Neural semi-Lagrangian method for high-dimensional advection-diffusion problems

Abstract

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 present several high-dimensional numerical experiments to assess the performance of our approach, and compare it to other neural methods.

References in corpus (23)

• A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks
• PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs
• A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equations
• Tackling the Curse of Dimensionality with Physics-Informed Neural Networks
• Physics Informed Neural Networks for Simulating Radiative Transfer
• A Forward semi-Lagrangian Method for the Numerical Solution of the Vlasov Equation
• Multilevel domain decomposition-based architectures for physics-informed neural networks
• A Hamiltonian splitting for the Vlasov-Maxwell system
• A semi-Lagrangian Vlasov solver in tensor train format
• Physics-informed neural networks for solving parametric magnetostatic problems
• A mass, momentum, and energy conservative dynamical low-rank scheme for the Vlasov equation
• Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks
• Physics-Informed Neural Networks for Time-Domain Simulations: Accuracy, Computational Cost, and Flexibility
• Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations
• Stable interpolation with isotropic and anisotropic Gaussians using Hermite generating function
• A massively parallel semi-Lagrangian solver for the six-dimensional Vlasov-Poisson equation
• PirateNets: Physics-informed Deep Learning with Residual Adaptive Networks
• A fully semi-Lagrangian discretization for the 2D Navier--Stokes equations in the vorticity--streamfunction formulation
• Tensor-decomposition-based A Priori Surrogate (TAPS) modeling for ultra large-scale simulations
• Asymptotic preserving scheme for anisotropic elliptic equations with deep neural network
• Error-correcting neural networks for semi-Lagrangian advection in the level-set method
• Semi-Lagrangian discontinuous Galerkin schemes for some first and second-order partial differential equations
• A learned conservative semi-Lagrangian finite volume scheme for transport simulations