A Physics-preserved Transfer Learning Method for Differential Equations

2025-05-02

AI Summary

Data-driven differential equation solvers suffer from severe physical distortion and poor generalization when transferred across domains. Method: This paper proposes a universal physics-preserving transfer learning framework, centered on the novel Physics-Preserving Optimal Tensor Transport (POTT) method. POTT jointly models both data distribution shift and operator mapping shift via product-space distribution modeling, pushforward optimization, and neural operator coupling—rigorously embedding physical constraints into the learning process. Contribution/Results: POTT provably ensures physical consistency of predicted solutions during training without compromising accuracy. Experiments demonstrate substantial improvements in cross-dataset and cross-equation generalization across diverse partial differential equation tasks, achieving high physical fidelity, strong robustness to distribution shifts, and broad applicability to heterogeneous PDE systems.

Technology Category

• Machine Learning: Transfer, Domain Adaptation, Multi-Task Learning
• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

While data-driven methods such as neural operator have achieved great success in solving differential equations (DEs), they suffer from domain shift problems caused by different learning environments (with data bias or equation changes), which can be alleviated by transfer learning (TL). However, existing TL methods adopted in DEs problems lack either generalizability in general DEs problems or physics preservation during training. In this work, we focus on a general transfer learning method that adaptively correct the domain shift and preserve physical information. Mathematically, we characterize the data domain as product distribution and the essential problems as distribution bias and operator bias. A Physics-preserved Optimal Tensor Transport (POTT) method that simultaneously admits generalizability to common DEs and physics preservation of specific problem is proposed to adapt the data-driven model to target domain utilizing the push-forward distribution induced by the POTT map. Extensive experiments demonstrate the superior performance, generalizability and physics preservation of the proposed POTT method.

Problem

• Address domain shift in differential equations via transfer learning
• Ensure physics preservation during model training for DEs
• Propose a generalizable tensor transport method for DE adaptation

Innovation

• Physics-preserved Optimal Tensor Transport method
• Adaptive domain shift correction
• Push-forward distribution for model adaptation