This work is devoted to the numerical approximation of high-dimensional advection-diffusion equations. It is well-known that classical methods, such as the finite volume method, suffer from the curse of dimensionality, and that their time step is constrained by a stability condition. The semi-Lagrangian method is known to overcome the stability issue, while recent time-discrete neural network-based approaches overcome the curse of dimensionality. In this work, we propose a novel neural semi-Lagrangian method that combines these last two approaches. It relies on projecting the initial condition onto a finite-dimensional neural space, and then solving an optimization problem, involving the backwards characteristic equation, at each time step. It is particularly well-suited for implementation on GPUs, as it is fully parallelizable and does not require a mesh. We provide rough error estimates, and present several high-dimensional numerical experiments to assess the performance of our approach, and compare it to other neural methods.

This work focuses on learning non-canonical Hamiltonian dynamics from data, where long-term predictions require the preservation of structure both in the learned model and in numerical schemes. Previous research focused on either facet, respectively with a potential-based architecture and with degenerate variational integrators, but new issues arise when combining both. In experiments, the learnt model is sometimes numerically unstable due to the gauge dependency of the scheme, rendering long-time simulations impossible. In this paper, we identify this problem and propose two different training strategies to address it, either by directly learning the vector field or by learning a time-discrete dynamics through the scheme. Several numerical test cases assess the ability of the methods to learn complex physical dynamics, like the guiding center from gyrokinetic plasma physics.

High-fidelity quantum control is a cornerstone of scalable quantum technologies. We introduce a shooting-based optimization framework that generates smooth, experimentally realistic control pulses for implementing quantum gates in discrete quantum systems. Through numerical simulations on realistic architectures inspired by single-molecule magnets, we demonstrate that our method efficiently decomposes target quantum operations into electric pulse sequences while outperforming the widely used GRAPE algorithm.

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.

Context. In current cosmological simulations, radiative transfer modules generally rely on the M1 approximation, which has some glaring flaws related to its fluid-like behaviour, such as spurious pseudo-sources and loss of directionality when radiation fronts from different directions collide. Pn, another moment-based model, may correct these issues. Aims. We aim to test Pn in an astrophysical setting and compare it to M1, in order to see if it can indeed correct M1’s issues. Also, we want to use Pn’s solutions to better pinpoint and quantify M1 errors. Methods. We implemented a Pn radiation transport method, coupled it to a photo-thermo-chemistry module to account for the interaction of ionising radiation with the hydrogen gas, and benchmarked it using tests for radiative transfer models comparison in astrophysics. Results. We implemented a Pn radiation transport method, coupled it to a photo-thermo-chemistry module to account for the interaction of ionising radiation with the hydrogen gas, and benchmarked it using tests for radiative transfer models comparison in astrophysics. Conclusions. We find that high-order Pn (e.g. P9) indeed corrects M1’s flaws, while faring as well or even better in some aspects in the tests, in particular when directionality is important or colliding radiation fronts occur. By comparing P9 and M1 radiation fields in an idealised cosmological test case, we highlight a new, thus far unreported artefact of M1, the ‘dark sombrero’. A dark sombrero appears as a spherical photon-deficit shell around the source, at typically one third to one quarter of the distance to the next source. The photon density in dark sombreros can be underestimated by a factor of up to 2–3. These artefacts occur in regions where a source’s radiation field connects with that of another source or group of sources. These basic properties (the position and amplitude) of the dark sombreros may depend on the sources’ relative intensities, positions, and spatial resolution, although we have not been able to test this in detail in this study. Moreover, the larger-scale M1 photon density also exhibits spurious features, enhancing or reducing photon density in various regions. We use a small reionisation-like test simulation to characterise the relative error in hydrogen neutral fractions between M1 and P9. The relative error is well represented by a Gaussian with a dispersion of 0.27 dex in log10(xHI). Both aspects are likely related to the photons’ collisional behaviour in M1.

In this work, we present a study combining two approaches in the context of solving PDEs: the continuous finite element method (FEM) and more recent techniques based on neural networks. In recent years, physics-informed neural networks (PINNs) have become particularly interesting for rapidly solving PDEs, especially in high dimensions. However, their lack of accuracy can be a significant drawback in this context, hence the interest in combining them with FEM, for which error estimates are already known. The complete pipeline proposed here consists in modifying the classical FEM approximation spaces by taking information from a prior, chosen as the prediction of a neural network. On the one hand, this combination improves and certifies the prediction of neural networks, to obtain a fast and accurate solution. On the other hand, error estimates are proven, showing that such strategies outperform classical ones by a factor that depends only on the quality of the prior. We validate our approach with numerical results performed on parametric problems with 1D, 2D and 3D geometries. These experiments demonstrate that to achieve a given accuracy, a coarser mesh can be used with our enriched FEM compared to the standard FEM, leading to reduced computational time, particularly for parametric problems.

