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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.23889 |
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| _version_ | 1866916472717049856 |
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| author | Koupaï, Armand Kassaï Benet, Jorge Mifsut Yin, Yuan Vittaut, Jean-Noël Gallinari, Patrick |
| author_facet | Koupaï, Armand Kassaï Benet, Jorge Mifsut Yin, Yuan Vittaut, Jean-Noël Gallinari, Patrick |
| contents | Solving parametric partial differential equations (PDEs) presents significant challenges for data-driven methods due to the sensitivity of spatio-temporal dynamics to variations in PDE parameters. Machine learning approaches often struggle to capture this variability. To address this, data-driven approaches learn parametric PDEs by sampling a very large variety of trajectories with varying PDE parameters. We first show that incorporating conditioning mechanisms for learning parametric PDEs is essential and that among them, $\textit{adaptive conditioning}$, allows stronger generalization. As existing adaptive conditioning methods do not scale well with respect to the number of parameters to adapt in the neural solver, we propose GEPS, a simple adaptation mechanism to boost GEneralization in Pde Solvers via a first-order optimization and low-rank rapid adaptation of a small set of context parameters. We demonstrate the versatility of our approach for both fully data-driven and for physics-aware neural solvers. Validation performed on a whole range of spatio-temporal forecasting problems demonstrates excellent performance for generalizing to unseen conditions including initial conditions, PDE coefficients, forcing terms and solution domain. $\textit{Project page}$: https://geps-project.github.io |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23889 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | GEPS: Boosting Generalization in Parametric PDE Neural Solvers through Adaptive Conditioning Koupaï, Armand Kassaï Benet, Jorge Mifsut Yin, Yuan Vittaut, Jean-Noël Gallinari, Patrick Machine Learning Artificial Intelligence Solving parametric partial differential equations (PDEs) presents significant challenges for data-driven methods due to the sensitivity of spatio-temporal dynamics to variations in PDE parameters. Machine learning approaches often struggle to capture this variability. To address this, data-driven approaches learn parametric PDEs by sampling a very large variety of trajectories with varying PDE parameters. We first show that incorporating conditioning mechanisms for learning parametric PDEs is essential and that among them, $\textit{adaptive conditioning}$, allows stronger generalization. As existing adaptive conditioning methods do not scale well with respect to the number of parameters to adapt in the neural solver, we propose GEPS, a simple adaptation mechanism to boost GEneralization in Pde Solvers via a first-order optimization and low-rank rapid adaptation of a small set of context parameters. We demonstrate the versatility of our approach for both fully data-driven and for physics-aware neural solvers. Validation performed on a whole range of spatio-temporal forecasting problems demonstrates excellent performance for generalizing to unseen conditions including initial conditions, PDE coefficients, forcing terms and solution domain. $\textit{Project page}$: https://geps-project.github.io |
| title | GEPS: Boosting Generalization in Parametric PDE Neural Solvers through Adaptive Conditioning |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2410.23889 |