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

2025/06/06 by W. Huo, Qiang He, Huo, Wenxuan +5
Physics and Astronomy · Mathematics · Materials Science · #Model Reduction and Neural Networks #Numerical methods for differential equations #Machine Learning in Materials Science

paper · pdf · doi:10.48550/arxiv.2506.05918

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.

Citations

• Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations
• Highly accurate protein structure prediction with AlphaFold
• Encoding physics to learn reaction-diffusion processes
• Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks
• Extended Physics-Informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition Based Deep Learning Framework for Nonlinear Partial Differential Equations
• hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition
• Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems
• Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems
• Physics-Guided Machine Learning for Scientific Discovery: An Application in Simulating Lake Temperature Profiles
• Understanding and mitigating gradient pathologies in physics-informed neural networks
• PPINN: Parareal physics-informed neural network for time-dependent PDEs
• DeepXDE: A Deep Learning Library for Solving Differential Equations
• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• Automatic differentiation in machine learning: a survey

Related

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• Revisiting PINNs: Generative Adversarial Physics-informed Neural Networks and Point-weighting Method