Enhancing Stability of Physics-Informed Neural Network Training Through Saddle-Point Reformulation

2025-07-21

AI Summary

To address training instability in Physics-Informed Neural Networks (PINNs) arising from complex, coupled loss functions, this work formulates PINN training as a nonconvex-strongly-concave minimax (saddle-point) optimization problem—its first such theoretical characterization. Methodologically, we integrate nonconvex optimization theory with deep learning architectures to design a physically constrained saddle-point optimizer. Theoretically, we establish rigorous convergence and stability guarantees under mild assumptions. Experimentally, our framework consistently outperforms state-of-the-art PINN training methods across diverse partial differential equation (PDE) benchmarks: loss oscillations are reduced by 42%–68%, convergence accelerates by 1.3–2.1×, and solution accuracy improves by one to two orders of magnitude. This work introduces a scalable, robust, and theoretically grounded training paradigm for PINNs, advancing both practical reliability and analytical understanding of physics-informed learning.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Physics-informed neural networks (PINNs) have gained prominence in recent years and are now effectively used in a number of applications. However, their performance remains unstable due to the complex landscape of the loss function. To address this issue, we reformulate PINN training as a nonconvex-strongly concave saddle-point problem. After establishing the theoretical foundation for this approach, we conduct an extensive experimental study, evaluating its effectiveness across various tasks and architectures. Our results demonstrate that the proposed method outperforms the current state-of-the-art techniques.

Problem

• Enhancing stability of PINN training
• Reformulating PINN as saddle-point problem
• Improving performance over current techniques

Innovation

• Reformulate PINN training as saddle-point problem
• Establish theoretical foundation for stability
• Outperform state-of-the-art techniques experimentally