A Physics-preserved Transfer Learning Method for Differential Equations

TL;DR

This paper introduces a physics-preserved transfer learning approach for differential equations that adaptively corrects domain shifts while maintaining physical laws, enhancing generalizability and accuracy.

Contribution

It proposes the POTT method, combining optimal tensor transport with physics preservation, addressing domain shift and physics retention in transfer learning for differential equations.

Findings

POTT outperforms existing methods in accuracy and generalizability.

The method effectively preserves physical laws during transfer.

Extensive experiments validate superior performance across diverse DE problems.

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.