Autoregressive PINNs for Time-Dependent PDEs

Mayank Nagda, Jephte Abijuru, Phil Ostheimer, Jan C. Aurich, Stephan Mandt, Marius Kloft, Sophie Fellenz

Keywords: Physics-Informed Neural Networks, Autoregressive Networks

TL;DR: We present a novel physics-informed autoregressive framework for time-dependent PDEs that conditions each prediction on prior states, improving stability and accuracy over existing methods.

Abstract: Accurately solving time-dependent partial differential equations (PDEs) is central to many areas of science and engineering. Physics-Informed Neural Networks (PINNs) use deep learning to solve PDEs, but their pointwise predictions ignore the autoregressive nature of dynamical systems, often leading to instability and error accumulation. We propose Physics-Informed Autoregressive Networks (PIANO), a novel framework that redefines PINNs for modeling dynamical systems. PIANO predicts future states conditioned on past ones, enforcing physical consistency through self-supervised rollouts. Our theoretical analysis shows that while PINNs suffer from temporal instability, PIANO achieves stability through autoregressive modeling. Across a range of challenging time-dependent PDEs, PIANO delivers state-of-the-art accuracy and stability, and it also surpasses existing methods in weather forecasting.

Email Sharing: We authorize the sharing of all author emails with Program Chairs.

Data Release: We authorize the release of our submission and author names to the public in the event of acceptance.

Submission Number: 50