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

• URL: http://arxiv.org/abs/2410.23889v1
• Date: Thu, 31 Oct 2024 12:51:40 GMT
• Title: GEPS: Boosting Generalization in Parametric PDE Neural Solvers through Adaptive Conditioning
• Authors: Armand Kassaï Koupaï, Jorge Misfut Benet, Yuan Yin, Jean-Noël Vittaut, Patrick Gallinari,
• Abstract summary: Data-driven approaches learn parametric PDEs by incorporating a very large variety of trajectories with varying PDE parameters. GEPS is a simple adaptation mechanism to boost GEneralization in Pde solvers. We demonstrate the versatility of our approach for both fully data-driven and for physics-aware neural solvers.
• Score: 14.939978372699084
• License:
• 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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