3.1.1 Optimizing parts of numerical solvers

High fidelity simulation of compressible fluids still remains an important challenge, and we are convinced that solving it would benefit from recent deep learning tools to optimize space and time discretizations, among which the Finite Volume, Discontinuous Galerkin or Lattice Boltzmann methods. For nonlinear problems such as the Euler system or the shallow water equations, theory has been developed to ensure that the numerical schemes capture the right solution. However, this theoretical framework does not always indicate how to adapt the schemes, especially high-order schemes that admit numerical instability problems, to increase accuracy or preserve certain physical properties. Choices are then sometimes made in a heuristic way by tuning some parameters in the schemes:

• numerical Finite Volumes flux corrections and slope limiters 77,
• artificial viscosity for Discontinuous Galerkin schemes 46,
• relaxation matrix/equilibrium function for multi-scale relaxation schemes and the Lattice Boltzman Method 71,
• well-balanced corrections 73, 50.

These choices could be greatly improved using machine learning techniques. Our aim is to design these key parameters automatically. This will simplify the use of these schemes (because the parameters will no longer have to be chosen by the user) and increase the accuracy of the methods (because the parameters will be chosen in a more optimal way). These strategies are particularly well-suited to high-order schemes, where the extra precision is offset by lower stability, which is often subject to correction. For that, we will consider numerical neural networks but also symbolic regression methods which make it possible to obtain analytical formulas.

The team has specialised in schemes based on relaxation approaches before avoiding or limiting the restriction of the CFL condition. These include the kinetic relaxation approach 11, 43, 47 and the Xin-Jin/Suliciu relaxation approaches for multiscale problems 91, 92. We will continue developing these approaches, in particular by extending them to more difficult problems and increasing the order of accuracy. The numerical methods thus obtained are then to be coupled with machine learning to make them more effective.

One of the main challenges is to incorporate these data-driven functions into the schemes, while still preserving the classical properties of the schemes: convergence, stability, entropy dissipation, positivity of the solutions, etc. To achieve this, the exact data-driven function, as well as the training procedure, must be chosen carefully. More precisely, the choice of the so-called loss function, which is optimized in the learning steps, is crucial. We will explore supervised learning where loss functions compare the learned function to reference ones, as well as unsupervised learning, where the loss functions do not refer to any reference functions, e.g., residual loss functions in PINNs (Physics-Informed Neural Networks 80) or discriminative loss functions that can detect defects in classical schemes (Generative Adversarial Networks 53).

3.1.2 Meshless and neural approaches

3.1.3 Optimizing the representation of the approximate solutions

3.1.4 Including data-driven predictions into numerical methods

3.2.1 Continual learning, sampling and likelihood

To obtain robust code that learns from the data produced by the code, we need to use continuous learning methods (data that arrives regularly) and to detect if the code is going to be used in a configuration where the neural network is not going to work in order to de-activate the network, for example. The second problem in learning is the detection of out-of-distribution examples (OOD) 95. We propose to study different methods for that and apply them to our hybrid simulation codes. We could use generative models (Variational auto-encoders 61, normalizing flows 62, Denoising Diffusion Probabilistic Model 58) in order to capture the probabilistic distribution of the inputs data of the code. This will allow us to test whether a new input has been produced by the distribution. If this is not the case, the network is likely to fail. Another important problem is to make the training continuous while a given simulation code is used, in order to become specialized to the data given by the user. We can model the parameters and initial state space $S_0$ by a time-dependent probability distribution where the evolution is assumed to be smooth over time. The aim will be to develop learning procedures that can capture $p_t$ using a large training set at $t=0$ and smaller data sets at later times $(t_1,...,t_n)$, in order to limit data storage requirements. These small sets of continuously arriving data correspond to the data produced each time the code is used to produce a simulation. This type of approaches are referred to as continual learning or life-long learning in the literature 59. Their use in the context of PDEs has not been explored, to the best of our knowledge. Continual learning could be used on all the supervised models developed in our project team, or specifically on generative models that could in turn be used to train other neural networks by means of generating examples following the relevant distribution.

3.2.2 Self-specialized numerical methods

Combining the data-driven solvers from the first axis with continual learning and Out-Of-Distribution Detection approaches, we wish to design complete prototypes of self-specialized codes. In practice, we would like to validate this approach for two particular numerical methods in simple configurations.

First use case: we construct a 2D or 3D Lattice Boltzmann scheme code for Euler, MHD or SHTC equations with some guarantees on the stability of all regimes. We will use a general relaxation matrix (multiple relaxation time method). Using a reinforcement method or the differentiable physical approach, we will learn the relaxation matrix such that the discrete residue or the error compared to a fine solution computed by the user will be minimal. To obtain a full prototype we must add a mechanism for continual learning and a mechanism of OOD detection. We must also include a safety mode of the method in this case.

Second use case: we write a hybrid finite element code for implicit time integration for strongly nonlinear equations (nonlinear anisotropic diffusion, reduced MHD). The idea will be to construct a first approximation of the solution using a parametric PINNs or a Neural operator (like Fourier Neural Operator) and integrate this approximation inside the numerical methods. This approach will be used to enhance the basis functions and accelerate the Newton convergence. One of the main questions is how to construct the learning process with data which arrive step by step. We consider an implicit scheme with a Newton method (neural networks which take data and previous time step input) with adaptive time step. If the Newton convergence is fast, we increase the time step at the following step. If there is no convergence, we recompute the time step with a smaller $\Delta t$. The more we make simulations, the more we collect data to increase the accuracy of the neural network for the prediction of Newton’s method.

3.3.1 Reduced models in asymptotic regimes

In some previous and ongoing works (ANR MILK), we have studied the construction of reduced models for kinetic equations in different asymptotic regimes using neural networks. Such reduced models are very interesting since kinetic equations describe the evolution of distribution in phase space (position-velocity), their full numerical resolution would require a lot of computing resources and thus reduced models are much cheaper to simulate. We consider two different types of asymptotic regimes.

The first asymptotic regime is the collisional regime. When the collision rate is high, the velocity distribution function tends to be Gaussian and analytical reduced models are well known: these are the Euler or Navier-Stokes systems. However, for weakly collisional regimes, there are no such analytical reduced models and this is where neural network can help. We have started a work on the non-local closure problem of the Euler equations for the Vlasov-Poisson dynamics 54, 33 and we would like to generalize it to more complex physical problems: complex collisional operator, two species coupling, Vlasov-Maxwell dynamics with method able to be invariant to the spatial grid and potentially interpretable. We can also consider generalized moment models 79, 55 and aim for a network to learn the choice of moment and the local closure. In this context, we propose to ensure the entropy stability of the models obtained since local models are easier to study. This type of method will also be used to design macroscopic biological models using particle simulations (ANR Mapeflu). The company AxesSim is very interested in Vlasov-Maxwell simulations and has optimized codes for hyperbolic equations. The construction of reduced models within this framework would be an important axis of collaboration.

The second asymptotic regime is the strong magnetic field regime. Indeed, in this regime, the charged particles will rotate rapidly around a slow trajectory. Since following the highly oscillating trajectories would be very computationally demanding, we would like a model for the slow trajectory only. In sophisticated physical configurations (e.g., for non-periodic magnetic fields), deriving an analytical solution is difficult if not impossible. Using neural networks, we would like to filter the fast oscillating dynamics and then devise a reduced model. In addition, separately reconstructing the fast rotation dynamics would make it possible to add relevant corrections to the slow dynamics model. This would allow us to construct valid schemes in different magnetic field magnitude regimes 44.

