Jean-Paul Caltagirone
email: calta@ipb.fr

Mimetic approach of nuclear spin in discrete mechanics

Keywords: Discrete Mechanics, Nuclear Spin, Mass-Energy Equivalence, Conservation of acceleration, Helmholtz-Hodge Decomposition, Flow control

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Introduction

Atoms are composed of a nucleus of protons and neutrons that are held together by strong interactions and a cloud of electrons that rotate around the nucleus held together by electrical forces. The electrons that also rotate around their own axis produce a magnetic dipole. One of the fundamental properties of the electron is that it has a magnetic dipole moment, that is, it behaves like a small magnet. In magnetic materials, atomic and molecular dipoles have magnetic moments not only because of their orbital angular momentum, but also because of the spin of the elementary particles that compose them, the rotation of the particles on themselves [START_REF] Landau | The Classical Theory of Fields, Third Revised English Edition[END_REF][START_REF]Quantum field theory -Wikipedia, the free encyclopedia[END_REF]. This intrinsic property of quantum mechanics has no equivalent in classical mechanics.

Moreover, the notion of a continuous medium is not adapted to the integration of this physical phenomenon insofar as each physical quantity must be defined at a point, whereas the direction of the axis of rotation is a necessary datum as much for an orbital angular motion as for the spin. Discrete mechanics [START_REF] Caltagirone | On primitive formulation in fluid mechanics and fluid-structure interaction with constant piecewise properties in velocity-potentials of acceleration[END_REF] introduces a notion which cannot be reduced to a point based on a segment of finite length. All the modeling of physical phenomena in mechanics and more precisely in fluid mechanics is based on the projection of translational and rotational actions on this segment. This mechanics opens the way to the integration of fundamental concepts of electromagnetism [START_REF] Maxwell | A dynamical theory of the electromagnetic field[END_REF], of the theory of relativity [START_REF] Caltagirone | Physique discrète et relativité[END_REF] or of quantum mechanics.

The discrete formulation is based on a differential geometry approach where the global inertial frame of reference is replaced by a local frame of reference consisting of two nested primal and dual geometrical structures. The derivation of the equation of motion is based on the conservation of total energy per unit of mass i.e. an acceleration. The intrinsic acceleration of a particle or a material medium is equal to the sum of the accelerations applied to it, the latter representing the effects of compression and rotation-shear. This law is a formal Helmholtz-Hodge decomposition [START_REF] Denaro | On the application of the Helmholtz-Hodge decomposition in projection methods for incompressible flows with general boundary conditions[END_REF][START_REF] Bhatia | The Helmholtz-Hodge Decomposition -A Survey[END_REF][START_REF] Ranocha | Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators[END_REF] where the intrinsic acceleration is the sum of a curl-free term and a divergence-free term.

The notion of nuclear spin is naturally integrated in the discrete formulation by mimicking local magnetic moments around the primal structure defined by a segment. The fluid mechanics equivalent would be a vortex, both irrotational and incompressible, represented by a singularity at the origin. Of course the discrete mechanics approach remains in a macroscopic framework where the motions of the material medium are represented by velocity vectors or by components of this vector.

2 Discrete mechanics framework

Physical principles

The fundamental principles of discrete mechanics have already been described in [START_REF] Caltagirone | Discrete Mechanics, concepts and applications[END_REF][START_REF] Caltagirone | Application of discrete mechanics model to jump conditions in two-phase flows[END_REF]. A reduced description allows however to introduce the new concept of nuclear spin.

One dimensional model

First of all, the notion of continuous medium is abandoned; this historical notion has many advantages, including differential calculation, but also has some limitations due to the necessity to restrict all physical variables to a point. The spatial variations of these variables are then described by differential operators.

Discrete mechanics is associated with a vision coming from differential geometry based on two geometrical architectures, the primal and dual interchangeable structures presented on the figure [START_REF] Landau | The Classical Theory of Fields, Third Revised English Edition[END_REF]. The primal structure is a straight Γ segment of length dh = [a, b] and the dual structure corresponds to a closed Σ contour delimiting a S planar surface. The segment Γ has a unit vector t and the dual surface is oriented along the normal n with the structural condition t • n = 0. The scalars are carried by the a or b vertex of the primal structure while the vectors are associated with the Γ segment itself. In fact, space vectors or tensors of order greater than one are absent from the formulation. A vector defined on Γ is indifferently, a scalar carried by the oriented segment and the projection of the space vector V such that v = (V • t) t; v represents the component on the segment Γ of the space vector, u the component of displacement and γ the component of acceleration. At no time will it be necessary to use the space vectors to derive the discrete equation of motion. All the physics represented is integrally associated with this one-dimensional vision of space.

The notion of discrete time or rather of time interval dt = t -t o allows to introduce a current time t and an initial time t o . It is the second and only fundamental quantity strongly linked to the distance dh = c dt which represents the horizon as it is defined in cosmology: an event which occurs in a cannot be perceived at a distance greater than dh. The celerity c can be c 0 the celerity of light or the celerity of the material medium, for example for acoustic waves.

The global frame of reference of classical mechanics is abandoned in favor of the local frame of reference defined by the Γ segment. Of course the notion of change of reference frame does not exist anymore, the interactions between the local reference frames are realized from cause to effect. Insofar as the velocities are variable in space, the change of reference frame loses its interest in predicting distant phenomena. The Galilean or Lorentz transformations are no longer applicable in the new context.

In discrete mechanics, the analysis is put aside, the derivation at a point, the integration, the passage to the limits, ... are not applicable anymore. Only four discrete operators are constructed to realize the passage between the two geometrical structures. The operator ∇φ is the restriction of the gradient of a scalar on the segment Γ, the gradient vector does not exist in space. The primal rotational, ∇ × v is constructed from the velocity flow on the primal contour defined by the surface S. The divergence of the velocity ∇ • v has the same meaning as in continuum mechanics and represents the flows on the dual surface around a vertex a. Finally, the dual curl, ∇ d × ψ, computed from the circulation on the dual contour Σ with Stokes's theorem projects the result on the oriented segment Γ. Some properties of discrete mechanics mimic those of the continuous medium in particular ∇×(∇φ) = 0 and ∇• ∇ d × ψ = 0. These important identities are verified for all primal and dual structures of polygonal or polyhedral type [START_REF] Shashkov | Conservative Finite-Difference Methods on General Grids[END_REF][START_REF] Hyman | The orthogonal decomposition theorems for mimetic finite difference methods[END_REF].

