PIANO: Physics Informed Autoregressive Network
This addresses a critical limitation in PINNs for modeling dynamical systems across science and engineering, representing a novel method rather than an incremental improvement.
The paper tackles the problem of temporal instability in Physics-Informed Neural Networks (PINNs) for solving time-dependent PDEs by introducing PIANO, which uses autoregressive modeling to condition future predictions on the past. The result is state-of-the-art performance with significant improvements in accuracy and stability, demonstrated on challenging PDEs and weather forecasting.
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.