  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.

Numerical Analysis;Machine Learning;Numerical Analysis