Piecewise Neural Network Method for Solving Large Interval Solutions to Initial Value Problems of Ordinary Differential Equations

Abstract

Traditional numerical methods often provide local solutions for initial value problems of differential equations, even though these problems may have solutions over larger intervals. Current neural network algorithms and deep learning methods also struggle to ensure solutions across these broader intervals. This paper introduces a novel approach employing piecewise neural networks to address this issue. The method involves dividing the solution interval into smaller segments and utilizing neural networks with a uniform structure to solve sub-problems within each segment. These solutions are then combined to form a piecewise expression representing the overall solution. The approach guarantees continuous differentiability of the obtained solution over the entire interval, except for finite end points of those sub-intervals.To enhance accuracy, parameter transfer and multiple rounds of pre-training are employed. Importantly, this method maintains a consistent network size and training data scale across sub-domains, unlike existing neural network algorithms. Numerical experiments validate the efficiency of the proposed algorithm.

Keywords

• piecewise neural network
• initial value problem
• ordinary differential equations
• extension of local solution
• parameter transfer