The work with D. Han (Dartmouth Math) Hierarchical Learning to Solve PDEs Using Physics-Informed Neural Networks is published in Computational Science - ICCS 2023, Lecture Notes in Computer Science.
Abstract: The neural network-based approach to solving partial differ- ential equations has attracted considerable attention. In training a neu- ral network, the network learns global features corresponding to low- frequency components while high-frequency components are approxi- mated at a much slower rate. For a class of equations in which the solu- tion contains a wide range of scales, the network training process can suffer from slow convergence and low accuracy due to its inability to capture the high-frequency components. In this work, we propose a sequential training based on a hierarchy of networks to improve the convergence rate and accu- racy of the neural network solution to partial differential equations. The proposed method comprises multi-training levels in which a newly intro- duced neural network is guided to learn the residual of the previous level approximation. We validate the efficiency and robustness of the proposed hierarchical approach through a suite of partial differential equations.