Extension of physical model to other dimensions

The abandonment of mass leads to a law of dynamics γ = g for gravitational effects; this law is extended as a postulate for any other phenomenon:

γ = h ( 1
)
where h is the sum of the accelerations, those of the effects of compression and shear but also all the other potential source terms: gravitation, capillary acceleration, etc. The physical modeling of the discrete mechanics is realized only on the Γ segment associated to the δ dual contour of the figure [START_REF] Landau | The Classical Theory of Fields, Third Revised English Edition[END_REF]. The direct action defined by -∇φ and the acceleration induced by ∇ d × ψ are both projected onto the segment Γ; the law of dynamics thus becomes:

γ = -∇φ + ∇ d × ψ (2)
where γ c = -∇φ is the longitudinal acceleration and γ r = ∇ d × ψ is the angular acceleration carried by Γ. The equation ( 2) is a formal Helmholtz-Hodge decomposition of the acceleration and φ and ψ are its scalar and vector potential. Possible divergence-free and curl-free terms are filtered by the equation and correspond to the rigid translation and rotation motions. The extension to several dimensions of space is done naturally by associating the local reference frames by the vertices a and b of the figure [START_REF] Landau | The Classical Theory of Fields, Third Revised English Edition[END_REF] to form a closed contour Γ * and the flat surface S of the primal structure. By extension volumes are created to form dual structures in polyhedra with planar surfaces.

From the physical point of view the interactions are realized from cause to effect, from one segment to another. Even if the notion of change of reference frame is lost, the advantage of the formulation is to avoid any superfluous constraint. For example, the limitation of the celerity to a defined value a priori, that of light for light waves. The values of the velocity defined on each vertex thus ensure the representativeness of the change of refractive index between two different media. In fluid mechanics, the subsonic then supersonic flow beyond the throat of a Laval nozzle translates this causal phenomenon where the supersonic state can only be obtained if the flow is sonic at the throat.

Mass -energy equivalence

Consider a mass of fluid m contained within the dual volume Ω centered around a vertex a of the primal structure that we follow in its motion. The volume of Ω remains fixed while a flow of material is injected through its surface Σ. The calculation can be carried out in the same way for the conservation of mass and the conservation of compression energy. The compression energy per unit mass initially contained in Ω is noted φ o and its variation with time is a function of the injected flow rate represented by the divergence of the velocity, ∇ • v, orthogonal to volume limit. This variation is thus equal to dφ for a time lapse of dt. The compression energy is a function of the compressibility of the medium χ S and of the density ρ whose grouping c l = 1/(ρχ S ) is none other than the longitudinal celerity. As the energy of compression per unit of mass is equal to φ = c 2 l the laws of conservation of the mass and that of the energy of compression are given by the laws:

         dm dt = -m ∇ • v dφ dt = -φ ∇ • v (3)
The first law of (3) reflects the conservation of mass in continuum mechanics and of course remains valid in special relativity and discrete mechanics. In the case where c l = c 0 the celerity of light in vacuum we get φ = c 2 0 , that is the energy available if the mass m is completely transformed into energy. We thus find the famous law of special relativity E = m c 2 0 . This conformity of the two laws (3) translates the equivalence between the mass and the energy of the theory of relativity which concerns only the compression energy.

As the divergence of the velocity ∇ • v is constant between the times t o and t corresponding to two mechanical equilibria the evolution of the compression energy can be integrated directly:

φ = φ o e -dt ∇ • v (4)
The dimensionless grouping dt ∇ • v can be equated to the CFL (Courant-Friedrichs-Lewy condition) and compared to unity but it is here a parameter due to physical modeling alone. Moreover, the compression and rotation energies must be advected by the velocity field v. This Lagrangian advection phase can be carried out by various methodologies.

The mass-energy equivalence of special relativity is naturally found in discrete mechanics. The consequences for the recent development of fluid mechanics have not been drawn because, amazingly, the Navier-Stokes equation is always associated with the law of mass conservation. This position leads a priori to a duplication of certain laws of mechanics.

Equivalence moment of inertia -rotation energy

The mass-energy equivalence of the Special Relativity theory concerns only the compressional energy and c 0 is the longitudinal wave celerity c l . The symmetry of the discrete equation of motion [START_REF] Caltagirone | On primitive formulation in fluid mechanics and fluid-structure interaction with constant piecewise properties in velocity-potentials of acceleration[END_REF] shows that each term representing the compression effects is doubled by a term corresponding to the rotation. Similarly, the form of the acceleration [START_REF]Quantum field theory -Wikipedia, the free encyclopedia[END_REF] in two terms at curlfree and divergence-free of a Helmholtz-Hodge decomposition suggests to extend the equivalence between mass and translation energy to rotation.

In classical mechanics the angular acceleration around an axis ∆ is associated to the moment of inertia J ∆ and to the rotational energy. In an analogous way in discrete mechanics, the energy of rotation per unit mass ψ around an axis n orthogonal to each facet of the primal structure is associated to the rate of rotation characterized by the vortex vector ω = ∇ × v; the potential vector ψ and the vortex vector ω are, of course, colinear. Thus, the conservations of angular momentum and energy and rotation are given by the laws:

         dJ ∆ dt = -J ∆ ∇ × V dψ dt = -ψ ∇ × v (5)
where the vectors J ∆ and ∇ × V are collinear just like ψ and ∇ × v, scalars on oriented segments are also vectors. The potential vector ψ of the total acceleration γ is an energy per unit mass and, like the scalar potential, it is equal to the squared celerity ψ = c 2 t defined at the barycenter of the primal surface S associated with the normal n.

This second equivalence (5) between moment of inertia and rotational energy is not present in relativity theory, it is however the counterpart to the translation effects. In classical mechanics the resultant of the forces is associated with another law that is added to Newton's laws, the angular momentum. In discrete mechanics the intrinsic acceleration has both translational and rotational components, or in other words preserves the linear momentum and the angular momentum.

An equation of motion without mass

Let us consider a particle or a material medium accelerating on the segment Γ its total energy variation per unit of mass between a and b is given by the line integral:

Φ = Γ γ • t dl (6)
The total energy is in fact the sum of the compression energy φ and the rotation energy ψ, the two scalar and vector potentials where γ l = -∇φ is the translation acceleration and γ r = ∇ d × ψ rotation acceleration. The acceleration γ is considered in discrete mechanics as an absolute quantity, measurable at any point and at any instant from which are updated the velocity v = v o + γ dt and the displacement u = u o + v dt. At time t o the velocity and the displacement are respectively v o and u o .

The modern formulation of fluid mechanics, the Navier-Stokes equation, establishes the conservation of the momentum p = m V or rather its form per unit volume q = ρ V. In fact, the mass or density is found in many equations of physics, Bernoulli, Lagrange, Euler, Lamé and even in the equations of quantum mechanics like that of Dirac. At the same time, all the quantities of physics which make mass appear do so at order -1 or 1. It is therefore possible to define an equivalent quantity per unit of mass. This possibility also applies to the universal constants of physics.