3.3.2 Continuous Reduced Order Modeling (CROM) for strongly nonlinear PDEs and kinetic equations

The Reduced Order Modeling (ROM) and reduced basis methods 57 have demonstrated their powerful capabilities for many problems. They are based on a “projection" of the model onto a reduced basis obtained by singular value decomposition of snapshots of the solutions 39. However, they have difficulties in reducing hyperbolic PDE dynamics and highly nonlinear problems. Since some years the main alternative is based on auto-encoder neural networks 72, 63. Imposing a symplectic structure in these methods is part of the ANR project MILK, where we investigate reduction with manifold learning or auto-encoder approaches. In this project, we propose to focus on a very recent approach called Continuous ROM41, 40, 96, which is based on the implicit neural representation paradigm. The classical ROM approach discretizes the PDE to obtain a large dimensional ODE and compresses this ODE with a Convolutional Auto-encoder or POD. Here, the idea to represent the solution (decoder) using a MLP coordinate based network which depends also on latent variables which are obtained by an auto-encoder. As in the ROM approach, the aim is to write the dynamics on the latent variable. All the works on physics-informed neural networks will be used for the decoder. For the encoder process, we will investigate some permutation-invariant neural networks or greedy approaches. Here, we propose to study multiple strategies to learn the reduced model (Reduced Galerkin projection, differentiable physics learning, classical supervised methods). We will also study how to incorporate the properties of the PDE in these reduced models (asymptotic limits, well-balancing, symplectic properties, Poisson brackets 56 or entropy dissipation, which is essential for stability). We will mainly apply these news approaches on nonlinear conservation laws, wave equations and kinetic models. These approaches could also be interesting to compute a reduced basis only in the velocity space for kinetic equations. To obtain explicable and easy to disseminate reduced models, we will couple these methodologies with symbolic regression methods (see e.g. 90, 38) which allows to learn analytic formula. Comparing to the classical approaches, the reduced models obtained with CROM approach can be used on any meshes and consequently coupled with any code without additional interpolation step and can be used with symbolic approaches. Obtaining analytical models is not possible with traditional ROM approaches.

3.4.1 Reinforcement Learning methods for PDEs and high dimensional action spaces

In the last decade, many RL algorithms have been written by the machine learning community. This approach allows constructing feedback loops that are essential for real-time control. Some of these methods can handle a continuous action (control) space. However these methods are more difficult to use in large-dimensional action spaces and for long time problem. Indeed, depending on the application, there may be problems with exploration (we do not know the action space correctly) or with precision and regularity (it is more difficult to obtain precise control compared to classical gradient approaches). For such problems, the control is a spatial function (discretized in general) so the dimension is large. In this case we are not able to correctly define an admissible/realistic action and sample the action space. For this reason, these algorithms generally make use of a basic parametrization of the action function which is very restrictive. We propose to use generative models/operators to construct probabilistic policies able to explore large-dimensional structured actions, and to couple this with gradient methods (adjoint approach, PINNs) to guide the exploration towards interesting areas. These methods will be an alternative to differential physics to train large networks (which is a very costly, and sometimes unstable, approach) present in numerical schemes or physical model taking into account of the long time stability.

3.4.2 Optimal control and physics-informed ML

Recent work, such as 75 have used PINNs to solve open loop optimal control problems. These methods could be very interesting for solving high dimensional problems, closed loop problems, or shape optimization, which is equivalent to a neural implicit representation. Indeed, for closed loop problems, standard approaches based on the Hamilton Jacobi Bellman equation are very cumbersome to implement. We will study these methods from a theoretical point of view on simple cases and their practical improvement. We will also investigate the extension of these approaches with physics-informed neural operators, which seem to have a greater approximation capacity than PINNs and could be efficient to treat inverse problems in high dimension. Another strategy we propose is to use generative models. Generative models, like diffusion models, make it possible to sample high-dimensional probability distributions. Recently, a new approach to robot control has been proposed. It consists in training a diffusion model to build an efficient control from a random control. This amounts to concentrating a probability law of controls on the most optimal control using physics-informed loss. We propose to couple this with Neural encoder approaches and apply it to control PDEs and inverse problems.

3.4.3 Accelerated open loop optimal control by ML

Optimal control methods that are based on the Pontryagin maximum principle and gradient-based methods are very computationally intensive. Reducing the computational cost of these methods is an important issue, that could be solved using machine learning. In a similar way to the case of Newton's method, Neural operator/PINNs approaches can be used to obtain an approximate solution of the control problem which will be used as an initial guess to accelerate the convergence of the iterative methods. We will also study another approach which consists of using reduced modeling to accelerate each step of the gradient method. We have recently constructed a method where we control a complex problem by constructing a reduced model which is checked and corrected according to the effect of the control obtained on the full model. We wish to focus on correcting the reduced model (which may be invalid in some areas) as the control algorithm proceeds. These approaches could be viewed as model-based Reinforcement learning and will use our work on reduced modeling in Section 3.3. This project is a lower priority than the other optimal control projects.

3.4.4 Inverse problem and super-resolution

We consider the following inverse problem: given a discrete-time measurement of the propagation of a sound impulse from a pointwise, omnidirectional source to a microphone array (called RIR or Room Impulse Response), can we estimate the geometry of the room? As part of Tom Sprunck's PhD, which started on November 1, 2021, this question has been partially addressed theoretically and numerically, in the case of cuboid rooms. To formalize this problem and deal with numerical aspects, we are investigating the use of the so-called image source method, that allows replacing the boundary of the room to be reconstructed by a constellation of image sources corresponding to iteratively reflected copies of the original sound source with respect to the walls of the room. The question can then be formulated as an optimization problem which consists of identifying the positions of a linear combination of Dirac masses in space — the image sources — from discrete time observations of the solution of the wave equation at the microphones (the RIRs), which are imaged by a linear operator. Knowing the positions of the image sources up to order 1 is then sufficient to reconstruct the geometry of the room. This problem is thus part of the recent framework of super-resolution, which aims at reconstructing the positions and amplitudes of the peaks of a sparse measurement from linear observations. We consider here a convex relaxation of the problem by extending it to the set of Radon measurements (BLASSO). We are currently considering the implementation of efficient numerical methods exploiting this relaxation. To make the method applicable to real world data that feature complex frequency- and angle-dependent responses of sources, microphones and reflecting surfaces inside the room, we intend to hybridize these methods with data driven techniques. These could be achieved by training generative models on real databases of source, microphone and wall responses, or by using deep unrolling on parts of the optimization scheme, in order to optimize them end-to-end on real or realistically simulated data.

6.1.1 acoustic-sfw

• Name:
  Hearing the Shape of a Shoebox Room
• Keywords:
  Acoustic Model, Acoustics, Inverse problem, Super-resolution
• Functional Description:

  1) An adaptation of the Sliding Frank-Wolfe algorithm for the gridless 3D recovery of image sources from room impulse responses recorded with a compact microphone array. The algorithm is described in details in this article: [1] Sprunck, T., Deleforge, A., Privat, Y., & Foy, C. (2022). Gridless 3d recovery of image sources from room impulse responses. IEEE Signal Processing Letters, 29, 2427-2431. HAL: https://inria.hal.science/hal-03763838/

  2) An algorithms that recovers the 18 input parameters of the shoebox image source method from given such an estimated image source point cloud, namely: - The room's width, depth and height - The 6 DoF room translation and rotation in the array's coordinate frames - The 3D source position - One absorption coefficient for each of the room boundary.

  The algorithm is described in details in this article: [2] Sprunck, T., Deleforge, A., Privat, Y., & Foy, C. (2025). Fully reversing the shoebox image source method: From impulse responses to room parameters. IEEE Transactions on Audio, Speech and Language Processing. HAL: https://inria.hal.science/hal-04567514/

• Release Contributions:
  First version.
• URL:
  https://­github.­com/­Sprunckt/­acoustic-sfw
• Contact:
  Antoine Deleforge

6.1.2 opla

• Name:
  One Page Layout Automator
• Keywords:
  Python, Web
• Functional Description:
  - Generate a professional webpage from a single markdown file - Handle publication lists coming from HAL database or from a bibtex file - Highly customizable thanks to the use of jinja templates, shortcodes and custom styles
• Contact:
  Matthieu Boileau

6.1.3 LLG3D

• Name:
  LLG3D
• Keywords:
  3D, Python, MPI, Micromagnetism, Landau-Lifshitz-Gilbert equation
• Functional Description:
  LLG3D is a solver for the stochastic Landau-Lifshitz-Gilbert equation in 3D. It is written in Python and utilizes the MPI library to parallelize computations.
• URL:
  https://­llg3d.­pages.­math.­unistra.­fr/­llg3d/
• Contact:
  Matthieu Boileau
• Partner:
  IPCMS

7.1.1 Implicit-explicit solver for a two-fluid single-temperature model

Participants: Andrea Thomann.

