Over-PINNs: Enhancing Physics-Informed Neural Networks via Higher-Order Partial Derivative Overdetermination of PDEs

TL;DR

Over-PINNs enhances physics-informed neural networks by incorporating higher-order derivatives as additional constraints, significantly improving accuracy in solving PDEs with minimal extra computational cost.

Contribution

The paper introduces a novel Over-PINNs framework that uses higher-order derivatives to impose extra physical constraints, improving PDE solution accuracy.

Findings

Achieves higher accuracy in PDE solutions.

Maintains low additional computational costs.

Demonstrates versatility across different PDE types.

Abstract

Partial differential equations (PDEs) serve as the cornerstone of mathematical physics. In recent years, Physics-Informed Neural Networks (PINNs) have significantly reduced the dependence on large datasets by embedding physical laws directly into the training of neural networks. However, when dealing with complex problems, the accuracy of PINNs still has room for improvement. To address this issue, we introduce the Over-PINNs framework, which leverages automatic differentiation (AD) to generate higher-order auxiliary equations that impose additional physical constraints. These equations are incorporated as extra loss terms in the training process, effectively enhancing the model's ability to capture physical information through an "overdetermined" approach. Numerical results illustrate that this method exhibits strong versatility in solving various types of PDEs. It achieves a significant improvement in solution accuracy without incurring substantial additional computational costs.