Victor Michel-Dansac

Stéphane Clain, Emmanuel Franck, Victor Michel-Dansac

We present an adaptation of the so-called structural method \cite{CMM23} for Hamiltonian systems, and redesign the method for this specific context, which involves two coupled differential systems. Structural schemes decompose the problem into two sets of equations: the physical equations, which describe the local dynamics of the system, and the structural equations, which only involve the discretization on a very compact stencil. They have desirable properties, such as unconditional stability or high-order accuracy. We first give a general description of the scheme for the scalar case (which corresponds to e.g. spring-mass interactions or pendulum motion), before extending the technique to the vector case (treating e.g. the $n$-body system). The scheme is also written in the case of a non-separable system (e.g. a charged particle in an electromagnetic field). We give numerical evidence of the method's efficiency, its capacity to preserve invariant quantities such as the total energy, and draw comparisons with the traditional symplectic methods.

Victor Fournet, Martin Campos Pinto, Emmanuel Franck et al.

We present a novel δf particle-in-cell (PIC) method for the kinetic simulation of electrostatic plasmas in which the bulk density, acting as a control variate, is evolved using symplectic neural networks (SympNets). The SympNets are used as an approximation of the backward flow and trained using the particle trajectories. We introduce a periodic variant of the SympNet architecture that encodes the spatial periodicity of the problem into the network itself. We validate the approach with numerical results in 1D1V and 3D3V for the Vlasov-Poisson system.

León Ávila, Manuel Castro, Victor Michel-Dansac et al.

This work presents a novel family of well-balanced numerical schemes for hyperbolic systems of balance laws based on the kinetic relaxation approach. The method begins by transforming the original non-linear system into a linearized kinetic system with an increased number of variables. In this framework, non-linearities are shifted to the source term, and the connection to the macroscopic variables is maintained via a projection operator related to the Maxwellian equilibrium states. These relaxed systems are typically solved using splitting techniques, where the evolution is decomposed into two distinct steps: first, the transport stage solving the source-free kinetic system, and second, the relaxation stage governed by an ordinary differential equation where the time derivative is balanced by the kinetic source term. The primary objective is to design schemes that accurately preserve the steady-state solutions of the original macroscopic equations, noting that these do not necessarily coincide with the trivial equilibria of the relaxed system. This well-balanced property is achieved through an appropriate formulation of the kinetic system combined with consistent reconstruction operators. This approach ensures that the balance is maintained throughout both the transport stage and the projection stage, where the treatment of the macroscopic source term is carefully integrated. We demonstrate that this framework can be implemented using both Finite Volume (FV) and Semi-Lagrangian (SL) techniques in the transport step. We develop and analyze these schemes for the Finite Volume case in both explicit and implicit formulations. Results are validated through numerical benchmarks, including the Burgers equation with a source term, 1D Shallow Water Equations (SWE) with bathymetry preserving non-trivial moving water equilibria and the Euler equations under external potentials.

Camilla Fiorini, Clément Flint, Louis Fostier et al.

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.

Victorita Dolean, Daria Hrebenshchykova, Stéphane Lanteri et al.

Accurately simulating wave propagation is crucial in fields such as acoustics, electromagnetism, and seismic analysis. Traditional numerical methods, like finite difference and finite element approaches, are widely used to solve governing partial differential equations (PDEs) such as the Helmholtz equation. However, these methods face significant computational challenges when applied to high-frequency wave problems in complex two-dimensional domains. This work investigates Finite Basis Physics-Informed Neural Networks (FBPINNs) and their multilevel extensions as a promising alternative. These methods leverage domain decomposition, partitioning the computational domain into overlapping sub-domains, each governed by a local neural network. We assess their accuracy and computational efficiency in solving the Helmholtz equation for the homogeneous case, demonstrating their potential to mitigate the limitations of traditional approaches.