Reference: 14

In this work, we have derived an analyzed a new implicit-explicit finite volume (RS-IMEX FV) scheme for a single-temperature SHTC model. We note that the two-fluid model allows two velocities and pressures. Further, it includes two dissipative mechanisms: phase pressure and velocity relaxations. In the proposed scheme these processes are treated differently. The relative velocity relaxation term is linear and is resolved as a part of the implicit sub-system, whereas the pressure relaxation is strongly nonlinear and therefore is treated separately by the Newton method. Our RS-IMEX FV method is constructed in such a way that acoustic-type waves are linearized around a suitably chosen reference state (RS) and approximated implicitly in time and by means of central finite differences in space. The remaining advective-type waves are approximated explicitly in time and by means of the Rusanov FV method. The RS-IMEX FV scheme is suitable for all Mach number flows, but in particular it is asymptotic preserving in the low Mach number flow regimes. Many multi-phase flows, such as granular or sediment transport flows, can be modeled within the single-temperature approximation. In turn, many of these flows are weakly compressible and therefore impose severe time step restrictions if solved with a time-explicit numerical scheme. Therefore, the proposed RS-IMEX FV scheme is suitable to model various environmental flows. The proposed method was tested on a number of test cases for low and moderately high Mach number flows demonstrating the capability of the scheme to properly capture both regimes. The theoretical second order accuracy of the scheme was confirmed on a stationary vortex test case. We compared the second order scheme against its first order variant which showed that the second order scheme yields more accurate approximations of discontinuities. Finally, the theoretically proven asymptotic preserving property was verified numerically.

7.1.2 A structure-preserving semi-implicit IMEX finite volume scheme for ideal magnetohydrodynamics at all Mach and Alfvén numbers

Participants: Andrea Thomann.

Reference: 5

In this work, we present a divergence-free semi-implicit finite volume scheme for the simulation of the ideal magnetohydrodynamics (MHD) equations which is stable for large time steps controlled by the local transport speed at all Mach and Alfvén numbers. An operator splitting technique allows to treat the convective terms explicitly while the hydrodynamic pressure and the magnetic field contributions are integrated implicitly, yielding two decoupled linear implicit systems. The linearity of the implicit part is achieved by means of a semi-implicit time linearization. This structure is favorable as second-order accuracy in time can be achieved relying on the class of semi-implicit IMplicit–EXplicit Runge–Kutta (IMEX-RK) methods. In space, implicit cell-centered finite difference operators are designed to discretely preserve the divergence-free property of the magnetic field on three-dimensional Cartesian meshes. The new scheme is also particularly well suited for low Mach number flows and for the incompressible limit of the MHD equations, since no explicit numerical dissipation is added to the implicit contribution and the time step is scale independent. Likewise, highly magnetized flows can benefit from the implicit treatment of the magnetic fluxes, hence improving the computational efficiency of the novel method. The convective terms undergo a shock-capturing second order finite volume discretization to guarantee the effectiveness of the proposed method even for high Mach number flows. The new scheme is benchmarked against a series of test cases for the ideal MHD equations addressing different acoustic and Alfvén Mach number regimes where the performance and the stability of the new scheme is assessed.

7.1.3 Convergence of a hyperbolic thermodynamically compatible finite volume scheme for the Euler equations

Participants: Andrea Thomann.

Reference: 23

We study the convergence of a novel family of thermodynamically compatible schemes for hyperbolic systems (HTC schemes) in the framework of dissipative weak solutions, applied to the Euler equations of compressible gas dynamics. Two key novelties of our method are i) entropy is treated as one of the main field quantities and ii) the total energy conservation is a consequence of compatible discretization and application of the Abgrall flux.

7.1.4 A fully well-balanced hydrodynamic reconstruction

Participants: Victor Michel-Dansac.

Reference: 4

The present work focuses on the numerical approximation of the weak solutions of the shallow water model over a non-flat topography. In particular, we pay close attention to steady solutions with nonzero velocity. The goal of this work is to derive a scheme that exactly preserves these stationary solutions, as well as the commonly preserved lake at rest steady solution. These moving steady states are solution to a nonlinear equation. We emphasize that the method proposed here never requires solving this nonlinear equation; instead, a suitable linearization is derived. To address this issue, we propose an extension of the well-known hydrostatic reconstruction. By appropriately defining the reconstructed states at the interfaces, any numerical flux function, combined with a relevant source term discretization, produces a well-balanced scheme that preserves both moving and non-moving steady solutions. This eliminates the need to construct specific numerical fluxes. Additionally, we prove that the resulting scheme is consistent with the homogeneous system on flat topographies, and that it reduces to the hydrostatic reconstruction when the velocity vanishes. To increase the accuracy of the simulations, we propose a well-balanced high-order procedure, which still does not require solving any nonlinear equation. Several numerical experiments demonstrate the effectiveness of the numerical scheme.

7.1.5 Parallel kinetic schemes for conservation laws, with large time steps

Participants: Philippe Helluy, Victor Michel-Dansac.

Reference: 11

We propose a new parallel Discontinuous Galerkin method for the approximation of hyperbolic systems of conservation laws. The method remains stable with large time steps, while keeping the complexity of an explicit scheme: it does not require the assembly and resolution of large linear systems for the time iterations. The approach is based on a kinetic representation of the system of conservation laws previously investigated by the authors. In this paper, the approach is extended with a subdomain strategy that improves the parallel scaling of the method on computers with distributed memory.

7.1.6 An entropy-stable and fully well-balanced scheme for the Euler equations with gravity

Participants: Victor Michel-Dansac, Andrea Thomann.

References: 19, 28

19 concerns the derivation of a numerical scheme to approximate weak solutions of the Euler equations with a gravitational source term. The designed scheme is proved to be fully well-balanced since it is able to exactly preserve all moving equilibrium solutions, as well as the corresponding steady solutions at rest obtained when the velocity vanishes. Moreover, the proposed scheme is entropy-preserving since it satisfies all fully discrete entropy inequalities. In addition, in order to satisfy the required admissibility of the approximate solutions, the positivity of both approximate density and pressure is established. Several numerical experiments attest the relevance of the developed numerical method.

In addition, in 28, we extended this work to general equations of states. When the system is equipped with a convex entropy and associated entropy inequality, it is also entropy-stable and positivity-preserving for all thermodynamic variables. An extension to high order accuracy is presented. Numerical test cases illustrate the performance of the new scheme, using six different equations of state as examples, four analytic and two tabulated ones.

7.1.7 A high-order, fully well-balanced, unconditionally positivity-preserving finite volume framework for flood simulations

Participants: Victor Michel-Dansac.

Reference: 20

In this work, we present a high-order finite volume framework for the numerical simulation of shallow water flows. The method is designed to accurately capture complex dynamics inherent in shallow water systems, particularly suited for applications such as tsunami simulations. The arbitrarily high-order framework ensures precise representation of flow behaviors, crucial for simulating phenomena characterized by rapid changes and fine-scale features. Thanks to an ad-hoc reformulation in terms of production-destruction terms, the time integration ensures positivity preservation without any time-step restrictions, a vital attribute for physical consistency, especially in scenarios where negative water depth reconstructions could lead to unrealistic results. In order to introduce the preservation of general steady equilibria dictated by the underlying balance law, the highorder reconstruction and numerical flux are blended in a convex fashion with a well-balanced approximation, which is able to provide exact preservation of both static and moving equilibria. Through numerical experiments, we demonstrate the effectiveness and robustness of the proposed approach in capturing the intricate dynamics of shallow water flows, while preserving key physical properties essential for flood simulations.

7.1.8 Composite unstructured meshes with source term

Participants: Emmanuel Franck.

Reference: 6

In this work we focus on an adaptation of the composite method in order to deal with source term in the 2D Euler equations. This method extends classical 1D solvers (such as VFFC, Roe, Rusanov) to the two-dimensional case on unstructured meshes. The resulting schemes are said to be composite as they can be written as a convex combination of a purely node-based scheme and a purely edge-based scheme. We combine this extension with the ideas developed by Alouges, Ghidaglia and we propose two attempts at discretizing the source term of the Euler equations in order to better preserve stationary solutions. We compare these discretizations with the “usual” centered discretization on several numerical examples.

7.1.9 Maximum principle for the mass fraction in a system with two mass balance equations

Participants: Philippe Helluy.

Reference: 13

In this paper written with EDF colleagues, we show how to improve the robustness of the computation of the mass fraction in the thermohydraulic EDF THYC software.

7.2.1 Reducing the memory usage of lattice-Boltzmann schemes with a DWT-based compression

Participants: Clément Flint, Philippe Helluy.