Amazingly, the theory of special relativity which establishes the equivalence between energy and mass has not encouraged the mechanistic community to choose one or the other of these quantities, why keep both? The relativistic energy is E = γ m 0 c 2 0 and p = γ m 0 V where m 0 is the rest mass and γ = 1/ » 1 -v 2 /c 2 0 the Lorentz factor and c 0 the celerity of light in vacuum. The relativistic invariant is the mass energy of the particle:

E 2 -p 2 c 2 0 = m 2 0 c 4 0 (7)
Unlike the Newtonian case, the relativistic energy of massive particles is generally an irrational function of the particle momentum. For zero mass photons E 2p 2 c 2 0 = m 2 0 c 4 0 = 0; this difficulty is overcome by fixing a value p = E/c 0 of the norm of the momentum. The consequence is the attribution to the photon of a constant celerity always equal to c 0 , the maximum velocity of the special relativity framework. Discrete mechanics proposes to put aside the notion of mass to derive an alternative law based on acceleration and which conserves the total energy on a segment as shown by the form [START_REF] Denaro | On the application of the Helmholtz-Hodge decomposition in projection methods for incompressible flows with general boundary conditions[END_REF]. In the absence of a global reference frame and of a change of reference frame the energy per unit of mass is simply written:

φ = c v ( 8
)
This energy is linked to the segment Γ but it can be defined on the vertices of this one and will take the form φ a = c v a on the vertex a. Like many physical quantities it is a relative quantity which will have to be updated from its value at time t o . The maximum energy that can be extracted from the mass m is always equal to E = m c 2 0 . The expression ( 8) is not only justified for the propagation of light with c 0 but also for other physical phenomena such as the propagation of acoustic waves in fluids.

Discrete equation of motion

The equation of motion is then derived as a conservation of the total Φ on the segment Γ by an integration:

Γ γ • t dl = - Γ ∇φ • t dl + Γ ∇ d × ψ • t dl (9)
where γ is the intrinsic acceleration associated to the oriented segment, φ the scalar potential defined on each vertex and ψ the oriented vector potential along the normal n of the primal surface S. All contributions are thus projected onto the segment Γ. The intermediate variable is the component v of the velocity from which the other quantities are expressed, notably the γ = dv/dt. These are the scalars of the average quantities on the segment oriented by t.

The actual potentials φ and ψ are given in terms of their values at time t o , φ o and ψ o named retarded potentials by analogy to the electromagnetic potentials [START_REF] Liénard | Champ électrique et magnétique produit par une charge électrique concentrée en un point et animée d'un mouvement quelconque[END_REF] from the energy balances where the potentials are expressed by c l and c t , the longitudinal and transverse celerities. The equation of motion then takes the form:

dv dt = -∇ φ o -c 2 l dt ∇ • v + ∇ d × ψ o -c 2 t dt ∇ × v + h s (10)
where h s represents the possible source terms. Solving this equation in terms of velocity v allows to update the potentials and eventually the displacement:

updates            α l φ o -c 2 l dt ∇ • v -→ φ o α t ψ o -c 2 t dt ∇ × v -→ ψ o u -v dt -→ u o (11)
where the symbol -→ represents the update of the retarded potentials. The quantities α l and α t are respectively the attenuation factors of longitudinal and transverse waves. The material derivative includes the time derivative and the two inertia terms as defined in discrete mechanics [START_REF] Caltagirone | On Helmholtz-Hodge decomposition of inertia on a discrete local frame of reference[END_REF]:

γ ≡ dv dt = ∂v ∂t + ∇ Å 1 2 |v| 2 ã -∇ d × Å 1 2 |v| 2 n ã (12)
The intrinsic acceleration contains the two terms of a Helmholtz-Hodge decomposition of the inertial potential φ i = |v| 2 /2. The second term differs significantly from the Lamb vector L = -V ×∇×V of continuum mechanics where the inertia is of the form

V •∇V = ∇(|V| 2 /2)- V × ∇ × V.

Integration of nuclear spin

The equation of motion [START_REF] Caltagirone | Application of discrete mechanics model to jump conditions in two-phase flows[END_REF] associated with updates [START_REF] Shashkov | Conservative Finite-Difference Methods on General Grids[END_REF] and inertia [START_REF] Hyman | The orthogonal decomposition theorems for mimetic finite difference methods[END_REF] represents an alternative physical model to the Navier-Stokes equations. Many differences are to be noted but one of the main ones the necessary addition of the conservation mass to the Navier-Stokes equation; without it the conservation of mass is not intrinsically assured. Given its symmetries the discrete equation ( 10) has a number of mathematical properties stated by Noether's theorems [START_REF] Kosmann-Schwarzbach | Noether Theorems. Invariance and Conservations Laws in the Twentieth Century[END_REF] even if the momentum is replaced by the conservation of acceleration. In spite of the notable formal differences the classical solutions, analytical or numerical of the Navier-Stokes equation, are found exactly [START_REF] Caltagirone | Application of discrete mechanics model to jump conditions in two-phase flows[END_REF]. Important consequences due to the form of the inertia can however appear in particular conditions, especially in turbulence.

The discrete formulation based on a geometrical approach using the two primal and dual structures allows us to foresee a potential harmonization of physical concepts which are not compatible with continuum mechanics. The abandonment of mass is already a first step towards concepts used in electromagnetism.

The transformation of the notion of point in continuum mechanics into a finite dimensional segment dh opens the perspective of an assignment of a new intrinsic property, the nuclear spin. Of course it is not a question here of establishing a strict coherence with this phenomenon of quantum mechanics, the framework of discrete mechanics is part of a classical field theory; the equation of motion itself translates the propagation of longitudinal and transverse waves of the more general framework of the duality hypothesis between waves and particles emitted by Louis de Broglie in 1924. Spin is the only quantum quantity that has no classical equivalent, unlike, for example, the position, the momentum or the energy of a particle. It is not a question here of unifying the physical laws of discrete mechanics and quantum mechanics but rather of mimicking the phenomenon of nuclear spin in the framework defined above.

Let us consider the segment dh and a rotation of this one around its axis (figure 2). Spin is in no way related to a dimension of the primal or dual structures, so its representation in terms of simple rotation has been abandoned. W. Pauli had already shown that, given the estimated dimensions of the electron, a rotation of the electron would require a tangential velocity of rotation at its equator which would be greater than the celerity of light, a velocity which is in principle unachievable according to the theory of special relativity. It is thus necessary to define an intrinsic quantity called spin energy σ representing the energy of rotation per unit of mass expressed in [σ] = m 2 s -2 , the analogue of the vector potential ψ for the rotation of the fluid at large scale. The quantity σ is the equivalent of the magnetic moment in quantum mechanics and the notion defined here two phenomena; indeed, atomic and molecular dipoles have magnetic moments not only because of their quantized orbital angular momentum, but also because of the spin of the elementary particles that compose them.

