value of ∆E. Consequently, the DCAEE relation can be extended by adding only a dependency in V [log(ν/ν 0 )|∆E].

FIG. 5 :

 5 FIG. 5: Drawing the enthalpy-entropy compensation relation for the Si amorphous database. Figure 5.(a) shows values of ∆E and log(ν/ν 0 ) computed with ARTn method. Figure 5.(b) shows predicted values of ∆E and log(ν/ν 0 ) with linear model. The color gradient represents data distribution; yellow corresponds to dense data zones for both types of points. Adjusted EEC relations, following Eq. (29), for both direct and surrogate models are emphasized in (a) and (b) by white points with black contour. Both models, the direct and the surrogate, have distributions with very close correlation indicators. Marginal variance distribution for both dataset is presented in Figure 5.(c) and Figure 5.(d) for direct and surrogate data respectively. The marginal variance is quantitavely almost the same for both dataset.

  tium and has received funding from Euratom Research and Training Programme 2019-2020 under Grant Agreement No. 633053 and Grant Agreement No. 900018 from the ENTENTE project. TDS gratefully recognizes support from the Agence Nationale de Recherche, via the MEMOPAS project ANR-19-CE46-0006-1, and access to the HPC resources of IDRIS under the allocation A0090910965 attributed by GENCI. CL and MCM acknowledge the support from GENCI -(CINES/CCRT) computer center under Grant No. A0130906973.


Françoise Adeline Kerner
email: kerner@mnhn.fr

F Debrenne

The role of Archaeocyatha in Cambrian biostratigraphy and biogeography

Keywords: Archaeocyatha, Porifera, Cambrian biozonations, palaeoecology, reefs, palaeogeography, migration pathways

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Archaeocyatha represent the oldest calcified sponges and the first metazoans to build bioconstructions in association with calcimicrobes. They are a key group in biological and evolutionary studies, biostratigraphy, palaeoecology and the palaeogeography of the early Cambrian.

