A Physics-preserved Transfer Learning Method for Differential Equations

2025/05/02 by Hao-Ran Yang, Yang, Hao-Ran, Chuan-Xian Ren +1
Computer Science · #Artificial Intelligence (cs.AI) #Educational Technology and Assessment #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning and ELM #Neural Networks and Applications

paper · pdf · doi:10.48550/arxiv.2505.01281

openalex publication_date 2025/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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Related

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• Towards Universal Neural Operators through Multiphysics Pretraining
• TransNet: Transferable Neural Networks for Partial Differential Equations