Work

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

Year: 2024

Type: article

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.

Source: Symmetry

DOI: 10.3390/sym16111490

Authors: Dongpeng Han, Chaolu Temuer

Institution: Shanghai Maritime University

Language: English

Topic: Numerical methods for differential equations

Subfield: Numerical Analysis

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $2,728

Funder: National Natural Science Foundation of China