Title: Beyond Robustness: Probing Physical Systems with Regularized PINNs
Primary area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
Keywords: Physics-Informed Neural Networks, Robust Training, Uncertainty Quantification, Regularization, Computational Efficiency

Abstract: Physics-Informed Neural Networks (PINNs) are a powerful paradigm for solving physical systems, but their susceptibility to noise poses a significant challenge for real-world applications. While robust Bayesian methods exist, their prohibitive computational cost and practical instability can create a scalability bottleneck. To address this robustness-scalability trade-off, we introduce **Robustness-Regularized Physics-Informed Training (R-PIT)**, a lightweight framework that achieves significant noise robustness with minimal computational overhead. Our extensive validation shows R-PIT is remarkably effective on a wide range of problems—achieving **orders-of-magnitude** performance gains on engineering problems with underlying smooth solutions—with only a minor increase in training time. Crucially, this work offers **more than just an algorithm**; it provides a principled analysis of how a model's inductive bias interacts with a problem's physical characteristics. We demonstrate that R-PIT's performance is governed by an explicit smoothness assumption. This finding reframes the framework's application: its success or failure on a given PDE directly reflects the alignment between the model's bias and the problem's intrinsic properties (e.g., smooth solutions vs. shock formations). By establishing this connection between a model's design and a system's physics, R-PIT serves not only as a practical tool but also as a clear case study for analyzing model-problem alignment, guiding the future development of more specialized scientific machine learning methods.

TL;DR: R-PIT is a lightweight regularized PINN whose performance on noisy data acts as a probe to diagnose the smoothness of a physical system.