PIANO: Physics Informed Autoregressive Network

TL;DR

PIANO introduces an autoregressive framework for physics-informed neural networks, improving stability and accuracy in solving time-dependent PDEs and weather forecasting by explicitly modeling dynamical systems.

Contribution

The paper presents PIANO, a novel autoregressive neural network approach that enhances stability and accuracy in physics-informed PDE solutions compared to traditional PINNs.

Findings

PIANO achieves state-of-the-art accuracy on challenging PDEs.

PIANO demonstrates improved stability over PINNs.

Outperforms existing methods in weather forecasting.

Abstract

Solving time-dependent partial differential equations (PDEs) is fundamental to modeling critical phenomena across science and engineering. Physics-Informed Neural Networks (PINNs) solve PDEs using deep learning. However, PINNs perform pointwise predictions that neglect the autoregressive property of dynamical systems, leading to instabilities and inaccurate predictions. We introduce Physics-Informed Autoregressive Networks (PIANO) -- a framework that redesigns PINNs to model dynamical systems. PIANO operates autoregressively, explicitly conditioning future predictions on the past. It is trained through a self-supervised rollout mechanism while enforcing physical constraints. We present a rigorous theoretical analysis demonstrating that PINNs suffer from temporal instability, while PIANO achieves stability through autoregressive modeling. Extensive experiments on challenging time-dependent PDEs demonstrate that PIANO achieves state-of-the-art performance, significantly improving accuracy and stability over existing methods. We further show that PIANO outperforms existing methods in weather forecasting.