References: 8, 9

We develop a new on-the-fly compression strategy for accelerating CFD computations. The resulting scheme is optimized for GPU computations. In a standard lattice-Boltzmann simulation, it allows reducing the memory footprint by an order of magnitude with a moderate impact on the computation time.

7.2.2 Stability analysis of the vectorial lattice-Boltzmann method (LBM)

Participants: Philippe Helluy.

References: 12

We provide a fully rigorous entropy stability analysis of the vectorial Lattice-Boltzmann method. We also develop a new equivalent equation approach for studying the high-order consistency of the method.

7.2.3 Fourth-order entropy-stable lattice Boltzmann schemes for hyperbolic systems

Participants: Thomas Bellotti, Philippe Helluy, Laurent Navoret.

References: 18

We construct a fourth-order entropy stable LBM scheme and apply it to compressible flows.

7.3.1 Accelerating the convergence of Newton's method for nonlinear elliptic PDEs using Fourier neural operators

Participants: Joubine Aghili, Victor Michel-Dansac, Vincent Vigon, Emmanuel Franck.

Reference: 2

It is well known that Newton's method can struggle to converge if the initial guess is too far from the solution. Such a problem occurs in particular when this method is used to solve nonlinear elliptic PDEs discretized by finite differences. This work focuses on accelerating the convergence of Newton's method in this context. We use Fourier Neural Operator (FNO) to construct an approximation of the discrete solution of a PDE from its parameters. This approximation is then used as an initial guess for Newton's method. Numerical results, in one and two dimensions, show that the proposed initial guess accelerates the convergence of Newton's method by a significant margin compared to a naive initial guess, especially for highly nonlinear and anisotropic problems, with larger gains on coarse grids.

7.3.2 Approximately well-balanced discontinuous Galerkin methods using bases enriched with physics-informed neural networks

Participants: Emmanuel Franck, Victor Michel-Dansac, Laurent Navoret.

Reference: 10

This work concerns the enrichment of Discontinuous Galerkin (DG) bases, so that the resulting scheme provides a much better approximation of steady solutions to hyperbolic systems of balance laws. The basis enrichment leverages a prior – an approximation of the steady solution – which we propose to compute using a Physics-Informed Neural Network (PINN). To that end, after presenting the classical DG scheme, we show how to enrich its basis with a prior. Convergence results and error estimates follow, in which we prove that the basis with prior does not change the order of convergence, and that the error constant is improved. To construct the prior, we elect to use parametric PINNs, which we introduce, as well as the algorithms to construct a prior from PINNs. We finally perform several validation experiments on four different hyperbolic balance laws to highlight the properties of the scheme. Namely, we show that the DG scheme with prior is much more accurate on steady solutions than the DG scheme without prior, while retaining the same approximation quality on unsteady solutions.

7.4.1 A dynamical neural Galerkin scheme for filtering problems

Participants: Joubine Aghili.

References: 17

This work explores reconstructing the state of a dynamic system from partial observations, using a model based on a physical PDE with unknown parameters. We introduce a filtering method where a neural network, whose parameters are updated with observational data, approximates the dynamics. This not only estimates the system's state but also tracks changes in PDE parameters over time. We demonstrate this with a one-dimensional Korteweg-de Vries (KdV) equation, highlighting how the placement and number of sensors significantly impact results, suggesting the need for dynamic sensor positioning.

7.4.2 Volume-preserving geometric shape optimization of the Dirichlet energy using variational neural networks

Participants: Victor Michel-Dansac, Emmanuel Franck.

Reference: 3

In this work, we explore the numerical solution of geometric shape optimization problems using neural network-based approaches. This involves minimizing a numerical criterion that includes solving a partial differential equation with respect to a domain, often under geometric constraints like a constant volume. We successfully develop a proof of concept using a flexible and parallelizable methodology to tackle these problems. We focus on a prototypical problem: minimizing the so-called Dirichlet energy with respect to the domain under a volume constraint, involving Poisson's equation in ${ℝ}^2$. We use variational neural networks to approximate the solution to Poisson's equation on a given domain, and represent the shape through a neural network that approximates a volume-preserving transformation from an initial shape to an optimal one. These processes are combined in a single optimization algorithm that minimizes the Dirichlet energy. A significant advantage of this approach is its inherent parallelizability, which makes it easy to handle the addition of parameters. Additionally, it does not rely on shape derivative or adjoint calculations. Our approach is tested on Dirichlet and Robin boundary conditions, parametric right-hand sides, and extended to Bernoulli-type free boundary problems. The source code for solving the shape optimization problem is open-source and freely available here.

7.5.1 Micromagnetic simulations of the size dependence of the Curie temperature in ferromagnetic nanowires and nanolayers

Participants: Clémentine Courtès, Mathieu Boileau.

Reference: 7

We solved the Landau-Lifshitz-Gilbert equation in the finite-temperature regime, where thermal fluctuations are modeled by a random magnetic field whose variance is proportional to the temperature. We obtained Curie temperatures that are in line with the experimental values for cobalt, iron and nickel and for finite-sized objects such as nanowires (1D) and nanolayers (2D), we study the variances of the Curie temperature with respect to the smallest size of the system. Moreover, optimization and parallelization of the python code solving the Landau-Lifshitz-Gilbert equation in the finite-temperature regime led to a 100-time acceleration of the code. We were therefore able to study the effect of the size of the material on the magnetization, in particular the value of the Curie temperature.

7.5.2 Existence and uniqueness of domain walls for notched ferromagnetic nanowires

Participants: Clémentine Courtès.

Reference: 22

In this work we investigate a simple model of notched ferromagnetic nanowires using tools from calculus of variations and critical point theory. Specifically, we focus on the case of a single unimodal notch and establish the existence and uniqueness of the critical point of the energy. This is achieved through a lifting argument, which reduces the problem to a generalized Sturm-Liouville equation. Uniqueness is demonstrated via a Mountain-Pass argument, where the assumption of two distinct critical points leads to a contradiction. Additionally, we show that the solution corresponds to a system of magnetic spins characterized by a single domain wall localized in the vicinity of the notch. We further analyze the asymptotic decay of the solution at infinity and explore the symmetric case using rearrangement techniques.

7.6.1 Generalizing the SINDy approach with nested neural networks

Participants: Clément Flint, Emmanuel Franck, Victor Michel-Dansac.

Reference: 25

Symbolic Regression (SR) is a widely studied field of research that aims to infer symbolic expressions from data. A popular approach for SR is the Sparse Identification of Nonlinear Dynamical Systems (SINDy) framework, which uses sparse regression to identify governing equations from data. This study introduces an enhanced method, Nested SINDy, that aims to increase the expressivity of the SINDy approach thanks to a nested structure. Indeed, traditional symbolic regression and system identification methods often fail with complex systems that cannot be easily described analytically. Nested SINDy builds on the SINDy framework by introducing additional layers before and after the core SINDy layer. This allows the method to identify symbolic representations for a wider range of systems, including those with compositions and products of functions. We demonstrate the ability of the Nested SINDy approach to accurately find symbolic expressions for simple systems, such as basic trigonometric functions, and sparse (false but accurate) analytical representations for more complex systems. Our results highlight Nested SINDy's potential as a tool for symbolic regression, surpassing the traditional SINDy approach in terms of expressivity. However, we also note the challenges in the optimization process for Nested SINDy and suggest future research directions, including the designing of a more robust methodology for the optimization process. This study proves that Nested SINDy can effectively discover symbolic representations of dynamical systems from data, offering new opportunities for understanding complex systems through data-driven methods.

7.7.1 Fully reversing the shoebox image source method: from impulse responses to room parameters

Participants: Antoine Deleforge, Tom Sprunck.

Reference: 30

We developped an algorithm that fully reverses the shoebox image source method (ISM), a popular and widely used room impulse response (RIR) simulator for cuboid rooms introduced by Allen and Berkley in 1979. More precisely, given a discrete multichannel RIR generated by the shoebox ISM for a microphone array of known geometry, the algorithm reliably recovers the 18 input parameters. These are the 3D source position, the 3 dimensions of the room, the 6-degrees-of-freedom room translation and orientation, and an absorption coefficient for each of the 6 room boundaries. The approach builds on a recently proposed gridless image source localization technique combined with new procedures for room axes recovery and first-order-reflection identification. Extensive simulated experiments reveal that near-exact recovery of all parameters is achieved for a 32-element, 8.4-cm-wide spherical microphone array and a sampling rate of 16 kHz using fully randomized input parameters within rooms of size 2x2x2 to 10x10x5 meters. Estimation errors decay towards zero when increasing the array size and sampling rate. The method is also shown to strongly outperform a known baseline, and its ability to extrapolate RIRs at new positions is demonstrated. Crucially, the approach is strictly limited to low-passed discrete RIRs simulated using the vanilla shoebox ISM. Nonetheless, it represents to our knowledge the first algorithmic demonstration that this difficult inverse problem is in principle fully solvable over a wide range of configurations.