Hamiltonian particle-based simulations of plasma dynamics are inherently computationally intensive, primarily due to the large number of particles required to obtain accurate solutions. This challenge becomes even more acute in many-query contexts, where numerous simulations must be conducted across a range of time and parameter values. Consequently, it is essential to construct reduced order models from such discretizations to significantly lower computational costs while ensuring validity across the specified time and parameter domains. Preserving the Hamiltonian structure in these reduced models is also crucial, as it helps maintain long-term stability. In this paper, we introduce a nonlinear, non-intrusive, data-driven model order reduction method for the 1D-1V Vlasov–Poisson system, discretized using a Hamiltonian Particle-In-Cell scheme. Our approach relies on a two-step projection framework: an initial linear projection based on the Proper Symplectic Decomposition, followed by a nonlinear projection learned via an autoencoder neural network. The reduced dynamics are then modeled using a Hamiltonian neural network. The offline phase of the method is split into two stages: first, constructing the linear projection using full-order model snapshots; second, jointly training the autoencoder and the Hamiltonian neural network to simultaneously learn the encoder-decoder mappings and the reduced dynamics. We validate the proposed method on several benchmarks, including Landau damping and two-stream instability. The results show that our method has better reduction properties than standard linear Hamiltonian reduction methods.

The reduction of Hamiltonian systems aims to build smaller reduced models, valid over a certain range of time and parameters, in order to reduce computing time. By maintaining the Hamiltonian structure in the reduced model, certain long-term stability properties can be preserved. In this paper, we propose a non-linear reduction method for models coming from the spatial discretization of partial differential equations: it is based on convolutional auto-encoders and Hamiltonian neural networks. Their training is coupled in order to simultaneously learn the encoder-decoder operators and the reduced dynamics. Several test cases on non-linear wave dynamics show that the method has better reduction properties than standard linear Hamiltonian reduction methods.

It is well known that Newton’s method can have trouble converging if the initial guess is too far from the solution. Such a problem particularly occurs when this method is used to solve nonlinear elliptic partial differential equations (PDEs) discretized via finite differences. This work focuses on accelerating Newton’s method convergence in this context. We seek to construct a mapping from the parameters of the nonlinear PDE to an approximation of its discrete solution, independently of the mesh resolution. This approximation is then used as an initial guess for Newton’s method. To achieve these objectives, we elect to use a Fourier neural operator (FNO). The loss function is the sum of a data term (i.e., the comparison between known solutions and outputs of the FNO) and a physical term (i.e., the residual of the PDE discretization). Numerical results, in one and two dimensions, show that the proposed initial guess accelerates the convergence of Newton’s method by a large margin compared to a naive initial guess, especially for highly nonlinear and anisotropic problems, with larger gains on coarse grids.

We present an adaptation of the so-called structural method 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.

In this paper, we propose a method for constructing a neural network viscosity in order to reduce the non-physical oscillations generated by high-order Discontiuous Galerkin (DG) methods. To this end, the problem is reformulated as an optimal control problem for which the control is the viscosity function and the cost function involves comparison with a reference solution after several compositions of the scheme. The learning process is strongly based on gradient backpropagation tools. Numerical simulations show that the artificial viscosities constructed in this way are just as good or better than those used in the literatur

This paper proposes a reduced model to simulate the one-dimensional Vlasov-Poisson equation with the non-linear Fokker-Planck operator. The model provides the space-time dynamics of a few macroscopic quantities constructed following the Reduced Order Method (ROM) in the velocity variable: the compression is thus applied to the semi-discretization of the Vlasov equation. To gain efficiency, a Discrete Empirical Interpolation Method (DEIM) is applied to the compressed non-linear Fokker-Planck operator. The size of the resulting reduced model is chosen empirically according to the Knudsen number. Furthermore, we propose a correction to the reduced collision operator that ensures the reduced moments to satisfy an Euler-type system. Numerical simulations of the reduced model show that the model can capture the plasma dynamics in different collisional regimes and initial conditions at a low cost.

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.

In this work we focus on an adaptation of the method described in [1] 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 Tajchman in an unpublished work [2] – focused mainly on the 1D case – 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.

Modelling epidemics using classical population-based models suffers from shortcomings that so-called individual-based models are able to overcome, as they are able to take into account heterogeneity features, such as super-spreaders, and describe the dynamics involved in small clusters. In return, such models often involve large graphs which are expensive to simulate and difficult to optimize, both in theory and in practice. By combining the reinforcement learning philosophy with reduced models, we propose a numerical approach to determine optimal health policies for a stochastic individual-based model taking into account heterogeneity in the population. More precisely, we introduce a deterministic reduced population-based model involving a neural network, designed to faithfully mimic the local dynamics of the more complex individual-based model. Then the optimal control is determined by sequentially training the network until an optimal strategy for the population-based model succeeds in also containing the epidemic when simulated on the individual-based model. After describing the practical implementation of the method, several numerical tests are proposed to demonstrate its ability to determine controls for models with contact heterogeneity.

How to free a road from vehicle traffic as efficiently as possible and in a given time, in order to allow for example the passage of emergency vehicles? We are interested in this question which we reformulate as an optimal control problem. We consider a macroscopic road traffic model on networks, semi-discretized in space and decide to give ourselves the possibility to control the flow at junctions. Our target is to smooth the traffic along a given path within a fixed time. A parsimony constraint is imposed on the controls, in order to ensure that the optimal strategies are feasible in practice. We perform an analysis of the resulting optimal control problem, proving the existence of an optimal control and deriving optimality conditions, which we rewrite as a single functional equation. We then use this formulation to derive a new mixed algorithm interpreting it as a mix between two methods: a descent method combined with a fixed point method allowing global perturbations. We verify with numerical experiments the efficiency of this method on examples of graphs, first simple, then more complex. We highlight the efficiency of our approach by comparing it to standard methods. We propose an open source code implementing this approach in the Julia language.

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.

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 prototypal problem: minimizing the so-called Dirichlet energy with respect to the domain under a volume constraint, involving Poisson’s equation in R2. 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.