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：本文提出一种针对有限神经元法的新型算法——神经元级并行子空间校正方法（NPSC），该方法利用神经网络函数逼近偏微分方程（PDEs）的数值解。尽管神经网络在数值PDE求解中的应用研究极为广泛，但即便对于一维问题，目前仍缺乏能实现足够精度的有效训练算法。基于线性层谱性质与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的子空间校正方法，该方法分别对线性层和非线性层中的每个神经元进行优化。针对一维问题，我们提出了一种能解决线性层病态条件的最优预处理器，使得线性层能在与神经元数量无关的均匀迭代次数内完成训练。在每个单神经元问题中，通过超线性收敛算法找到能规避平坦能量区域的优质局部极小值。针对函数逼近问题和PDE的数值实验表明，该方法的性能优于其他基于梯度的算法。