7.8.1 Phi-FD : A well-conditioned finite difference method inspired by Phi-FEM for general geometries on elliptic PDEs

Participants: Vincent Vigon.

References: 24

This work presents a new finite difference method, called Phi-FD, inspired by the Phi-FEM approach for solving elliptic PDEs on general geometries. The proposed method uses Cartesian grids, ensuring simplicity in implementation. Moreover, contrary to the previous finite difference scheme on non-rectangular domains, the associated matrix is well-conditioned. The use of a level-set function for the geometry description makes this approach relatively flexible. We prove the quasi-optimal convergence rates in several norms and the fact that the matrix is well-conditioned. Additionally, the paper explores the use of multigrid techniques to further accelerate the computation. Finally, numerical experiments in both 2D and 3D validate the performance of the Phi-FD method compared to standard finite element methods and the Shortley-Weller approach.

7.8.2 Latent watermarking of audio generative models

Participants: Antoine Deleforge.

Reference: 29

The advancements in audio generative models have opened up new challenges in their responsible disclosure and the detection of their misuse. In response, we introduced a method to watermark latent generative models by a specific watermarking of their training data. The resulting watermarked models produce latent representations whose decoded outputs are detected with high confidence, regardless of the decoding method used. This approach enables the detection of the generated content without the need for a post-hoc watermarking step. It provides a more secure solution for open-sourced models and facilitates the identification of derivative works that fine-tune or use these models without adhering to their license terms. Our results indicate for instance that generated outputs are detected with an accuracy of more than 75% at a false positive rate of less than 0.1%, even after fine-tuning the latent generative model.

9.1.1 Inria associate team not involved in an IIL or an international program

Participants: Clémentine Courtès.

Inria associate team PANDA (2024-2026). Coordinator: A. de Laire (Université de Lille). MACARON participant: Clémentine Courtès. Study of dispersive PDE systems for wave propagation.

9.2.1 Visits to international teams

9.3.1 Other european programs/initiatives

9.4.1 ANR MOSICOF (MOdeling and SImulation of COmplex Ferromagnetic systems)

Participants: Clémentine Courtès.

Dates: 10/2021 – 10/2025.

Coordinator: S. Labbé, Sorbonne Université.

Partners: Sorbonne université, Université de Pau et des Pays de l'Adour, Université de Strasbourg

Decription: During the last decade, promising applications of ferromagnetic materials have emerged in the domains of nanoelectronics (spintronic) and data storage: complex ferromagnetic systems are increasingly used for digital data recording and logic devices. They reduce the energy storage cost while improving the performance of the devices. The goal of this proposal is to bring together mathematicians and physicists around the understanding of the properties of ferromagnetism. One of the main objectives is to highlight and treat new multi-physics models, allowing for optimization and control of the magnetizations, and to simulate the phenomena in a more efficient and less expensive way. We wish to develop approaches leading to mathematically justified and physically relevant solutions for the analysis and optimization of these materials, and which could ultimately lead to implementation on devices.

9.4.2 ANR JCJC SMEAGOL (Méthode structurelle – Application à des systèmes hyperboliques généraux)

Participants: Victor Michel-Dansac.

Dates: 11/2024 – 10/2028.

Funding: 283 k€

Coordinator: Victor Michel-Dansac

Decription: The structural method, introduced over the last two years, develops high-order numerical schemes for solving PDEs on compact stencils. This finite difference approach is unique in that it approximates both the solution and its derivatives with the same order of accuracy by defining two independent sets of discrete equations: the physical equations (PEs) and the structural equations (SEs). The PEs represent the problem's physics, while SEs ensure the accuracy of the discretization. This separation provides a high degree of flexibility, allowing for the modification of PEs to include specific constraints (e.g., ensuring a vector field is divergence-free) and the adjustment of SEs to handle non-smooth solutions or enhance stability. The ANR project SMEAGOL seeks to extend this method to hyperbolic systems of balance laws in multiple spatial dimensions, where continuous initial conditions often lead to non-smooth solutions and multiscale regime changes. This makes the method particularly suitable for complex problems in fluid mechanics or electromagnetism. The separation between physical and structural equations in this framework allows for the dynamic adaptation of the scheme to the local problem, switching PEs or SEs on or off as needed. SMEAGOL's goals include both constructing and adapting the structural method to these advanced applications, to develop well-balanced, asymptotic-preserving and robust schemes.

9.4.3 ANR MAPEFLU (Modelling the effect of Apoptosis on Epithelium FLUidity)

Participants: Laurent Navoret.

Dates: 03/2023–02/2027

Coordinator: Laurent Navoret

Partners: Institut Pasteur, Université Paris Université de Strasbourg (IRMA, IGBMC)

Decription: Epithelia have a viscoelastic behaviour: they respond as solids over short times and as fluids over large times. This fluidity plays an essential role in morphogenesis and tissue deformation. At the cellular scale, fluidity is achieved by the remodelling of junctions between cells due to their interactions but also by cell division and death. However, the contribution of apoptosis to fluidity has been little studied and remains unclear since cell death is also associated with local elastic constraints. Our project first aims at developing a novel particle model, describing cell cycles and the polarities interactions (Vicsek-like model), to assess the impact of cell death rate on tissue fluidity. The construction of this model will be strongly guided by comparisons with in vitro (MDCK cells) and in vivo (Drosophila pupa) experiments. From this particle model, a hydrodynamic model will be rigorously derived and simulations based on this new macroscopic description will be utilized to improve the understanding of tissue dynamics. The present study will thus provide a generic model, consistent with the experimental data and allowing one of the first systematic assessments of the role of apoptosis in tissues.

9.4.4 ANR SIMBADNESTICOST (Simulation Based Network Structure Inference Constrained by Observed Spike Trains)

Participants: Vincent Vigon.

Dates: 2023 - 2026

Grant: 159 k€

Coordinator: Christophe Pouzat

Partners: IRMA (Institut de recherche en mathématique avancées) and University of Strasbourg

Decription: Neurophysiologists are now able to record from a large number of electrodes the sequences of action potentials generated by many neurons. Unfortunately, these "many neurons" still represent only a tiny fraction of the neuronal population that constitutes the network. By using association statistics such as the estimation of cross-correlation functions, they attempt to infer the structure of the network formed by the recorded neurons. This produces a "network image" generally called a "functional network" whose characteristics strongly depend on the recording conditions. We consider that reconstructing the network formed by the recorded neurons is an ill-posed problem. We propose to focus instead on the "generative probability distribution" of the network: what is the probability of having a connection from a type A neuron to a type B neuron?

9.4.5 ANR PHI-FEM (Développement d'une méthode aux éléments finis pour la conception de jumeaux numériques temps réel en chirurgie)

Participants: Vincent Vigon.

Dates: 2023 - 2026

Grant: 324 k€

Coordinator: Michel Duprez

Partners: Inria teams MACARON and MIMESIS.

Decription: Phi-FEM is a recently proposed finite element method for the efficient numerical solution of partial differential equations in domains defined by level-set functions. The main objective of this project is to develop Phi-FEM into an efficient, patient-specific, and real-time simulation tool for human organs. To achieve this goal, we will adapt Phi-FEM to equations relevant to biomechanics, provide an efficient implementation allowing the use of real organ geometries, and finally combine it with convolutional neural networks to make it real-time after training.

9.4.6 ANR HAIKUS (Artificial Intelligence applied to augmented acoustic Scenes)

Participants: Antoine Deleforge.

Dates: 12/2019 - 12/2024

Coordinator: Olivier Warsufel (Ircam, Paris)

Partners: Ircam (Paris), Inria (Nancy), IJLRA (Paris)

Decription: HAIKUS aims to achieve seamless integration of computer-generated immersive audio content into augmented reality (AR) systems. One of the main challenges is the rendering of virtual auditory objects in the presence of source movements, listener movements and/or changing acoustic conditions.