In discrete mechanics all actions are projected in fine on the Γ segment through differential operators. As the dual curl d × ψ is linked to a spatial dimension of the primal structure it is necessary to define a projector named spin-curl ∇ s × σ which will act as a positive or negative acceleration on the intrinsic acceleration of the material medium γ. From the physical point of view each segment behaves like a magnet by specifying that γ s = ∇ s × σ is only a component of the spin acceleration vector which remains indeterminate like all other vectors in space.

Since the rotation velocity is not defined as a function of the distance r to the axis of rotation, the rotation is modeled from a vortex of velocity v s = C/r and intensity Γ such that:

Γ = 2π 0 v s (r) r dr = 2 π C ( 13
)
where C is a constant and Γ the circulation. This divergence-free and curl-free vortex model has the advantage of eliminating the notion of distance to an axis and of conceiving the spin as an intrinsic quantity associated with the singularity of v s (r) on the axis. The magnetic moment of classical mechanics is equivalent to the spin energy thus equal to σ = Γ/dt. Its spin curl allows express the component on Γ of the spin acceleration ∇ s × σ; this acceleration represents both the orbital and intrinsic spin moments which act on the acceleration of the material medium in an unsteady dynamics. The discrete equation of motion integrating the notion of nuclear spin becomes:

dv dt = -∇ φ o -c 2 l dt ∇ • v + ∇ d × ψ o -c 2 t dt ∇ × v + ∇ s × σ (14)
For a static equilibrium as in the case of a permanent magnet the equation of motion ( 14) reduces to:

-∇φ s + ∇ s × σ = 0 (15)
where φ s is the scalar potential which ensures mechanical equilibrium; in the general case this can only be obtained by the unsteady solution of the equation of motion when t → ∞. The two potentials of the equation ( 15) represent two orthogonal fields.

Examples

The two examples presented are simple illustrations of the flexibility of the discrete equation of motion, they are not complete studies of the simulated phenomena. They allow to promote the concept of nuclear spin, introduces and formalizes the adaptation of discrete mechanics to a notion assigned to the atomic scale.

Magnetic field around a permanent magnet

First of all, it is necessary to note that the discrete equation ( 10) is a law of propagation of acoustic waves but also of electromagnetic waves in general. To show this we must consider the displacement u = v dt and the two equal longitudinal and transverse velocities c l = c t = c. Taking into account ∇ 2 u = ∇(∇ • u) -∇ × ∇ × u it is simple to show that the discrete law becomes:

1 c 2 0 d 2 u dt 2 -∇ 2 u = -∇φ o + ∇ × ψ o (16)
where φ o and ψ o are the retarded potentials assimilated to those of electromagnetism and c 0 the celerity of light.

In a Lagrangian view the first member of this expression is the d'Alembertian u but here this relation integrates the inertia (1) equivalent to the nonlinear Lorentz force. If E and B are electromagnetic fields find the laws E 0 and B = 0. There is thus a strong equivalence between the mechanical quantities and those of electromagnetism (i) the velocity v replaces the current v = j/ρ m where ρ m is current density, (ii) the scalar potential is identified with the electric potential φ = (ρ m /ρ) e, (iii) the vector potential is replaced by the magnetic induction ρ m B/(ρ µ m σ m ) where µ m the magnetic permeability and σ the electrical conductivity. The solution of the equation [START_REF] Caltagirone | Application of discrete mechanics model to jump conditions in two-phase flows[END_REF] allows to find the solution of the electromagnetic problem using these equivalences.

The first example of magnetostatics is to make a permanent magnet and calculate the residual magnetic field. The equation ( 10) is a continuous memory model that includes the possibility of representing the permanent magnetization of a ferromagnetic metal. The first phase consists in creating a permanent magnetization from an electromagnet of magnetic field B on a piece of ferromagnetic material shown on the figure [START_REF] Caltagirone | On primitive formulation in fluid mechanics and fluid-structure interaction with constant piecewise properties in velocity-potentials of acceleration[END_REF]. The system of equations [START_REF] Caltagirone | Application of discrete mechanics model to jump conditions in two-phase flows[END_REF] is solved by maintaining a constant field B and by ensuring the upgrade of the equilibrium vector potential ψ o with the transverse attenuation factor α t = 1. At the end of this phase the magnet is permanent and ψ o maintains a constant field inside the magnet.

The second phase of the experiment consists in looking for the induced magnetic field B outside the magnet in the vacuum or in the air, always with [START_REF] Caltagirone | Application of discrete mechanics model to jump conditions in two-phase flows[END_REF], by maintaining a spin field σ = y e z generating an acceleration equal to γ s = ∇ s × σ = 1 e x inside the solid domain. The scalar potential φ s updated over time by the divergence of the velocity is not zero but it does not influence the magnetic field; this velocity field becomes divergence-free (Coulomb gauge) inside but also outside the magnet.

Figure (3) shows the magnetic field generated by a permanent magnet in a vacuum. This magnetic spin field σ is indeed with zero divergence and corresponds to that which gave birth to it if the material is perfectly persistent; the steady state agrees with equation [START_REF] Kosmann-Schwarzbach | Noether Theorems. Invariance and Conservations Laws in the Twentieth Century[END_REF]. We can see that the current lines of the magnetic flux σ and the equipotential lines of φ s are indeed orthogonal. Once the stationary regime is reached, no more energy is exchanged and the translation acceleration defined by ∇φ s is equal to the rotation acceleration ∇ s × σ.

Convection in a ferromagnetic liquid

Other potential uses be found in the of fluid mechanics or magnetohydrodynamics without having to resort to equation coupling. For viscous fluids the attenuation factor α t ≈ 0 which is justified by the very small time constant for which transverse waves are attenuated, τ ≈ 10 -11 s. The grouping dtc 2 t must then be replaced by ν the kinematic viscosity of the fluid and the instantaneous vector potential by ψ = ν ∇ × v. The equation of motion of a viscous fluid in the presence of a nuclear spin σ reads:

dv dt = -∇ φ o -c 2 l dt ∇ • v -∇ d × (ν ∇ × v) + ∇ s × σ (17)
Obviously, the viscous dissipation transforms part of the mechanical energy into heat in an irreversible way and a flow cannot remain for long if external energy is not supplied to it. The problem posed here is therefore very different from the example of the magnet where the storage of magnetic energy is almost permanent. The mechanisms are more complex: it is necessary that an external magnetic field B is maintained to orient the micro-magnets simulating the nuclear spin whose orientation is modified by the fluid flow. In the case of a non-magnetic medium, the B field would have no influence.

