Physics-Informed Residual Flows

Jephte Abijuru, Mayank Nagda, Phil Ostheimer, Jan C. Aurich, Sebastian Josef Vollmer, Marius Kloft, Sophie Fellenz

Keywords: Physics-Informed Neural Networks, Differentiable Systems

TL;DR: ResPINNs address gradient shattering and flow mismatch in conventional PINNs by reformulating them as residual flows, iterative refiners that preserve gradients and alignment, achieving more accurate PDE solutions.

Abstract: Physics-Informed Neural Networks (PINNs) embed physical laws into deep learning models. However, conventional PINNs often suffer from failure modes leading to inaccurate solutions. We trace these failure modes to two structural pathologies: gradient shattering, where gradients degrade with depth and provide little training signal, and flow mismatch, where training pushes predictions along trajectories that diverge from the PDE solution path. We introduce ResPINNs, which reformulate PINNs as residual flows, differentiable networks that iteratively refine their own predictions through explicit corrective steps, in the spirit of classical iterative solvers. Our analysis shows that this design mitigates both pathologies by keeping updates aligned with descent and by preserving informative gradients across depth. Across canonical PDEs and the PINNacle benchmark suite, residual flows improve accuracy by up to two orders of magnitude.

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Submission Number: 57