PIANO: Physics Informed Autoregressive Network

2025-08-22

AI Summary

Standard physics-informed neural networks (PINNs) for time-dependent partial differential equations (PDEs) neglect the system’s autoregressive temporal structure, leading to unstable and inaccurate long-term predictions. To address this, we propose AR-PINN—the first autoregressive PINN framework explicitly designed for time-dependent PDEs. AR-PINN incorporates an autoregressive mechanism into the PINN architecture to model historical state dependencies, introduces a physics-integrated loss function enforcing governing PDE constraints, and employs a self-supervised rollout training strategy that theoretically guarantees temporal stability of the learned dynamics. Evaluated on canonical time-dependent PDEs—including Burgers and Navier–Stokes equations—as well as real-world weather forecasting tasks, AR-PINN achieves substantial improvements in long-horizon prediction accuracy and numerical robustness, establishing new state-of-the-art performance.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Multiagent Systems: Adversarial Agents
• Machine Learning: Deep Neural Architectures and Foundation Models

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Solving time-dependent partial differential equations
• Addressing PINNs' instability in dynamical systems
• Improving accuracy in autoregressive physical predictions

Innovation

• Autoregressive modeling for dynamical systems
• Self-supervised rollout training mechanism
• Physics constraints enforcement in neural networks