The equation (17) associated with the nonlinear terms [START_REF] Caltagirone | On Helmholtz-Hodge decomposition of inertia on a discrete local frame of reference[END_REF] without the spin source term has already shown its ability to find the theoretical or numerical solutions on many examples in fluid mechanics [START_REF] Caltagirone | Application of discrete mechanics model to jump conditions in two-phase flows[END_REF]. The ∇ s × σ term has the same properties as the dual curl ∇ d × ψ o , in particular it is divergence-free whatever the type of polyhedral geometric structure chosen. As the considered state is stationary, the acceleration is null and the law of motion becomes -∇φ o + ∇ d × ψ o = 0 where ψ o = ν∇ × v is the energy dissipated by viscosity of fluid. Thus, the energy supplied from outside by the field B and then assigned to the fluid through the spin is transformed into heat by maintaining the acceleration ∇ d × ψ o = ∇ s × σ. In the case where the velocities are important the inertia terms ∇φ i -∇ d × ψ i are no longer negligible and the equality given above becomes -∇φ o B + ∇ d × ψ o B = 0 which make the Bernoulli potentials appear. It is indeed possible to introduce the inertia term within the scalar potential φ o B = φ o + |v| 2 /2. The scalar φ o and vector ψ o potentials of the intrinsic acceleration of the fluid are shown in figure [START_REF] Maxwell | A dynamical theory of the electromagnetic field[END_REF]. The flow calculated from equation (17) tends to a stationary convective state in the form of a vortex.

The introduction of the notion of spin in the framework of discrete mechanics is consistent with the objective of remaining in the macroscopic context of classical field theory. The result is the definition of a term compatible with those already existing in the equation of motion in the form of a Helmholtz-Hodge decomposition.

Conclusions

The discrete formulation where the notion of continuous medium has been abandoned has allowed the introduction of a concept not naturally present in classical mechanics, that of nuclear spin by mimicking the phenomena at the atomic scale. The spin has been associated to the rotation of the Γ segment of the primal formulation as an intrinsic property while remaining in a continuous approach. This extension contributes to the progressive integration of contemporary physics within the older laws of mechanics. The inescapable results of the theory of relativity, of electromagnetism, of quantum mechanics must necessarily be integrated into an appropriate formalism, here that of discrete mechanics. It is quite possible that limitations will be detected in the future, only the experiment will allow to judge the validity of this approach for nonconventional applications.

The main interest of the entanglement of different physical phenomena represented by terms within the same equation of motion is to ensure their temporal and spatial consistency. The formulation in a Helmholtz-Hodge decomposition of all phenomena, inertia, compression, rotation and even the source terms which are also written in the same way, consolidates the cohesion of the equation of motion.

Figure 1 .

 1 Figure 1. Entangled primal (blue) and dual (red) geometric structures; the Γ segments of unit vectors t of the primal structure form the Γ * collection bounding the primal S planar surface. The segments δ oriented along n form the dual contour whose face ∆ is also flat. The two vectors t and n are orthogonal by construction. The scalar potential φ is defined on the vertices of the primal contour and the vector potential ψ is associated to the barycenter of the surface S.

Figure 2 .

 2 Figure 2. Mimetic simulation of the nuclear spin by the rotation of the dual vector σ around segment Γ; the circulation of this vector around the oriented axis t induces an acceleration directly on Γ

Figure 3 .

 3 Figure 3. A magnet made of a ferromagnetic material represented by the rectangle produces a magnetic field outside. The residual field in the magnet is equal to σ = y e z . The magnetic field is represented by a gradient of colors and the scalar potential by lines.

Figure 4 .

 4 Figure 4. Convection in a square cavity induced by the application of a magnetic field B on a viscous ferromagnetic fluid; scalar potential φ o and vector potential ψ o


Adrien Pupier

Maximin Coavoux

Benjamin Lecouteux

Jérôme Goulian

End-to-End Dependency Parsing of Spoken French

Keywords: speech recognition, dependency parsing, end-to-end, raw signal, spontaneous speech

Research efforts in syntactic parsing have focused on written texts. As a result, speech parsing is usually performed on transcriptions, either in unrealistic settings (gold transcriptions) or on predicted transcriptions. Parsing speech from transcriptions, though straightforward to implement using out-of-the-box tools for Automatic Speech Recognition (ASR) and dependency parsing has two important limitations. First, relying on transcriptions will lead to error propagation due to recognition mistakes. Secondly, many acoustic cues that are important for parsing (prosody, pauses, . . . ) are no longer available in transcriptions.

To address these limitations, we introduce wav2tree, an end-to-end dependency parsing model whose only input is the raw signal. Our model builds on a pretrained wav2vec2 encoder with a CTC loss to perform ASR. We extract token segmentation from the CTC layer to construct vector representations for each predicted token. Then, we use these token representations as input to a generic parsing algorithm. The whole model is trained end-to-end with a multitask objective (ASR, parsing) to reduce error propagation. Our experiments on the Orféo treebank of spoken French show that direct parsing from speech is feasible: wav2tree outperforms a pipeline approach based on wav2vec (for ASR) and FlauBERT (for parsing).

Introduction

Progress on syntactic parsing of written text has been remarkable in the last decade, thanks to the use of bi-LSTM [START_REF] Kiperwasser | Simple and accurate dependency parsing using bidirectional LSTM feature representations[END_REF] and pretrained language models such as BERT [START_REF] Devlin | BERT: Pretraining of deep bidirectional transformers for language understanding[END_REF]. Most of the work on syntactic parsing of speech has been done on transcribed speech, either from a human annotator (gold) leading to unrealistic parsing settings or from automatic speech recognition tools [START_REF] Béchet | Robust dependency parsing for spoken language understanding of spontaneous speech[END_REF][START_REF] Béchet | Adapting dependency parsing to spontaneous speech for open domain spoken language understand-ing[END_REF]. Spoken language, as opposed to written language, exhibits specific phenomena, such as disfluencies, hesitations, false starts, that arguably make syntactic parsing harder.

