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.

翻译：物理信息神经网络（PINNs）近年来因缓解了传统方法中出现的维度灾难问题，在求解偏微分方程（PDEs）方面引起了广泛关注。然而，PINNs最显著的缺点在于每个神经网络仅对应一个PDE。在实际应用中，我们通常需要求解一类PDEs，而不仅仅是单个方程。随着深度学习的爆炸式增长，通用深度学习任务中的许多实用技术同样适用于PINNs。迁移学习方法可降低PINNs求解一类PDEs的计算成本。本文提出了一种基于保持奇异向量并优化奇异值的PINNs迁移学习方法（即SVD-PINNs）。针对高维PDEs（10维线性抛物型方程和10维Allen-Cahn方程）的数值实验表明，SVD-PINNs能够有效求解具有不同但相近右端项函数的一类PDEs。