9.4.7 ANR JCJC DENISE (Tackling hard problems in audio using Data-Efficient Non-linear InverSe mEthods)

Participants: Antoine Deleforge.

Dates: 10/2020 – 12/2024

Coordinator: Antoine Deleforge

Decription: DENISE aims to explore the applicability of recent breakthroughs in the field of nonlinear inverse problems to audio signal reparation and to room acoustics, and to combine them with compact machine learning models to yield data-efficient techniques.

9.4.8 PEPR IA/PC IA-EDP

Participants: Philippe Helluy, Joubine Aghili, Clémentine Courtès, Emmanuel Franck, Victor Michel-Dansac, Vincent Vigon, Laurent Navoret.

Dates: 01/09/2023 – 31/08/2027

Coordinator: A. Chambolle (Univ. Paris-Dauphine)

Decription: The PEPR IA is a large national project on artificial intelligence (AI). The PC IA-PDE is a project funded by the ANR, which gathers ten major French institutions involved in developing the mathematical analysis of AI, the study of optimization in machine learning, as well as in developing machine learning for numerical analysis and scientific computing. We will study the link between modern AI methods and optimal control, optimal transport, PDE and numerical analysis. The team is involved in the optimal control aspect.

9.4.9 PEPR Numpex/PC Exa-MA

Participants: Philippe Helluy, Joubine Aghili, Clémentine Courtès, Emmanuel Franck, Victor Michel-Dansac, Vincent Vigon, Laurent Navoret.

Dates: 10/2021 – 10/2025.

Coordinator: Christophe Prud'homme

Decription: The Exa-MA project focuses on the Exascale aspects of digital methods, guaranteeing their adaptability to existing and future hardware. It is also a cross-disciplinary project, proposing methods and tools in which modelling, data and AI, through algorithms, are central. The team is mainly involved in the WP2 on reduced modeling and ML technics but also in the WP1 on numerical methods

10.1.1 Scientific events: organisation

• Member of organizing commitee of PANDA-Hyp workshop, Arcachon, France, oct. 2024 (A. Thomann). More information.
• Co-organization of the 21st International Congress on Mathematical Physics (ICMP) : July 2024, Palais des Congrès, Strasbourg (C. Courtès). More information.
• Co-organization of two mini-symposia "Models and asymptotics for ferromagnetic materials" : May 2024, CANUM2024, Ile de Ré (C. Courtès). More information.
• Organization of the workshop on Scientific Machine Learning, Strasbourg, approx. 60 participants, June 2024 (V. Michel-Dansac, L. Navoret, E. Franck). More information.
• Organization of the summer school « New trends in computing » (L. Navoret). More information.
• Co-organization of the 6th workshop IRMA-EDF on compressible multiphase flows, June 2024 (P. Helluy). More information.
• Co-organization of the workshop "Methodological advances for Audio Augmented Reality and its applications" at IRCAM, Paris, France, approx. 40 participants, December 2024 (A. Deleforge). More information.
• Co-organization of the regular ITI IRMIA++ seminars since Sept. 2024, Strasbourg university (C. Courtès, L. Navoret).
• Co-organization of the regular sem'in (internal seminar of the laboratory) since Sept. 2023, Strasbourg university (C. Courtès).
• Organization of the weekly MACARON/MOCO seminar since Sept. 2021 (C. Courtès, V. Michel-Dansac, J. Aghili).

10.1.2 Scientific events: selection

10.1.3 Journal

10.1.4 Invited talks

10.1.5 Leadership within the scientific community

• Leader of the Research Group (GdR) "Calcul" (M. Boileau)

10.1.6 Scientific expertise

• One project review for the Swiss scientific foundation (P. Helluy)

10.1.7 Research administration

• Nominated member of the IRMA Laboratory Council, Strasbourg university (C. Courtès)
• Elected member of the Mathematicians Commission of the university of Strasbourg (C. Courtès, L. Navoret)
• Nominated member at the Inria Nancy - Grand Est center committee (C. Courtès)
• Referent, co-responsible and member on laboratory parity for INSMI, the mathematics and computer science faculty (UFR math-info) and IRMA (C. Courtès)
• Responsible of training for Institut Thématique Interdisciplinaire IRMIA++, Strasbourg university (L. Navoret)
• Elected member of the expert committee, UFR Math-Info, Université de Strasbourg (L. Navoret, J. Aghili)
• Head of the IRMA MOCO team (P. Helluy)
• Referent on research data for the Inria Nancy - Grand Est center (A. Deleforge)
• Board member of the Research Group (GdR) "Calcul" (E. Franck)
• Member of the COMIPERS, Inria Nancy - Grand Est (E. Franck)

10.2.1 Teaching

• 28h Methodes numeriques pour les EDP 2, Master 2 on Scientific Computing (M2 CSMI) at UFR Maths-Info, Strasbourg university (A. Thomann)
• 8h TP Numerical resolution techniques for engineering, M1 CSMI at UFR Maths-Info, Strasbourg university (A. Thomann)
• 37h of CM on computer science in L3 magistère at UFR Maths-Info, Strasbourg university (C. Courtès)
• 28h of TD on computer science in L3 magistère at UFR Maths-Info, Strasbourg university (C. Courtès)
• 15h of CM on numerical analysis in L3 mathématiques at UFR Maths-Info, Strasbourg university (C. Courtès)
• 36h of TD on numerical analysis in L3 mathématiques at UFR Maths-Info, Strasbourg university (C. Courtès)
• 8h of TD on real analysis in L2 informatique at UFR Maths-Info, Strasbourg university (C. Courtès)
• 8h of TP on real analysis in L2 informatique at UFR Maths-Info, Strasbourg university (C. Courtès)
• Responsible of the option "mathématiques et statistiques pour préparer aux concours administratifs" of L3, UFR Maths-Info and Institut de Préparation à l'Administration Générale, Strasbourg (C. Courtès)
• 6h of "textes" in M2 Agrégation at UFR Maths-Info, Strasbourg university (V. Michel-Dansac)
• 14h of CM in Intro to Object-Oriented Programming in L2 informatique at UFR Maths-Info, Strasbourg university (V. Michel-Dansac)
• 14h of TP in Scientific Computing in M1 CSMI at UFR Maths-Info, Strasbourg university (V. Michel-Dansac)
• 36h of TD, C++ L3 informatique at UFR Maths-Info, Strasbourg university (E. Franck)
• 28 of CI, SciML2 M2 CSMI at UFR Maths-Info, Strasbourg university (E. Franck)
• 16h of CM in “Basics in mathematics” in M2 Cell Physics at Strasbourg university (L. Navoret)
• 4h of CM in “Math for living matter” in M2 Cell Physics at Strasbourg university (L. Navoret)
• 12h of CM in “Scientific computing” in Master 2 Agrégation at Strasbourg university (L. Navoret)
• 9h of TD in “Scientific computing” in Master 2 Agrégation at Strasbourg university (L. Navoret)
• 28h of CI in "Scientific computing (sparse linear algebra)" in Master 2 Scientific Computing (M2 CSMI) at Strasbourg university (L. Navoret)
• 6h of TD in "Interdisciplinary seminars" in Diplôme d’université at Strasbourg university (L. Navoret)
• 4h of TD in « Interdisciplinary project » in Diplôme d’université at Strasbourg university (L. Navoret)
• 56h of CI in Mathematics for sciences II in L1 at UFR Maths-Info, Strasbourg university (J. Aghili)
• 7h of CI in Projet management in M1 CSMI at UFR Maths-Info, Strasbourg university (J. Aghili)
• 63h of CI in Numerical Analysis and Computer sciences at UFR Physics and Engineering, Strasbourg university (J. Aghili)
• 10h of TD in Numerical Analysis in Prépa Agregation of Mathematics at UFR Maths-Info, Strasbourg university (J. Aghili)
• 63h of CI in Mathematics for sciences I in L1 at UFR Maths-Info, Strasbourg university (J. Aghili)
• 7h of CI in Projet management in M2 CSMI at UFR Maths-Info, Strasbourg university (J. Aghili)
• 8h of TP in Numerical Methods in M1 CSMI at UFR Physics and Engineering, Strasbourg university (J. Aghili)
• 6h of course in the Summer school « New trends in computing », ITI IRMIA++, Strasbourg university (M. Boileau)
• 18h of Python for Data Science, M2 CSMI at UFR Math-Info, Strasbourg university (M. Boileau)
• 60h of CI. Modelisation Probabiliste en M2 Agreg, UFR Maths-Info, Strasbourg university (V. Vigon)
• 35h of CI. Traitement et exploitation des données. M1 CSMI, UFR Maths-Info, Strasbourg university (V. Vigon)
• 35h of CI. Traitement du signal et des images 1. M1 CSMI, UFR Maths-Info, Strasbourg university (V. Vigon)
• 35h of CI. Traitement du signal et des images 2. M2 CSMI, UFR Maths-Info, Strasbourg university (V. Vigon)
• 35h of CI. Science des données pour l’actuariat, M2 Actuariat, UFR Maths-Info, Strasbourg university (V. Vigon)
• 16h of TP on computer assisted proof in M1 CSMI at UFR Math-Info, Strasbourg university (P. Helluy)
• 20h of CM in C++ in L3 informatique, UFR Math-Info, Strasbourg university (P. Helluy)
• 28h of CI in scientific computing in M1 CSMI, UFR Math-Info, Strasbourg university (P. Helluy)
• 10,5h of CM on Artificial Intelligence, Machine Learning, Deep Learning in M1 at Telecom-Physique Strasbourg (A. Deleforge)
• 12h of TP on Artificial Intelligence, Machine Learning, Deep Learning in M1 at Telecom-Physique Strasbourg (A. Deleforge)
• 12h of TP on Artificial Intelligence, Machine Learning, Deep Learning in M1 at Telecom-Physique Strasbourg (A. Deleforge)
• 12h of TP on Unsupervized Machine Learning in L3 at Telecom-Physique Strasbourg (A. Deleforge)
• 7h of CM on Artificial Intelligence and Statistical Learning in M1 at Telecom-Physique Strasbourg (A. Deleforge)

