Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information of the target PDEs such as its formulation and/or data of its solution for pre-training. In this work, we propose to construct transferable neural feature spaces from purely function approximation perspectives without using PDE information. The construction of the feature space involves re-parameterization of the hidden neurons and uses auxiliary functions to tune the resulting feature space. Theoretical analysis shows the high quality of the produced feature space, i.e., uniformly distributed neurons. Extensive numerical experiments verify the outstanding performance of our method, including significantly improved transferability, e.g., using the same feature space for various PDEs with different domains and boundary conditions, and the superior accuracy, e.g., several orders of magnitude smaller mean squared error than the state of the art methods.

翻译：针对偏微分方程（PDE）的迁移学习旨在开发预训练神经网络，使其能够求解多类PDE问题。现有迁移学习方法需要大量目标PDE信息（如问题公式化和/或解的预训练数据）。本文提出纯函数逼近视角下无需PDE信息即可构建可迁移神经特征空间的方法。该特征空间的构建涉及对隐含层神经元进行重参数化，并利用辅助函数调整所生成的特征空间。理论分析证明了所生成特征空间的高质量特性（即神经元均匀分布）。大量数值实验验证了本方法的卓越性能，包括显著提升的可迁移性（例如，对具有不同定义域和边界条件的各类PDE使用统一特征空间）及更优的精度（例如，均方误差比现有最优方法低数个数量级）。