History of Systematic Studies

Archaeocyathans were first discovered in the middle of the 19th Century in the oldest fossiliferous rocks of Labrador (Canada); their geographical distribution is world-wide including Russia, Kazakhstan, Uzbekistan, Poland, Mongolia, China, Greenland, Canada, USA, Mexico, Argentina, France, Germany, Sardinia, Serbia, Spain, Morocco, South Africa, Antarctica & Australia. Following their discovery by Captain Bayfield, a hydrographer [START_REF] Bayfield | On the Junction of the Transition and Primary Rocks of Canada and Labrador[END_REF], sporadic new finds essentially formed the basis for discussions on the affinities of this enigmatic group. Only when large collections were made, authors had enough data to propose a coherent systematic approach: Bornemann (Sardinia) coined the name Archaeocyatha for a class within the Coelenterata [START_REF] Bornemann | Über die Fortsetzung seiner Untersuchungen cambrischer Archaeocyathus-Formen und verwandter Organismen von der Insel Sardinien[END_REF] [START_REF] Bornemann | Archaeocyathinae, in Die Versteinerungen des Cambrischen Schichtensystems der Insel Sardinien, nebst vergleichenden Untersuchungen über analogue Vorkomnisse aus anderen Ländern[END_REF])"Über die Fortsetzung seiner Untersuchungen cambrischer Archaeocyathus-Formen und verwandter Organismen von der Insel Sardinien.," 1884) (Anon, 1884) and [START_REF] Taylor | The Archaeocyathinae from the cambrian of South Australia: with an account of the morphology and affinities of the whole class: Memoirs of the Royal Society of South Australia[END_REF], working on material from Australia, adopted the name and was the first to consider Archaeocyatha as an independent group intermediate between the Porifera and Coelenterata at the same taxonomic level [START_REF] Taylor | The Archaeocyathinae from the cambrian of South Australia: with an account of the morphology and affinities of the whole class: Memoirs of the Royal Society of South Australia[END_REF]. The investigations world-wide really began in the 1930s with Vologdin and the Eastern school, Okulitch and the Bedfords and the Western school. Both groups based their classification on ontogenic stages but Okulitch's classification [START_REF] Okulitch | North American Pleospongia[END_REF] was not accepted by other palaeontologists. A more logical pattern was established by Russian specialists based on abundant material with wide geographical and stratigraphical ranges. Vologdin [START_REF] Vologdin | Arkheotsiaty i rezul'taty ikh izucheniya v SSSR [Archaeocyaths and the results of their study in the USSR[END_REF] established two classes, whereas [START_REF] Vologdin | Morfologiya pravil'nykh arkheotsiat [Morphology of regular archaeocyaths[END_REF] proposed a separate phylum for the Archaeocyatha. The modern systematic approach established by Zhuravleva [START_REF] Zhuravleva | Arkheotsiaty Sibirskoy Platformy [Archaeocyaths of the Siberian Platform[END_REF] was based on the order of appearance and complexity of skeletal structures, determining key characters and consequently the hierarchy of systematics ranks, still in use today. Intensive studies have been carried out through international cooperation since 1970. At that time most of palaeontologists accepted the concept of an independent phylum and thereafter Archaeocyatha were seldom compared with Porifera except by some far-sighted zoologists. Technical progress in marine research by scuba-divers and the discovery of sponges capable of secreting massive calcareous skeletons with or without spicules, forced a reconsideration of the nature of a number of related fossils groups, e.g., stromatoporoids, chaetetids and archaeocyaths. There is now a consensus on the phylogenetic relationships of these enigmatic organisms: Archaeocyatha represent an extinct class of the phylum Porifera, close to the Demospongiae [START_REF] Debrenne | Archaeocyatha: is the sponge model consistent with their structural organization?: 4th International Symposium on Fossil Cnidaria[END_REF] [START_REF] Debrenne | Irregular Archaeocyaths : morphology, ontogeny, systematics, biostratigraphy, palaeoecology = Archéocyathes irréguliers : morphologie, ontogénie, systématique, biostratigraphie, paléoécologie[END_REF].

Biostratigraphy

Subdivision of the Cambrian System was traditionally based on trilobite ranges. Zhuravleva [START_REF] Zhuravleva | Arkheotsiaty Sibirskoy Platformy [Archaeocyaths of the Siberian Platform[END_REF] established the first stage subdivision based on Archaeocyatha on the Siberian Platform. [START_REF] Rozanov | Biostratigrafiya i fauna nizhnikh gorizontov kembriya [Biostratigraphy and fauna of the lower horizons of the Cambrian[END_REF] discovered a rich archaeocyathan fauna on the Siberian Platform in some strata below the first appearance data (FAD) of trilobites & named a new stage, the Tommotian. Since then, archaeocyathans biozones have been used in key Cambrian areas such as Australia, Canada, Mongolia, Morocco, Siberia & Spain. The distribution of Archaeocyatha in time, mainly the early Cambrian with few relicts in the middle and late Cambrian, limits their use to stages 2 to 4 of the International Stratigraphic Chart. As the regional Siberian chronostratigraphical scale was the reference scheme for archaeocyathan specialists to establish palaeobiostratigraphical correlations, this scale is used in this article: i.e. Fortunian, Tommotian, Atdabanian, Botomian and Toyonian. When possible for many Cambrian localities where archaeocyathan and trilobites were well studied, two parallel scales, one based on trilobites the other on archaeocyathans are utilized. Under certain circumstances, archaeocyaths may actually provide finer biozones than do trilobites (Table 1).

Palaeoecology

Archaeocyathans were adapted to a narrow range of depths, salinity and temperature, but in association with calcimicrobes were the first widespread metazoan reef builders. Archaeocyaths were stenothermal organisms, corresponding to intertropical climatic conditions except to the South in South Morocco and South Europe occurrences & with some Chinese and possible Korean occurrences to the North [START_REF] Debrenne | Repartition paleogeographique des archeocyathes et delimitation des zones intertropicales au Cambrien inferieur[END_REF]) (Fig. 1).

