Various traditional numerical methods for solving initial value problems of differential equations often produce local solutions near the initial value point, despite the problems having larger interval solutions. Even current popular neural network algorithms or deep learning methods cannot guarantee yielding large interval solutions for these problems. In this paper, we propose a piecewise neural network approach to obtain a large interval numerical solution for initial value problems of differential equations. In this method, we first divide the solution interval, on which the initial problem is to be solved, into several smaller intervals. Neural networks with a unified structure are then employed on each sub-interval to solve the related sub-problems. By assembling these neural network solutions, a piecewise expression of the large interval solution to the problem is constructed, referred to as the piecewise neural network solution. The continuous differentiability of the solution over the entire interval, except for finite points, is proven through theoretical analysis and employing a parameter transfer technique. Additionally, a parameter transfer and multiple rounds of pre-training technique are utilized to enhance the accuracy of the approximation solution. Compared with existing neural network algorithms, this method does not increase the network size and training data scale for training the network on each sub-domain. Finally, several numerical experiments are presented to demonstrate the efficiency of the proposed algorithm.

翻译：求解微分方程初值问题的各种传统数值方法通常仅能获得初值点附近的局部解，而问题本身往往具有更大区间的解。即便是当前流行的神经网络算法或深度学习方法，也无法保证为这些问题生成大区间解。本文提出一种分段神经网络方法，用于获得微分方程初值问题的大区间数值解。该方法首先将待求解的初始问题所在解区间划分为若干子区间，然后在每个子区间上采用统一结构的神经网络求解相应的子问题。通过组合这些神经网络解，构建问题大区间解的分段表达式，称为分段神经网络解。通过理论分析并采用参数传递技术，证明了整个区间上除有限点外解具有连续可微性。此外，利用参数传递和多轮预训练技术提高了近似解的精度。与现有神经网络算法相比，本方法无需增加各子域训练网络的网络规模和训练数据规模。最后，通过多个数值实验验证了所提算法的有效性。