However, parsing transcriptions is unsatisfactory since listeners use prosodic cues to help solve ambiguity [START_REF] Price | The use of prosody in syntactic disambiguation[END_REF] and transcriptions no longer provide access to prosodic information. Indeed, using acoustic cues has proven to be a viable way to improve parsing on transcribed speech [START_REF] Pate | Unsupervised dependency parsing with acoustic cues[END_REF]. Thus, previous work focused mainly on integrating acoustic cues from the signal and using a multimodal model with both text and features extracted from the audio such as word duration and fundamental frequency [START_REF] Tran | Parsing speech: a neural approach to integrating lexical and acoustic-prosodic information[END_REF]. Another line of work has focused on automatically detecting disfluencies and ASR mistakes while parsing transcriptions [START_REF] Charniak | Edit detection and parsing for transcribed speech[END_REF][START_REF] Jørgensen | The effects of disfluency detection in parsing spoken language[END_REF][START_REF] Honnibal | Joint incremental disfluency detection and dependency parsing[END_REF][START_REF] Yoshikawa | Joint transition-based dependency parsing and disfluency detection for automatic speech recognition texts[END_REF][START_REF] Lou | Neural constituency parsing of speech transcripts[END_REF]. The disfluencies detection mitigates the error propagation due to the ASR model. Moreover, it allows transition-based parsers to prune disfluency nodes with specific repair actions [START_REF] Honnibal | Joint incremental disfluency detection and dependency parsing[END_REF].

We focus on dependency parsing, where each word in a sentence is attached to its head word with a syntactic function (subject, object, modifier,. . . ). In this paper, we introduce wav2tree, an end-to-end neural model for speech parsing. Wav2tree jointly predicts a transcription and its syntactic analysis (a labeled dependency tree and a sequence of part-of-speech tags), using only the raw signal as input. The motivations for designing an end-to-end model for this task are manifold. End-to-end models are known to reduce error propagation thanks to shared error optimization, unlike pipeline approaches, where an error in a previous component will trigger errors in downstream tasks. We compare wav2tree to several pipeline baselines that use pretrained models for both the speech and text modalities. To the best of our knowledge, this is the first attempt to only use signal representations for syntactic parsing. In summary, our contributions are threefold:

• We introduce a new architecture to parse speech directly from the signal; • We provide an evaluation on our model on a spoken treebank for French; • We publicly release the code of this architecture online along with pretrained models.1 2. Wav2tree: an end-to-end architecture for parsing speech

The objective of our model is to take the raw signal corresponding to a sentence2 as input and compute the sequence of words and the dependency tree contained within this signal. The architecture of our model is divided into three parts: (i) the representation extractor module, (ii) the ASR module, and (iii) the parsing module, as illustrated in Figures 1 and2.

The first module of our architecture is the representation extractor. From a signal S, we compute a feature matrix X as follows:

X =FNNenc(f(S)), (1)
where f is any feature extraction method (MFCC, wav2vec, . . . ) and FNNenc is a feedforward neural network. Another possible instantiation for FNNenc is a convolutional recurrent (CRNN) encoder [START_REF] Shi | An end-to-end trainable neural network for image-based sequence recognition and its application to scene text recognition[END_REF], but any neural architecture can be used.

ASR module

The automatic speech recognition module is composed of a fully connected layer with a softmax activation. The model predicts a character for each audio feature, the sequence of characters is then collapsed through Connectionist Temporal Classification [START_REF] Graves | Connectionist temporal classification: Labelling unsegmented sequence data with recurrent neural networks[END_REF][START_REF] Graves | Towards end-to-end speech recognition with recurrent neural networks[END_REF]CTC]. We do not use any language model or beam search, despite what is usually done in ASR. Indeed, we are not aware of any pre-trained language model for transcriptions of spontaneous French.

Using a language model trained on French books or data crawled from the web would introduce bias, due to linguistic differences between spoken and written French.

Dependency parsing module

The dependency parsing module is composed of two parts: (i) the audio word embedding extraction and (ii) the parsing model. For the first one, we use the CTC segmentation of the audio features, through the prediction of the whitespace characters, to compute an embedding for each predicted token. Each audio feature, except for the ones corresponding to whitespaces, from the encoder module is part of one word, and every feature between two whitespace characters belongs to the same word. Thanks to this property, we map each feature to a word, and we use this mapping to compute our audio word embedding through an LSTM. This process is described in Figure 2 and is formally described in eqs. ( 2) and (3):

Xw i =X[xj,...,x k ], (2)
ew i =LSTM hidden (Xw i ), (3)
where X is the representation matrix from the encoder defined in eq. ( 1), j and k are the boundary of the word i, and ew i is the word embedding corresponding to token i.

Once we have computed an "audio word embedding" for each token in a sentence, we can use any generic dependency parsing method. For the sake of simplicity and speed, we used the dep2Label method [START_REF] Strzyz | Viable dependency parsing as sequence labeling[END_REF], that reduces the parsing problem into a sequence tagging problem. Each token is assigned three labels for (i) its part of speech tag, (ii) the relative position of its head, and (iii) its syntactic function. For example, in Figure 3a, the gold labels for token j' (I) are: CLS, 2VPP, subj, meaning j' is the subject of the second past participle verb (VPP) on its right. Our model computes scores for each token and each subtask as follows:

Z =biLSTMoutput(E), (4) Ppos =Softmax(FNNpos(Z)), (5) Phead =Softmax(FNN head (Z)), (6) Pdep =Softmax(FNN dep (Z)), (7)
where E is the matrix containing the sequence of audio word embeddings and FFNs are feedforward networks.

Training wav2tree

Training objective

To train our model, we minimize the sum of the negative log-likelihood on the three classification tasks, and we add the CTC loss for the automatic speech recognition task:

L(θ,S,c,p,h,d)=-α•logP (p|S;θ)-β•logP (h|S;θ) -γ•logP (d|S;θ)-δ•logP (c|S;θ) (8)
where θ are the model parameters, S is the signal, p is the target POS sequence, h is the target sequence of relative positions encoding the dependency tree, d is the target sequence of syntactic functions, c is the sequence of gold tokens, and (α,β,γ,δ) are hyperparameters.

Oracle

The token segmentation depends on the ASR frame-level predictions. An error in the segmentation makes the gold tree no longer attainable. As a result, the parsing models cannot always output the correct tree. The parsing model loss should not increase in these cases since the mistakes come from the ASR module. The most common problem is a mismatch in the number of tokens between the supervision and the prediction. In this case, giving the parsing model good supervision is crucial. Using the original supervision to the model would introduce noise when the segmentation is not correct as illustrated in Figure 3c. Thus, we need to compute the best possible dependency tree given (i) the gold tree and (ii) a noisy ASR segmentation. This process is similar to the method used in [START_REF] Yoshikawa | Joint transition-based dependency parsing and disfluency detection for automatic speech recognition texts[END_REF] to transfer the gold annotation to the noisy ASR output. Such a computation requires knowing which tokens are added or missing. To do so, we use an alignment tool: Sclite. 3 Sclite uses the transcriptions and the gold text to detect the added or missing tokens. We use the following heuristics to compute the best possible tree:

• Insertions are attached to the root (single root constraint).

