论文标题
SVD-PINN：通过奇异值分解的物理信息的转移学习
SVD-PINNs: Transfer Learning of Physics-Informed Neural Networks via Singular Value Decomposition
论文作者
论文摘要
近年来，物理知识的神经网络（PINN）引起了解决部分微分方程（PDE）的极大关注，因为它们减轻了传统方法中出现的维度的诅咒。但是，PINN的最不利地位是一个神经网络对应于一个PDE。实际上，我们通常需要解决一类PDE，而不仅仅是一个。随着深度学习的爆炸性增长，一般深度学习任务中的许多有用技术也适用于PINN。转移学习方法可能会降低PINN在解决一类PDE时的成本。在本文中，我们通过保持奇异向量和优化奇异值（即SVD-PINN）提出了PINN的转移学习方法。在高维PDE（10-D线性抛物线方程和10-D Allen-CAHN方程）上进行的数值实验表明，SVD-PINN可用于求解具有不同但右手侧函数的一类PDE。
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.