Archaeocyathans were stenohaline organisms. Increased salinity reduced the diversity of archaeocyathan communities; only forms, with simple walls and porous septa, tolerated higher salinities. During brief decreases in salinity, they were located in lime muds but not in evaporite environments [START_REF] Debrenne | Archaeocyatha, palaeoecology: a Cambrian sessile fauna, in Cherchi, A. Autecology of Selected Fossil Organisms: Achievements and Problems[END_REF]. Archaeocyaths were stenobathyal organisms, living anchored in soft substrates in intertidal to subtidal zones but probably not in the subphotic zone. They preferred shallow-water settings as supported by the following observations: 1, they mainly developed in carbonate-dominated lithofacies; 2, they are commonly associated with ooids and bioclastic limestones and with siltstones and quartz arenites indicating oscillating tidal-flat environments; 3, interbedded occurrences with a variety of hemispheroidal stromatolites and oncoids are observed and 4, they coexist with macroboring organisms in Labrador patch reefs (James & Kobluk 1978). In the deepest-water Tommotian bioherms of the Siberian Platform [START_REF] Kruse | Primordial Metazoan-Calcimicrobial Reefs: Tommotian (Early Cambrian) of the Siberian Platform[END_REF], archaeocyathan communities are characterized by a significant development of secondary skeletal structures. On the other hand, episodic erosional events can be demonstrated only in Sardinia (Debrenne et al., 1989a) and Khara-Ulakh Mountains, where a distinctive archaeocyath life strategy consists of Dictyocyathus encrusting the biohermal surface [START_REF] Debrenne | Archaeocyatha, palaeoecology: a Cambrian sessile fauna, in Cherchi, A. Autecology of Selected Fossil Organisms: Achievements and Problems[END_REF].

As passive filter feeders, archaeocyaths were better adapted to habitats with reduced turbulence and increased currents and nutrient supply, as are Recent calcified sponges. Certain morphological features are predominant within some archaeocyaths communities; it might be due to local environmental conditions such as turbulence, detrital input or volcanic ashfall [START_REF] Zhuravlev | Paleoecology of Cambrian reef ecosystems, in The history and sedimentology of ancient reef systems[END_REF]. They also were active bafflers at the periphery and on the top of reefs and constitute an abundant fauna in inter-reef environments.

Since their first occurrences, archaeocyathans were interpreted as being responsible for construction of the first metazoan reefs. The archaeocyathan-calcimicrobial bioconstructions have topographical relief, a biogenic framework & a certain capacity for wave resistance. They were seemingly initiated when mud input slowed down or ceased, thus allowing localized cementation or stabilization of seafloor mud. Archaeocyathans did not generally produce a real biological framework. Since the Tommotian, even when reefs were dominated by calcimicrobes, subordinate archaeocyathans have assisted reef building, supporting the builders and forming cavity roofs. Data on the analysis of biohermal communities and their architectures indicate that they reflect the relative development of the main reef-building components, dominant physical-chemical factors, mainly temperature and nutrient availability and the physiography of the depositional setting. The bioaccumulations show different reef building styles, 1, mud-supported simple to compound mounds locally with stromatactis-like cavities; 2, dendrolitic bioherms and crustose buildups with large shelter cavities and low synoptic relief; 3, cement supported skeletal reefs with wave resistant frameworks often associated with oolitic shoals & 4, bioclastic sands, developed at photic and shallow sub-photic depths on low-angle/low-energy ramps (1-2) or on high-energy conditions on platforms distally rimmed (3) or occasionally swept by storm currents (4) [START_REF] Gandin | Distribution of the archaeocyath-calcimicrobial bioconstructions on the Early Cambrian shelves[END_REF].