10.2.2 Supervision

• PostDoc Dinh Hung Truong, 2023-, "Design of Neural Operators based on PINNs; applications to wave propagation and fluid dynamics" (V. Michel-Dansac, E. Franck)
• PostDoc Leopold Tremant, 2023-2024, "Structure preserrving ML for ODE" (C. Courtès, E. Franck, L. Navoret)
• PostDoc Youssouf Nasseri, 2023-2024, "data driven reduced models for Vlasov equations" (C. Courtès, E. Franck, L. Navoret)
• PhD Thesis Roxana Sublet, 2023-, “Mathematical models for collective cell dynamics”(50% L. Navoret)
• PhD Thesis Amaury Bélières-Frendo, 2023-, "Shape optimization through learning" (25% V. Michel-Dansac)
• PhD Thesis Daria Hrebenshchykova, 2024-, "Building physics-based multilevel surrogate models from neural networks. Application to electromagnetic wave propagation" (33% V. Michel-Dansac)
• PhD Thesis Nicolas Pailliez, 2024-, "Implicit neural representation and opertator learning for multi-scale physical problems" (V. Michel-Dansac, E. Franck, L. Navoret)
• PhD Thesis Virgile Bertrand, 2024-, "Constructing the structural method for hyperbolic partial differential equations" (33% V. Michel-Dansac, 33% E. Franck)
• PhD Thesis Guillaume Steimer 2021-, "Hamiltonian data driven reduced models" (25% L. Navoret, 25% E. Franck, 25% V. Vigon)
• PhD Thesis Mei Pallanque 2022-, "New models for radiative transfer in reionization" (30% E. Franck)
• PhD Thesis Frédérique Lecourtier 2023-, "Hybrid ML and finit element methods for digital twins" (33% E. Franck)
• PhD Thesis Killian Lutz 2023-, "Control for quantum system" (50% E. Franck)
• PhD Thesis Claire Schnoebelen 2023-, "Sructure presering ML methods for symplectic PDEs" (33% E. Franck, 33% L. Navoret)
• PhD Thesis Lauriane Turelier, 2023- (50% C. Courtès)
• PhD Thesis Hassan Ballout 2024-, (33% J. Aghili)
• PhD Thesis Clément Flint, 2020-2024, (50 % P. Helluy)
• PhD Thesis Gauthier Lazare, 2022-, (50 % P. Helluy)
• PhD Thesis Stéphane Dilungana, 2020-2024, "Apprentissage automatique et optimisation pour la détermination des propriétés acoustiques d’une salle à partir de signaux audio", Link (33% A. Deleforge)
• PhD Thesis Tom Sprunck, 2021-2024, "Can one hear the shape of a room ? Room Geometry Reconstruction from Acoustic Measurements using Super-Resolution and Shape Optimization" 16 (33% A. Deleforge)
• PhD Thesis Robin San Roman 2022-, "Apprentissage autosupervisé de représentations audio désintriqués pour la compression et la génération" (33% A. Deleforge)
• Master Thesis (M2) Alexis Schmitt, "Introduction to the finite element method" (V. Michel-Dansac)
• Master Thesis (M2) Jérémy Pawlus "Learning Green functions and reduced modeling for the Helmholtz equation" (V. Michel-Dansac, E. Franck, A. Deleforge)
• Master Thesis (M2) Virgile Bertrand "ScimBa: Combining Numerical Methods and Machine Learning for Solving PDEs" (V. Michel-Dansac, E. Franck)
• Master Thesis (M2) B. Gaunard, Agreg project, "Symplectic schemes" (L. Navoret)
• Master Thesis (M2) Barnabé Miesch "Réflexion acoustique en incidence oblique et estimation de l’impédance de paroi par optimisation numérique." (50% A. Deleforge)
• Master Thesis (M1) Marie Sengler, co-supervised with Thales, "Multiscale electrostatics and magnetostatics simulations on complex domains" (V. Michel-Dansac, E. Franck)
• Master Thesis (M1) Rayen TLILI, "Interfacing Feel++ and Scimba" (J. Aghili)
• Master Thesis (M1) Adama DIENG, "The Discontinuous Galerking method in 1D" (J. Aghili)
• Master Thesis (M1) Hatim Ellabib, "Micro macro modeling for radiative transfer" (E. Franck)
• Master Thesis (M1) Antoine Regardin, "Adaptive implicit methods for hyperbolic scalar equation" (A. Thomann, E. Franck)
• Bachelor Thesis (L3) Elisa Cuoco (C. Courtès)
• Bachelor Internship (L2), "development of a personal researcher webpage generator" (M. Boileau)
• Apprentice developer, IRMA information system, web projects (M. Boileau)

10.2.3 Juries

• Reviewer and jury member of the PhD defense of Davide Ferrari, February, University Trento, Italy (A. Thomann)
• Member of the selection committee for associate professors (MCF), CNU 26, University of Lorraine (C. Courtès)
• Member of the recruitment committee for an AGPR, ENS de Cachan (C. Courtès)
• Jury member for the external aggregation of mathematics (C. Courtès)
• Jury member of the mathematics olympiad of 1ère (C. Courtès)
• Jury member for the defense of projects aiming to obtain the ITI++ University Diploma (J. Aghili)
• Jury member for the recruitment of three ATER positions and Contract Teachers at UFR Maths-Info (J. Aghili)
• Member of the Inria research engineer recruitment jury, June, MIMESIS (M. Boileau)
• Member of the regional commission for ranking CNRS engineers and technicians (M. Boileau)
• President of the PhD jury of Valentin Ritzenthaler, Toulouse university (P. Helluy)
• President of the PhD jury of Camille Poussel, Toulon university (P. Helluy)
• President of the PhD jury of Anil Gün, ICUBE, Strasbourg university (P. Helluy)
• HDR warrant of L. Navoret, Strasbourg university (P. Helluy)
• Reviewer for the PhD Thesis of Wageesha Manamperi, The Australian National University, Canberra (A. Deleforge)

10.3.1 Productions (articles, videos, podcasts, serious games, ...)

• Article: "Des modèles mathématiques pour simuler des problèmes d’ingénierie" to appear in the collective book M. Bouzeghoub, M. Daydé et C. Jutten (dir.) - Le calcul à découvert, CNRS Éditions (P. Helluy)

10.3.2 Participation in Live events

• Speed-meetings between high school girls and scientific women for the event "Sciences, un métier de femmes", March (C. Courtès)
• Scientific workshop "modelling a chocolate cake" for high school students, March (C. Courtès)
• Research talks aimed at high school students, June (C. Courtès)
• Co-organization of the mathematics and computing week "Les Cigognes" for high school girls, since 2022 (C. Courtès)
• Scientific computing workshop for middle school students during the "Semaine des Mathématiques 2024" (M. Boileau)
• 1h mediation sessions on digital science research in 7 classes of 2nd in the highschools of Bouxwiller and Bischwiller for the programm Chiche SNT (A. Deleforge)

