GEPS: Boosting Generalization in Parametric PDE Neural Solvers through Adaptive Conditioning
This work addresses the problem of poor generalization in data-driven PDE solvers for researchers and practitioners in computational science, representing an incremental improvement over existing adaptive conditioning methods.
The paper tackles the challenge of generalizing neural solvers for parametric PDEs by proposing GEPS, an adaptive conditioning method that improves generalization to unseen conditions, achieving excellent performance across various spatio-temporal forecasting problems.
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