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In this paper, we propose a novel algorithm called Neuron-wise ParallelSubspace Correction Method (NPSC) for the finite neuron method thatapproximates numerical solutions of partial differential equations (PDEs) usingneural network functions. Despite extremely extensive research activities inapplying neural networks for numerical PDEs, there is still a serious lack ofeffective training algorithms that can achieve adequate accuracy, even forone-dimensional problems. Based on recent results on the spectral properties oflinear layers and landscape analysis for single neuron problems, we develop aspecial type of subspace correction method that optimizes the linear layer andeach neuron in the nonlinear layer separately. An optimal preconditioner thatresolves the ill-conditioning of the linear layer is presented forone-dimensional problems, so that the linear layer is trained in a uniformnumber of iterations with respect to the number of neurons. In each singleneuron problem, a good local minimum that avoids flat energy regions is foundby a superlinearly convergent algorithm. Numerical experiments on functionapproximation problems and PDEs demonstrate better performance of the proposedmethod than other gradient-based methods.

A neuron-wise subspace correction method for the finite neuron method