10.3.3 Others science outreach relevant activities

• Journées des Universités et des formations post-bac, présentation des filières à l'UFR maths-info, Janvier (C. Courtès, J. Aghili)

International journals

• 2 articleJ.Joubine Aghili, R.Romain Hild, V.Victor Michel-Dansac, V.Vincent Vigon and E.Emmanuel Franck. Accelerating the convergence of Newton's method for nonlinear elliptic PDEs using Fourier neural operators.Communications in Nonlinear Science and Numerical Simulation1402January 2025, 108434HALDOIback to text
• 3 articleA.Amaury Bélières--Frendo, E.Emmanuel Franck, V.Victor Michel-Dansac and Y.Yannick Privat. Volume-Preserving Geometric Shape Optimization of the Dirichlet Energy Using Variational Neural Networks.Neural Networks184April 2025, 106957HALDOIback to text
• 4 articleC.Christophe Berthon and V.Victor Michel-Dansac. A fully well-balanced hydrodynamic reconstruction.Journal of Numerical MathematicsMarch 2024HALDOIback to text
• 5 articleW.Walter Boscheri and A.Andrea Thomann. A structure-preserving semi-implicit IMEX finite volume scheme for ideal magnetohydrodynamics at all Mach and Alfvén numbers.Journal of Scientific ComputingJuly 2024HALDOIback to text
• 6 articleM.Mohamed Boujoudar, E.Emmanuel Franck, P.Philippe Hoch, C.Clément Lasuen, Y.Yoann Le Henaff and P.Paul Paragot. A composite finite volume scheme for the Euler equations with source term on unstructured meshes.ESAIM: Proceedings and Surveys2024, 1-22In press. HALback to text
• 7 articleC.Clémentine Courtès, M.Matthieu Boileau, R.Raphaël Côte, P. A.Paul Antoine Hervieux and G.Giovanni Manfredi. Micromagnetic simulations of the size dependence of the Curie temperature in ferromagnetic nanowires and nanolayers.Journal of Magnetism and Magnetic Materials598April 2024, 172040HALDOISoftware Heritageback to text
• 8 articleC.Clément Flint and P.Philippe Helluy. Reducing the memory usage of Lattice-Boltzmann schemes with a DWT-based compression.ESAIM: Proceedings and Surveys77November 2024, 79-99HALDOIback to text
• 9 articleC.Clément Flint, A.Atoli Huppé, P.Philippe Helluy, B.Bérenger Bramas and S.Stéphane Genaud. Using the Discrete Wavelet Transform for Lossy On-the-Fly Compression of GPU Fluid Simulations.International Journal for Numerical Methods in FluidsOctober 2024HALDOIback to text
• 10 articleE.Emmanuel Franck, V.Victor Michel-Dansac and L.Laurent Navoret. Approximately well-balanced Discontinuous Galerkin methods using bases enriched with Physics-Informed Neural Networks.Journal of Computational PhysicsSeptember 2024HALDOIback to text
• 11 articleP.Pierre Gerhard, P.Philippe Helluy, V.Victor Michel-Dansac and B.Bruno Weber. Parallel kinetic schemes for conservation laws, with large time steps.Journal of Scientific Computing9912024HALDOIback to textback to text
• 12 articleK.Kévin Guillon, R.Romane Hélie and P.Philippe Helluy. Stability analysis of the vectorial lattice-Boltzmann method.ESAIM: Proceedings and Surveys77November 2024, 46-78HALDOIback to text
• 13 articleG.Gauthier Lazare, Q.Qingqing Feng, P.Philippe Helluy, J.-M.Jean-Marc Hérard, F.Frank Hülsemann and S.Stéphane Pujet. Maximum principle for the mass fraction in a system with two mass balance equations.Comptes Rendus. Mécanique3522024, 81-98HALDOIback to text
• 14 articleM.Mária Lukáčová-Medvid'ová, I.Ilya Peshkov and A.Andrea Thomann. An implicit-explicit solver for a two-fluid single-temperature model.Journal of Computational Physics498February 2024, 112696HALDOIback to text

Doctoral dissertations and habilitation theses

• 15 thesisL.Laurent Navoret. Models and numerical methods for multiscale transport problems.Université de StrasbourgDecember 2024HAL
• 16 thesisT.Tom Sprunck. Can one hear the shape of a room ?: Room Geometry Reconstruction from Acoustic Measurements usingSuper-Resolution and Shape Optimization.Université de strasbourgDecember 2024HALback to text

Reports & preprints

• 17 miscJ.Joubine Aghili, J. Z.Joy Zialesi Atokple, M.Marie Billaud-Friess, G.Guillaume Garnier, O.Olga Mula and D. N.Djahou Norbert Tognon. A Dynamical Neural Galerkin Scheme for Filtering Problems.January 2024HALback to text
• 18 miscT.Thomas Bellotti, P.Philippe Helluy and L.Laurent Navoret. Fourth-order entropy-stable lattice Boltzmann schemes for hyperbolic systems.March 2024HALback to text
• 19 miscC.Christophe Berthon, V.Victor Michel-Dansac and A.Andrea Thomann. An entropy-stable and fully well-balanced scheme for the Euler equations with gravity.June 2024HALback to textback to text
• 20 miscM.Mirco Ciallella, L.Lorenzo Micalizzi, V.Victor Michel-Dansac, P.Philipp Öffner and D.Davide Torlo. A high-order, fully well-balanced, unconditionally positivity-preserving finite volume framework for flood simulations.February 2024HALback to text
• 21 miscS.Stéphane Clain, E.Emmanuel Franck and V.Victor Michel-Dansac. Structural schemes for Hamiltonian systems.January 2025HAL
• 22 miscR.Raphaël Côte, C.Clémentine Courtès, G.Guillaume Ferriere, L.Ludovic Godard-Cadillac and Y.Yannick Privat. Existence and Uniqueness of Domain Walls for Notched Ferromagnetic Nanowires.December 2024HALback to text
• 23 miscM.Michael Dumbser, M.Mária Lukáčová-Medvid'ová and A.Andrea Thomann. Convergence of a Hyperbolic Thermodynamically Compatible Finite Volume scheme for the Euler equations.December 2024HALback to text
• 24 miscM.Michel Duprez, V.Vanessa Lleras, A.Alexei Lozinski, V.Vincent Vigon and K.Killian Vuillemot. Phi-FD : A well-conditioned finite difference method inspired by phi-FEM for general geometries on elliptic PDEs.October 2024HALback to text
• 25 miscC.Camilla Fiorini, C.Clément Flint, L.Louis Fostier, E.Emmanuel Franck, R.Reyhaneh Hashemi, V.Victor Michel-Dansac and W.Wassim Tenachi. Generalizing the SINDy approach with nested neural networks.April 2024HALback to text
• 26 miscE.Emmanuel Franck, I.Ibtissem Lannabi, Y.Youssouf Nasseri, L.Laurent Navoret, G.Giuseppe Parasiliti Rantone and G.Guillaume Steimer. Hyperbolic reduced model for Vlasov-Poisson equation with Fokker-Planck collision.April 2024HAL
• 27 reportS.Stéphane Lanteri, D.Daria Hrebenshchykova, V.Victor Michel-Dansac and V.Victorita Dolean. Multilevel and Distributed Physics-Informed Neural Networks for the Helmholtz Equation.RR-9571Inria & Université Cote d'Azur, CNRS, I3S, Sophia Antipolis, FranceOctober 2024HAL
• 28 miscV.Victor Michel-Dansac and A.Andrea Thomann. An entropy-stable and fully well-balanced scheme for the Euler equations with gravity. II: General equations of state.October 2024HALback to textback to text
• 29 miscR. S.Robin San Roman, P.Pierre Fernandez, A.Antoine Deleforge, Y.Yossi Adi and R.Romain Serizel. Latent Watermarking of Audio Generative Models.2024HALDOIback to text
• 30 miscT.Tom Sprunck, A.Antoine Deleforge, Y.Yannick Privat and C.Cédric Foy. Fully Reversing the Shoebox Image Source Method: From Impulse Responses to Room Parameters.May 2024HALback to text