Boosting Generalization in Parametric PDE Neural Solvers through Adaptive Conditioning
Keywords: Deep Learning, Parametric PDEs, Meta-Learning, physics-aware
TL;DR: We propose a $1^st$ order low-rank meta-learning model for generalizing Neural PDE dynamics solvers to unseen dynamics. Our approach is general, it can incorporate knowledge from known physics if available.
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. *Project page*: https://geps-project.github.io
Primary Area: Machine learning for physical sciences (for example: climate, physics)
Submission Number: 19446
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