Learning a Neural Solver for Parametric PDE to Enhance Physics-Informed Methods

TL;DR

This paper introduces a neural solver trained on data to improve the efficiency and stability of physics-informed deep learning for solving parametric PDEs, enabling faster convergence and broader applicability.

Contribution

It proposes a physics-informed iterative algorithm that learns to adapt to each PDE instance and extends to parametric PDEs, enhancing stability and generalization.

Findings

Accelerates PDE solving with learned iterative algorithms

Stabilizes training of physics-informed models

Effective on multiple datasets for parametric PDEs

Abstract

Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable training. These challenges arise particularly from the ill-conditioning of the optimization problem caused by the differential terms in the loss function. To address these issues, we propose learning a solver, i.e., solving PDEs using a physics-informed iterative algorithm trained on data. Our method learns to condition a gradient descent algorithm that automatically adapts to each PDE instance, significantly accelerating and stabilizing the optimization process and enabling faster convergence of physics-aware models. Furthermore, while traditional physics-informed methods solve for a single PDE instance, our approach extends to parametric PDEs. Specifically, we integrate the physical loss gradient with PDE parameters, allowing our method to solve over a distribution of PDE parameters, including coefficients, initial conditions, and boundary conditions. We demonstrate the effectiveness of our approach through empirical experiments on multiple datasets, comparing both training and test-time optimization performance. The code is available at https://github.com/2ailesB/neural-parametric-solver.