Studies of the worldwide distribution of Cambrian reefs have described and illustrated the lithology and biosedimentology of reef-building communities [START_REF] Debrenne | Lower Cambrian archaeocyathan bioconstructions[END_REF]. Features in common with Recent reefs: in situ organism-organism intergrowths, abundant marine cements, stromatactis structures, micro-and macroborings, primary cavities containing diverse cryptobionts and photosymbionts trophic web have been observed. Even if they occur in sedimentologically and climatically comparable environments, some studies of reef-dwelling species suggested that, ecologically, they were very different from the Recent reefs, in terms of nutrient availability, energy flow and their trophic nucleus [START_REF] Wood | [END_REF][START_REF] Zhuravlev | Lower Cambrian reefal cryptic communities[END_REF].

Palaeogeography

Cambrian palaeogeographic reconstructions are notoriously difficult for the Precambrian-Cambrian intervals due to sparse reliable palaeomagnetic data and a lack of any usual climate indicators. Lands distributions are constrained by the distribution of reefs established in associated epeiric seas. Global palaeogeographical reconstructions are based herein on Jaccard's similarity coefficient which calculates a relationship between the numbers of distinct taxa present at two localities at the same time. The Jaccard's similarity coefficient values are calculated for both Archaeocyatha and the total faunas of Tommotian (528-521Ma), early Atdabanian (521-515Ma) and Botomian (515-510 Ma) age. The resulting pathways of archaeocyathan migration suggest the early Cambrian existence of East and West Gondwana, the rifting of Laurentia from the Australian-Antarctic margin & the drift of the Altay-Sayan and Mongolia suspect terranes towards Siberia (Fig. 2) [START_REF] Debrenne | Faunal migrations of archaeocyaths and early Cambrian plate dynamics[END_REF]). An exhaustive compendium of archaeocyath genera may provide material for the further definition of provinces and realms by cluster analysis (e.g. [START_REF] Kruse | Palaeobiogeographic affinities of Australian Cambrian faunas in[END_REF]. Five provinces are recognized after an initial phase of cluster analysis of generic distribution data: A, Siberia-Mongolia (including Siberian Platform-Kolyma, Altai Sayan, Tuva, Transbaikalia Mongolia Far-East-Higgan); B, Europe-Morocco; C, Central East Asia (including Kazakhstan-Tajikistan-Turkmenistan-Uzbekistan-Tarim-Kirgiztan,Urals, South China); D, North America -Koryakia; E, Australia-Antarctica (and possibly South America) (Fig. 3). Two realms are defined by the second phase cluster analysis: Eurasia (the first three provinces) and Laurastal (the last two provinces) (Fig. 4). In the early Tommotian the first appearance of archaeocyaths (FAD) occurred within the transitional Anabar-Sinsk tract of the Siberian Platform (Fig. 5a). They spread to the Altay-Sayan and Russian Far-East only in the late Tommotian. Other regions with phosphatic-rich sedimentation, extensive evaporite basins or fluviatile and deltaic siliciclastic sediments were not amenable to archaeocyath settlement. In the early Atdabanian the major marine transgression generated widespread carbonate sedimentation in Altay-Sayan, Russian Far-East and Mongolia with corresponding dispersion of archaeocyaths (Fig. 5a) & in the mid Atdabanian, archaeocyaths of Siberian affinities reached Western Europe and Morocco where a new center of diversification developed and endemic forms emerged (Fig. 5b). By the late Atdabanian, continuous transgression initiated carbonate accumulation and archaeocyath proliferation on Yantsi and Australia from Siberia and Morocco (Fig. 5c). The peak of the transgression was reached at the beginning of the Botomian, leading to the relative isolation of various regions and to a maximum diversity of mostly endemic faunas. At that time archaeocyaths of Siberian and Australian affinities reached the North America-Koryakia Province (Fig. 5d). In the mid Botomian, Australia became a new center of diversification spreading to Antarctica and possibly South America [START_REF] Gonzalez | Early Cambrian archaeocyathan limestone blocks in low-grade meta-conglomerate from El Jaguelito Formation (Sierra Grande, Rio Negro, Argentina)[END_REF] whereas West America archaeocyaths populated East America (Fig. 5e).The medial Botomian transgressive peak was marked by the extensive accumulation of black shales, thin bedded limestones in low latitudes and an anoxic/dysoxic event. Consequently the archaeocyaths survived only as lowdiversity communities in localized refugia. The late Botomian-early Toyonian regression reduced shallow marine platform areas and thereby removed the archaeocyaths which virtually disappeared at the beginning of middle Cambrian (Fig. 5f). The only archaeocyathan species of post-early Cambrian age are found in Antarctica which was the only one to remain in low latitudes due to the counter-clock rotation of Gondwana.

