In this paper, we propose a novel algorithm called Neuron-wise Parallel Subspace Correction Method (NPSC) for the finite neuron method that approximates numerical solutions of partial differential equations (PDEs) using neural network functions. Despite extremely extensive research activities in applying neural networks for numerical PDEs, there is still a serious lack of effective training algorithms that can achieve adequate accuracy, even for one-dimensional problems. Based on recent results on the spectral properties of linear layers and landscape analysis for single neuron problems, we develop a special type of subspace correction method that optimizes the linear layer and each neuron in the nonlinear layer separately. An optimal preconditioner that resolves the ill-conditioning of the linear layer is presented for one-dimensional problems, so that the linear layer is trained in a uniform number of iterations with respect to the number of neurons. In each single neuron problem, a good local minimum that avoids flat energy regions is found by a superlinearly convergent algorithm. Numerical experiments on function approximation problems and PDEs demonstrate better performance of the proposed method than other gradient-based methods.

翻译：本文针对有限神经元方法提出一种新型算法——神经元并行子空间修正方法（NPSC），该方法利用神经网络函数近似求解偏微分方程（PDEs）的数值解。尽管将神经网络应用于数值PDE领域的研究活动极为广泛，但至今仍严重缺乏能够达到足够精度的有效训练算法，即使对于一维问题也是如此。基于线性层谱特性及单神经元问题景观分析的最新研究成果，我们开发了一种特殊类型的子空间修正方法，该方法分别优化线性层和非线性层中的每个神经元。针对一维问题，我们提出了一个能解决线性层病态条件的最优预处理器，使得线性层能够在与神经元数量一致的统一迭代次数内完成训练。在每个单神经元问题中，我们通过超线性收敛算法找到了能够避免平坦能量区域的优质局部最小值。关于函数逼近问题和PDEs的数值实验表明，所提出方法相较于其他基于梯度的方法具有更优的性能。