In this work, we propose a balanced multi-component and multi-layer neural network (MMNN) structure to approximate functions with complex features with both accuracy and efficiency in terms of degrees of freedom and computation cost. The main idea is motivated by a multi-component, each of which can be approximated effectively by a single-layer network, and multi-layer decomposition in a "divide-and-conquer" type of strategy to deal with a complex function. While an easy modification to fully connected neural networks (FCNNs) or multi-layer perceptrons (MLPs) through the introduction of balanced multi-component structures in the network, MMNNs achieve a significant reduction of training parameters, a much more efficient training process, and a much improved accuracy compared to FCNNs or MLPs. Extensive numerical experiments are presented to illustrate the effectiveness of MMNNs in approximating high oscillatory functions and its automatic adaptivity in capturing localized features.

翻译：本文提出了一种平衡的多组件多层神经网络（MMNN）结构，旨在以较少的自由度与计算成本高效且精确地逼近具有复杂特征的函数。其主要思想源于一种“分而治之”策略下的多组件与多层分解方法，其中每个组件均可由单层网络有效逼近，从而处理复杂函数。通过对全连接神经网络（FCNNs）或多层感知机（MLPs）引入平衡的多组件结构进行简易改进，MMNNs 相比 FCNNs 或 MLPs 显著减少了训练参数，实现了更高效的训练过程，并大幅提升了精度。大量数值实验展示了 MMNNs 在逼近高频振荡函数方面的有效性及其在捕捉局部特征时的自动适应性。