Archaeocyaths on line

Archaeocyathan database and website

Establishing a database is nowadays is one of the powerful means to manage and analyze complex information. The archaeocyathan database has been produced initially to provide a flexible tool to identify archaeocyath genera (Kerner et al., 2011b). This has been developed with Xper², a free software to store, edit and manage descriptive data and to provide a free access key. (http://www.infosyslab.fr/lis/?q=en/resources/software/xper2 [START_REF] Ung | Xper2: introducing etaxonomy[END_REF]. The database contains 307 valid genera, described with a set of standardized descriptive terms. Interactive identification requires the use of the same set of descriptors for each genus and an adapted terminology: terms are standardized and one term corresponds to a single morphological concept (Kerner et al. 2011a). An archaeocyathan description is composed of 85 morphological and ontogenetic characters. Specialists familiar with traditional terms can find a correspondence table between usual terminology and the computerized one. Each genus is illustrated by the type specimen of the type species; some additional specimens may be present. Each character and its states are associated with a definition and images. Additional data are present: Eight characters refer to stratigraphical and geographical information and 27 refer to traditional classification data. The database spans the entire Cambrian System and has world-wide coverage. As the regional Siberian stages were the traditional reference for archaeocyathan specialists before the establishment of the international stratigraphic chart, these stages are used in the database until correlations are refined.

The entire product is freely accessible on the internet within a website on Archaeocyatha: http://www.infosyslab.fr/archaeocyatha. This site is composed of three parts: 1, Archaeocyatha: introduction, role in Cambrian System, morphology and list of references; 2, knowledge base: introduction and some outputs like list of genera and their detailed sheets, list of characters, list of groups of characters and base properties; 3, interactive key: identification unit, user guide, matching terminologies and glossary.

Different tools available

Xper² offers a large range of tools to explore, analyze and interpreted the knowledge base with several approaches [START_REF] Ung | Xper2: introducing etaxonomy[END_REF]. The identification unit is a free access key. This kind of system is very flexible and is well adapted for palaeontological specimens that are often incomplete. At each step of the identification, the user can choose the adequate descriptor and one or several states. Doubt is possible and several tools are available. As the identification key is built from the database, it is not restricted to morphological information. The presence of stratigraphical and geographical data allows the rapid creation of some lists of genera for palaeobiogeographical or stratigraphical studies. For example, all genera from Canada present during the Botomian stage are found in few seconds contrary to traditional bibliographical studies. In the interactive key, two steps are sufficient: choosing the descriptor stratigraphical extension and the state Botomian and choosing geographical repartition and the state of Canada (Fig. 6a).

To obtain stratigraphical and geographical information for a particular genus it is possible to use another way. Each genus has a complete sheet with its descriptions, pictures and systematics information. Two ways allow access to the sheet. Firstly, the list of sheets is present on the page named "genera: 307" of the website. There, it is enough to click on the name of the genus and go to "other information" part of the sheet. Secondly, in the interactive key, a double-click on the genus name, opens a new window with the same genus sheet as the website one. For example, Archaeocyathus stratigraphical extension is Botomian 2 to Toyonian 3 and possibly during Atdabanian 4; its geographical distribution is Antarctica, Australia, Canada, China (South China), Europe (Iberia, Sardinia), Mexico, Mongolia, Uzbekistan, Russia (Altay Sayan, Far East, Siberian Platform, Transbaikal, Tuva), USA (Fig. 6b).

