SVD-PINNs: Transfer Learning of Physics-Informed Neural Networks via Singular Value Decomposition

TL;DR

This paper introduces SVD-PINNs, a transfer learning approach for physics-informed neural networks that leverages singular value decomposition to efficiently solve a class of high-dimensional PDEs with different right-hand sides.

Contribution

The paper proposes a novel transfer learning method for PINNs using SVD to keep singular vectors and optimize singular values, enabling efficient solving of PDE families.

Findings

SVD-PINNs effectively solve high-dimensional PDEs with different right-hand sides.

Numerical experiments demonstrate reduced computational cost and improved accuracy.

Method applicable to 10-dimensional PDEs like linear parabolic and Allen-Cahn equations.

Abstract

Physics-informed neural networks (PINNs) have attracted significant attention for solving partial differential equations (PDEs) in recent years because they alleviate the curse of dimensionality that appears in traditional methods. However, the most disadvantage of PINNs is that one neural network corresponds to one PDE. In practice, we usually need to solve a class of PDEs, not just one. With the explosive growth of deep learning, many useful techniques in general deep learning tasks are also suitable for PINNs. Transfer learning methods may reduce the cost for PINNs in solving a class of PDEs. In this paper, we proposed a transfer learning method of PINNs via keeping singular vectors and optimizing singular values (namely SVD-PINNs). Numerical experiments on high dimensional PDEs (10-d linear parabolic equations and 10-d Allen-Cahn equations) show that SVD-PINNs work for solving a class of PDEs with different but close right-hand-side functions.