Towards Generalizable PDE Dynamics Forecasting via Physics-Guided Invariant Learning

2025-09-29

AI Summary

To address the weak zero-shot out-of-distribution (OOD) generalization of partial differential equation (PDE)-based dynamical models, this paper proposes iMOOE, a physics-guided invariance learning framework. Methodologically, we introduce and formalize the *dual PDE invariance principle*: both the constituent differential operators and their compositional relationships remain invariant across diverse physical systems and temporal evolutions. Leveraging this principle, we design an invariance-aligned mixture-of-operators expert architecture and incorporate a frequency-domain-enhanced invariance learning objective, enabling tight integration of physical priors with data-driven modeling. Evaluated on multiple synthetic benchmarks and real-world fluid dynamics and heat conduction tasks, iMOOE achieves significant improvements in OOD zero-shot prediction accuracy while maintaining high in-distribution performance—without requiring test-time fine-tuning or domain adaptation.

Technology Category

• Machine Learning: Mixture of Experts (MoE)
• Computer Vision: Low Level & Physics-based Vision
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Advanced deep learning-based approaches have been actively applied to forecast the spatiotemporal physical dynamics governed by partial differential equations (PDEs), which acts as a critical procedure in tackling many science and engineering problems. As real-world physical environments like PDE system parameters are always capricious, how to generalize across unseen out-of-distribution (OOD) forecasting scenarios using limited training data is of great importance. To bridge this barrier, existing methods focus on discovering domain-generalizable representations across various PDE dynamics trajectories. However, their zero-shot OOD generalization capability remains deficient, since extra test-time samples for domain-specific adaptation are still required. This is because the fundamental physical invariance in PDE dynamical systems are yet to be investigated or integrated. To this end, we first explicitly define a two-fold PDE invariance principle, which points out that ingredient operators and their composition relationships remain invariant across different domains and PDE system evolution. Next, to capture this two-fold PDE invariance, we propose a physics-guided invariant learning method termed iMOOE, featuring an Invariance-aligned Mixture Of Operator Expert architecture and a frequency-enriched invariant learning objective. Extensive experiments across simulated benchmarks and real-world applications validate iMOOE's superior in-distribution performance and zero-shot generalization capabilities on diverse OOD forecasting scenarios.

Problem

• Generalizing PDE forecasting across unseen physical domains
• Capturing invariant physical laws in dynamical systems
• Enabling zero-shot out-of-distribution generalization without adaptation

Innovation

• Physics-guided invariant learning for PDE dynamics
• Two-fold PDE invariance principle definition
• Invariance-aligned Mixture Of Operator Expert architecture