Conclusion

The aim of this contribution is to present the current status concerning one of the most ancient fossil groups, the Archaeocyatha. Relevant studies have been published in various reviews ( [START_REF] Debrenne | Regular archaeocyaths : morphology, systematic, biostratigraphy, palaeogeography, biological affinities = Archéocyathes réguliers = morphologie, systématique, biostratigraphie, paléogéographie, affinités biologiques[END_REF] [START_REF] Debrenne | Irregular Archaeocyaths : morphology, ontogeny, systematics, biostratigraphy, palaeoecology = Archéocyathes irréguliers : morphologie, ontogénie, systématique, biostratigraphie, paléoécologie[END_REF], Debrenne et al. 2002, in press). A computer-aided identification tool for Archaeocyatha genera is now available online (http://www.infosyslab.fr/archaeocyatha). This database is firstly used as an identification key but can be also a driver for different areas of archaeocyathan research for instance, faunal geographical and stratigraphical distributions. The database will be an important tool for developing new palaeogeographical interpretations for the Cambrian Period.

Figures captions

Table 1. Comparison of trilobites and archaeocyaths biozones (Kerner et al. 2011a, modified after Rozanov & Sokolov 1984[START_REF] Mansy | Calcaires à Archéocyathes du Cambrien inférieur du Nord de la Colombie britannique (Canada). Implications paléogéographiques et précisions sur l'extension du continent Américano-Koryakien[END_REF][START_REF] Webster | Trilobite biostratigraphy and sequence stratigraphy of the upper (traditional Laurentian "lower Cambrian") in the southern Great Basin[END_REF][START_REF] Hollingsworth | Cambrian stratigraphy and paleontology of northern Arizona and southern Nevada[END_REF] P => A: Peachella iddingsi then Bristolia insolens then Bristolia mohaviensis and Arcuolenellus arcuatus. 3: First-phase cluster analysis of binary data matrix of presence/absence of 310 archaeocyathan and radiocyathan genera from 15 regions of the world. [START_REF] Kruse | Palaeobiogeographic affinities of Australian Cambrian faunas in[END_REF].

Figure 4: Second-phase cluster analysis of binary data matrix of presence/absence of 310 archaeocyathan and radiocyathan genera allocated to the five provinces deduced from the firstphase analysis. [START_REF] Kruse | Palaeobiogeographic affinities of Australian Cambrian faunas in[END_REF].

Figure 1 :

 1 Figure 1: Distribution of archaeocyathan bioconstructions on the Gondwana margins (Atdabanian-Botomian stages) (Debrenne and Courjault-Rade, 1994).

Figure 2 :

 2 Figure 2: Global paleaogeographical distribution for the early Cambrian and archaeocyathan distribution (violet). a, Tommotian Stage; b, Atdabanian Stage; c, Botomian Stage[START_REF] Debrenne | Faunal migrations of archaeocyaths and early Cambrian plate dynamics[END_REF].

Figure

  Figure3: First-phase cluster analysis of binary data matrix of presence/absence of 310 archaeocyathan and radiocyathan genera from 15 regions of the world.[START_REF] Kruse | Palaeobiogeographic affinities of Australian Cambrian faunas in[END_REF].

Figure 5 :

 5 Figure 5: Early Cambrian global palaeogeographical reconstruction and archaeocyathan migration paths. a, early Tommotian (ET) to early Atdabanian (EA) map; b, middle Atdabanian (MA) map; c, late Adtabanian (LA) map; d, early Botomian (EB) map; e, middle Botomian (MB) map; f, early Toyonian (EToy) map[START_REF] Kruse | Palaeobiogeographic affinities of Australian Cambrian faunas in[END_REF].

Figure 6

 6 Figure 6: a, list of Canadian genera during the Botoman. B, Archaeocyathus detailed sheet with geographical and stratigraphical ranges.

