用自回归机制提升物理约束神经网络的动态预测稳定性。
PIANO: Physics Informed Autoregressive Network
- 引入自回归结构，让未来预测依赖历史信息。
- 在多个时间依赖型PDE上实现更高精度与更强稳定性。
- 适合需要长期准确模拟的科学计算与气象预报场景。
求解随时间演化的偏微分方程（PDEs）是科学与工程中建模关键现象的基础。物理信息神经网络（PINNs）利用深度学习求解PDE，但其点对点预测忽略了动态系统的自回归特性，导致不稳定和预测误差。我们提出物理信息自回归网络（PIANO）——一种重新设计的框架，用于建模动态系统。PIANO采用自回归方式，显式地将未来预测依赖于历史状态，通过自监督滚动训练并施加物理约束。我们进行了严格的理论分析，证明了传统PINNs存在时间不稳定性，而PIANO通过自回归建模实现了稳定。大量实验表明，在复杂的时间依赖型PDE上，PIANO达到当前最优性能，显著优于现有方法。此外，我们在气象预报任务中也验证了其优越性。
原文摘要 · Abstract (English)
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.
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