• When a token is deleted, all the orphaned tokens are attached to the closest parent of the deleted tokens. Either the parent of the deleted token, or recursively until the root. • In the case of a missing root, the closest element to the root becomes the new root. If there are several possibilities, the leftmost candidate is chosen.

Thus, when computing the loss in eq. ( 8), we substitute p, d and h by the sequences computed by the oracle. The oracle is illustrated in Figure 3b built from the original tree in Figure 3a.

Experiments and result

Questions Our main objective is to assess whether using only the raw signal as input for the syntactic analysis task is feasible in a multitask setting where we simultaneously predict the transcription (ASR), POS tags, and the dependency tree.

Experimental settings We used a pretrained wav2vec2 [START_REF] Baevski | wav2vec 2.0: A framework for self-supervised learning of speech representations[END_REF] architecture as our feature extractor f for the raw signal, namely LeBenchmark/wav2vec2-FR-7K-large [START_REF] Evain | Lebenchmark: A reproducible framework for assessing self-supervised representation learning from speech[END_REF] available online. 4 Before we train the syntax model, we wait for the ASR module to reach a WER lower than 50 on the dev set (usually, one epoch is enough). Since the syntax module depends on the segmentation of the ASR, training it from the very beginning (with low-quality segmentations) could confuse or slow down learning. The FNNenc function has 3 fully connected layers with 1024 units and uses Leaky ReLu activation function. The audio frame LSTMhidden that computes audio word embeddings has 500 hidden units and 2 layers. The sentence level bi-LSTMoutput has 800 hidden units and 2 layers.

The implementation of our model uses the Speechbrain framework [START_REF] Ravanelli | SpeechBrain: A general-purpose speech toolkit[END_REF]. The loss function weight for each of the tasks is set to δ =0.4 for ASR (CTC) and α = β = γ = 0.2 for all the sequence labeling tasks (dependency parsing task) based on preliminary experiments. Due to limited resources, we could not tune the value of the hyperparameters, except for the number of epochs (to minimize the validation set WER). Thus, our result might improve with a hyperparameter search. The end-to-end model was trained for approximately 135 hours on a Quadro RTX 8000 GPU. Training the pipeline basline was quicker, with around 100 hours on the same GPU. However, the end-to-end model has several unoptimized computations, such as the call to Sclite during the oracle computation.

Dataset We evaluate our model on the Orféo treebank [START_REF] Benzitoun | Le projet orféo: un corpus d'étude pour le franc ¸ais contemporain[END_REF], composed of multiple corpora of spoken French with 5 dependency tree annotations, aligned with audio files at the sentence level. For this paper, we did not consider any sentence containing a multi-word expression, to ensure that the segmentations are comparable for ASR and dependency parsing. The multiple corpora include many domains, most of which feature spontaneous conversations. Domains range from prepared French fairy tales narrated by storytellers (Oral-Narrative) to recordings at checkouts of cheese shops (Clapi) or interviews with Parisian people about their life in the city (Cfpp). The diversity of speaking situations, along with the lower quality of some recordings, makes Orféo harder than standard 4 https://huggingface.co/LeBenchmark/wav2vec2-FR-7K-large 5 Around 5% of the Orféo treebank has been manually annotated (gold), the rest of the corpus is automatically annotated (good quality silver data). See the gitlab code release for more information on gold/silver breakdown statistics for each sub-corpus. We compute all evaluation metrics with a customized version of the 2018 shared task evaluation script 6 based on the token order and not on the form of each word, to account for minor errors (e.g. spelling) in the ASR. The script is also based on Sclite alignment, and allows us to evaluate our results even when the word forms do not match between the gold and predicted trees. We use word error rate (WER) and Character error rate (CER) for ASR, and the traditional metrics for dependency parsing: UPOS (Universal Part of Speech) accuracy, UAS (unlabeled attachment score), and LAS (labeled attachment score).

Results

For the first experiment, we train on all the subcorpora in Orféo and report the results in Table 2 broken down by subcorpus. The high WER on Orféo might be due to the type of speech; spontaneous speech is harder to recognize for ASR systems. Moreover, the recording conditions are of low quality for some corpora leading to very high WER in some cases (e.g. clapi). The baseline without the oracle has the worst results, showing that training on ASR output is important for the model and better matches the test settings. Wav2tree outperforms both baselines, showing that using only the signal as input for syntactic parsing is feasible. A notable fact is that this method naturally avoids the out-of-vocabulary problem since our model work on the character level for the ASR module, and our parser module only uses learnt representations extracted from a continuous span of audio representations. The only word-level information used is the boundary of the word. Thus, the only vocabulary needed for our model is the one used by the ASR. In this experiment, it is composed of the Latin alphabet and some diacritics. Moreover, the end-to-end model has around 350M parameters, whereas the pipelines use the same model for the ASR task and then use a pre-trained language model with 373M parameters (flaubert-large). The end-to-end model outperforms the pipeline while using less than half the parameters. The WER and the parsing results (UAS) are correlated (Pearson coefficient: -0.9648), meaning that noisy or hard-to-understand recordings are harder to parse. Thus, any improvement in the speech recognition process should have a positive effect on dependency parsing. For the second experiment, we focus on parsing only the corpus with the best WER of Orféo, namely French-oral-narrative. We train our model for 50 epochs on this corpus and the pipeline for 100 epochs. This corpus features prepared speech (readings) unlike other Orféo subcorpora; in this case, the corpus is composed of storytellers narrating fairy tales. The objective of this experiment is to see if our model still outperforms the pipeline approach on less noisy corpora. The results, as seen in Table 3, show that wav2tree still outperforms the baseline, even on the corpus favorable to the pipeline approach where the risk of error propagation is lower. Another notable result is that the model only trained on one corpus had worse results than the one trained on all of Orféo for 30 epochs. This is probably due to the sheer size difference between the two train sets. We conclude that the model can leverage information from other corpora.

Conclusion

We introduced a new architecture able to parse speech directly from the raw audio signal. We have shown that it outperforms a pipeline approach working on transcribed speech with twice the number of parameters. This model can use any generic parsing algorithm on top of the audio word embedding extractor, allowing for high modularity. We have shown that using only the raw signal for the dependency parsing task is feasible, meaning that the raw signal contains syntactic information. For this purpose, we designed a method to create audio word embedding from the CTC whitespace prediction usable in any task requiring word-level segmentation such as named entity recognition (NER), or word sense disambiguation (WSD). However, further research on the performance of these audio word embeddings is required, for instance comparing them to a baseline using both word embedding and prosody-acoustics features.

Extending this method to use multimodality is a promising idea to improve performance. We could use the predicted text as input for any Bert model and combine the word embeddings from Bert and the audio word embeddings. We also plan to use a language model on top of the CTC loss and experiment with other ASR methods.