We wish to thank the CNRS for funding the first author's PhD thesis. We also acknowledge the review of Dr Andrey Zhuravlev and especially for his help with the biostratigraphical table and David Harper for his linguistic correction of the manuscript.


Yacine Djabali

Melisande Blein-Nicolas

Annick Moing

Jorge Duarte

Stéphane Bernillon

Sylvain Prigent

Clément Mabire

Delphine Madur

Llorenç Cabrera-Bosquet

Claude Welcker

Presence/absence variations and SNPs equally contribute to the variations of protein and metabolite abundance

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Djabali Y. 1 , Blein-Nicolas M. 1 , Moing A. 2,3 , Duarte J. 4 , Berton T. 2 , Bernillon S. 2,3 , Prigent S. 2,3 , Fernandez O. 2 , Petriacq P. Presence Absence Variations (PAVs) are structural polymorphisms of the genome that can be induced by insertions or deletions of DNA sequences (InDels). Because InDels can occur in coding sequences or in gene regulatory regions, they are supposed to be involved in adaptation to the environment. However, their contribution to the genetic determinism of traits and genotype by environment (GxE) interactions remains largely unknown. Recently, Mabire et al. [1] published InDel genotyping data for a diversity panel of Dent maize lines. In addition, leaves samples taken on hybrids obtained from this same panel and grown under two watering conditions (well-watered, WW, and water deficit, WD) were analysed by proteomics [2] and metabolomics (unpublished). In this study, we took advantage of these data to evaluate to which extent InDels contribute to the abundance variations of proteins and metabolites as compared to SNPs. To address this issue, we performed genome-wide association studies (GWAS) to detect quantitative trait loci for protein abundance (pQTLs) and metabolite abundance (mQTLs) from both InDels and SNPs. Compared to SNPs, InDels allowed to detect 2061 and 2705 new pQTLs and mQTLs, respectively (Fig. 1). SNPs and InDels allowed to detect equivalent proportions of pQTLs and mQTLs (Fig. 3). In addition, water deficit had no influence on the number of pQTLs and mQTLs detected (Table 1). Table 1 : Number of pQTLs and mQTLs detected by InDels and SNPs according to the watering condition. The QTLs detected by the two types of polymorphism were assigned to the one that was the most strongly associated to the molecular phenotype. The proportion of variance explained by InDels and SNPs is similar in the two watering condition and for both proteins and metabolites (Fig. 2).

SNP and InDels have the same power of QTLsdetection

Fig. 1 :

 1 Fig. 1 : Manhattan plot obtained for (A) the protein "GRMZM2G358902" and (B) the metabolite signature "p3705". QTLs can be determined by InDels only (blue arrows), SNP only (red arrows) or both (pink arrows).

Fig. 3 :

 3 Fig. 3 : Proportion of (A) pQTLs and (B) mQTLs detected by InDels, SNPs or both. Result obtained after re-sampling the SNPs.

Fig. 2 :

 2 Fig. 2 : Proportion of variance explained by (A) pQTLs and (B) mQTLs in the two watering conditions

  2,3 , Zivy M. 1 , Charcosset A.1 , Nicolas S. 1 (1) Université Paris-Saclay, INRAE, CNRS, AgroParisTech, GQE -Le Moulon, 91190, Gif-sur-Yvette, France, (2) INRAE, Univ. Bordeaux, Biologie du Fruit et Pathologie, UMR 1332, Centre INRAE de Nouvelle Aquitaine -Bordeaux, F-33140 Villenave d'Ornon, France, (3) PMB-Metabolome, INRAE, 2018, MetaboHUB, Bordeaux, France, (4) Biogemma, Chappes, France, (5) LEPSE, INRAE, Univ Montpellier, SupAgro, Montpellier, France

This work was funded from the French Government managed by the Research National Agency under the French Stimulus Initiative call 2010 (AMAIZING project, ANR-10-BTBR-01) and from FranceAgrimer