RoPINN: Region Optimized Physics-Informed Neural Networks

TL;DR

RoPINN introduces a novel region-based optimization paradigm for physics-informed neural networks, improving their accuracy on PDE solutions by extending training from points to continuous regions, thereby reducing generalization error.

Contribution

This paper proposes and theoretically analyzes a new region optimization paradigm for PINNs, leading to the development of RoPINN, which enhances PDE solving accuracy through continuous neighborhood training.

Findings

RoPINN consistently improves PINN performance across various PDEs.

Region optimization reduces generalization error in PINNs.

The Monte Carlo sampling method effectively implements the region-based training.

Abstract

Physics-informed neural networks (PINNs) have been widely applied to solve partial differential equations (PDEs) by enforcing outputs and gradients of deep models to satisfy target equations. Due to the limitation of numerical computation, PINNs are conventionally optimized on finite selected points. However, since PDEs are usually defined on continuous domains, solely optimizing models on scattered points may be insufficient to obtain an accurate solution for the whole domain. To mitigate this inherent deficiency of the default scatter-point optimization, this paper proposes and theoretically studies a new training paradigm as region optimization. Concretely, we propose to extend the optimization process of PINNs from isolated points to their continuous neighborhood regions, which can theoretically decrease the generalization error, especially for hidden high-order constraints of PDEs. A practical training algorithm, Region Optimized PINN (RoPINN), is seamlessly derived from this new paradigm, which is implemented by a straightforward but effective Monte Carlo sampling method. By calibrating the sampling process into trust regions, RoPINN finely balances optimization and generalization error. Experimentally, RoPINN consistently boosts the performance of diverse PINNs on a wide range of PDEs without extra backpropagation or gradient calculation. Code is available at this repository: https://github.com/thuml/RoPINN.