Figure 1 :

 1 Figure 1: Wav2tree end-to-end architecture.

Figure 2 :

 2 Figure 2: Feature grouping using CTC segmentation.

( a )

 a Gold sentence dependency tree (b) New best tree from ASR output. (c) ASR tree without Oracle

Figure 3 :

 3 Figure 3: An example of the tree produced by the oracle with the relative encoding labeling.

Table 1 :

 1 Subcorpora in the Orféo treebank along with their types of speech and sizes in number of sentences and duration in hours.

	Corpus	Type	Train size	Dev size	Test size
	Cfpb	spontaneous	3030 (2.2h)	389 (0.3h)	362 (0.3h)
	Cfpp [21]	spontaneous	25500 (19.1h)	3175 (2.3h)	3232 (2.4h)
	Clapi [22]	spontaneous	7682 (5.3h)	1001 (0.7h)	967 (0.7h)
	Coralrom [23]	spontaneous	10889 (9.6h)	1342 (1.2h)	1376 (1.2h)
	Crfp [24]	spontaneous	17357 (15.3h)	2198 (2.0h)	2259 (2.0h)
	Fleuron [25]	spontaneous	1779 (1.4h)	207 (0.1h)	217 (0.2h)
	Oral-Narrative [26] prepared/read	8388 (7.3h)	1074(1.0h)	1050 (1h)
	Ofrom [27]	spontaneous	11665 (9.3h)	1461 (1.2h)	1476 (1.2h)
	Reunions	spontaneous	10067 (8.0h)	1283 (1h)	1245 (1h)
	Tcof [28]	spontaneous	16063 (11.6h)	1971 (1.5h)	1997 (1.5h)
	Tufs	spontaneous	35990 (24.3h)	4431 (3.0h)	4525 (3.0h)
	Valibel	mixed	21095 (17.5h)	2769 (2.3h)	2753 (2.3h)
	Total	mixed	169505 (130.9h) 21301 (16.6h) 21459 (16.8h)
	benchmarks for French ASR. The composition of the train, dev and
	test set is presented in

Table 1 .

 1 Evaluation We compare wav2tree to two pipeline baselines that first perform ASR, and then parse the transcriptions. We first train an ASR model on Orféo (train section) and predict transcriptions for the test section. The pipelines use the same parsing algorithm as wav2tree (see Section 2.2), but the inputs to the parser are embeddings from a pretrained language model representing the predicted tokens. Both pipelines use flaubert-large [29] to compute the embedding representation of each word (last embedding of each word in the last layer). We avoid overfitting by saving the model with the best UAS score on the validation set. The first model is trained on gold transcriptions from the training set and tested on the noisy ASR test file. The second baseline, pipeline+oracle, is trained on ASR transcriptions of the train set, whose trees have been processed by the oracle defined in Section 3.2.

Table 2 :

 2 Results by corpus with our model and baseline with and without the oracle. The model is trained on all the corpora and then tested on each of them. The model in the pipeline is dep2Label trained for 100 epochs.

			Wav2tree (end2end)				Pipeline				Pipeline + oracle
	Parser input			Signal				Train: Gold -Test: ASR			Train: ASR -Test: ASR
	Pre-trained			Wav2vec2				Wav2vec2 + FlauBERT			Wav2vec2 + FlauBERT
	Parameters			350M+33M				350M+373M+32M			350M+373M+32M
	Corpus	WER CER UPOS UAS LAS	WER CER UPOS UAS LAS	WER CER UPOS UAS LAS
	Cfpb	29.2	18.0	77.6	73.4	68.8	28.2	17.0	76.6	71.8	67.5	28.2	17.0	78.0	73.0	68.5
	Cfpp	35.7	24.2	71.8	66.6	61.8	35.5	24.2	69.9	64.8	60.2	35.5	24.2	70.7	65.0	60.4
	Clapi	53.6	35.4	58.8	54.4	47.6	53.2	35.0	56.4	53.5	46.7	53.2	35.0	57.4	53.75 47.0
	Coralrom	22.5	11.9	83.9	77.8	74.5	22.0	11.8	82.0	75.3	71.8	22.0	11.8	83.0	75.9	72.4
	Crfp	24.3	14.5	81.7	75.9	72.2	23.5	14.3	80.3	74.0	70.4	23.5	14.3	81.3	74.8	71.1
	Fleuron	36.1	23.1	71.9	65.3	60.5	35.5	21.6	71.0	65.1	61.3	35.5	21.6	72.1	66.0	61.1
	Oral-Narrative	11.1	4.7	93.2	87.9	85.7	10.2	4.3	92.1	86.1	83.4	10.2	4.3	93.0	86.6	84.2
	Ofrom	20.0	11.6	85.2	79.3	75.9	19.1	11.2	84.2	77.9	74.7	19.1	11.2	85.1	78.4	75.1
	Reunions	40.9	26.1	67.7	61.8	56.3	41.3	26.0	65.5	60.3	55.1	41.3	26.0	66.7	60.8	55.6
	Tcof	34.1	20.8	74.3	67.4	62.7	33.6	20.4	72.3	65.4	60.8	33.6	20.4	73.2	65.6	61.0
	Tufs	33.1	20.8	75.2	69.6	65.1	32.5	20.3	73.5	67.8	63.5	32.5	20.3	74.6	68.7	64.1
	Valibel	23.0	12.8	82.8	76.9	73.2	22.3	12.5	81.3	75.4	71.7	22.3	12.5	82.2	75.7	71.8
	Orféo full	31.0	19.4	77.4	71.7	67.5	29.1	17.9	75.8	70.0	65.8	29.1	17.9	76.7	70.5	66.2

Table 3 :

 3 Results with models trained only on the corpus with the best WER score: Oral-Narrative. The Pipeline model uses dep2Label and FlauBERT language model, and is trained for 100 epochs.

	Models	WER UPOS UAS	LAS
	Wav2tree (end2end)	10.9	93.18	86.60 84.31
	Pipeline	10.9	91.67	82.94 79.89
	Pipeline + Oracle	10.9	91.77	83.46 80.66
	Pipeline gold test	0	97.21	88.80 86.48

https://gricad-gitlab.univ-grenoble-alpes.fr/ pupiera/wav2tree_release

The term 'sentence' is questionable when referring to spontaneous speech. We keep it here by analogy to written text parsing.

https://github.com/usnistgov/SCTK

https://universaldependencies.org/conll18/ evaluation.html

Acknowledgements

This project was funded by the Agence National de la Recherche (ANR) through the project PROPICTO (ANR-20-CE93-0005). We thank Marco Dinarelli for feedback on an earlier version of this article.