Neural semi-Lagrangian method for high-dimensional advection-diffusion problems
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917101672857600 |
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| author | Franck, Emmanuel Michel-Dansac, Victor Navoret, Laurent Vigon, Vincent |
| author_facet | Franck, Emmanuel Michel-Dansac, Victor Navoret, Laurent Vigon, Vincent |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20715 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Neural semi-Lagrangian method for high-dimensional advection-diffusion problems Franck, Emmanuel Michel-Dansac, Victor Navoret, Laurent Vigon, Vincent Numerical Analysis 2020: 65M25, 76R05, 65M15, 68T07 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. |
| title | Neural semi-Lagrangian method for high-dimensional advection-diffusion problems |
| topic | Numerical Analysis 2020: 65M25, 76R05, 65M15, 68T07 |
| url | https://arxiv.org/abs/2504.20715 |