ProPINN: Demystifying Propagation Failures in Physics-Informed Neural Networks

2025-02-02

AI Summary

Physics-informed neural networks (PINNs) suffer from propagation failure when solving partial differential equations (PDEs)—a phenomenon wherein boundary and initial conditions fail to propagate effectively into the interior of the computational domain. This stems from low inter-point gradient correlation among neighboring collocation points and the absence of regional coordination in conventional pointwise architectures. This work formally defines propagation failure for the first time and proposes ProPINN, a novel regional gradient aggregation framework. ProPINN introduces a weighted regional gradient aggregation mechanism, establishes a quantifiable diagnostic criterion for propagation capability, and theoretically proves the intrinsic link between gradient correlation and information propagation efficacy. Experiments demonstrate that ProPINN reliably eliminates typical propagation failure and achieves a 46% relative accuracy improvement over state-of-the-art Transformer-based baselines across diverse PDE benchmarks.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Learning to Search

Application Category

• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Physics-informed neural networks (PINNs) have earned high expectations in solving partial differential equations (PDEs), but their optimization usually faces thorny challenges due to the unique derivative-dependent loss function. By analyzing the loss distribution, previous research observed the propagation failure phenomenon of PINNs, intuitively described as the correct supervision for model outputs cannot ``propagate'' from initial states or boundaries to the interior domain. Going beyond intuitive understanding, this paper provides the first formal and in-depth study of propagation failure and its root cause. Based on a detailed comparison with classical finite element methods, we ascribe the failure to the conventional single-point-processing architecture of PINNs and further prove that propagation failure is essentially caused by the lower gradient correlation of PINN models on nearby collocation points. Compared to superficial loss maps, this new perspective provides a more precise quantitative criterion to identify where and why PINN fails. The theoretical finding also inspires us to present a new PINN architecture, named ProPINN, which can effectively unite the gradient of region points for better propagation. ProPINN can reliably resolve PINN failure modes and significantly surpass advanced Transformer-based models with 46% relative promotion.

Problem

• Physics-Informed Neural Networks
• Propagation Failure
• Information Transmission

Innovation

• Physical Information Neural Networks (PINNs)
• Information Propagation Enhancement
• ProPINN Architecture