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arXiv · 2410.23889

GEPS: Boosting Generalization in Parametric PDE Neural Solvers through Adaptive Conditioning

Armand Kassaï Koupaï·Jorge Mifsut Benet·Yuan Yin·Jean-Noël Vittaut·Patrick Gallinari
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Abstract

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

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Armand Kassaï Koupaï, Jorge Mifsut Benet, Yuan Yin, Jean-Noël Vittaut, Patrick Gallinari. 2024-11-08. GEPS: Boosting Generalization in Parametric PDE Neural Solvers through Adaptive Conditioning. https://arxiv.org/abs/2410.23889

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