Pré-Publication, Document De Travail Année : 2025

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

Résumé

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.

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Dates et versions

hal-05051195 , version 1 (29-04-2025)
hal-05051195 , version 2 (03-06-2025)
hal-05051195 , version 3 (07-10-2025)
hal-05051195 , version 4 (09-12-2025)

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  • HAL Id : hal-05051195 , version 3

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Emmanuel Franck, Victor Michel-Dansac, Laurent Navoret, Vincent Vigon. Neural semi-Lagrangian method for high-dimensional advection-diffusion problems. 2025. ⟨hal-05051195v3⟩
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