Collocation Polynomial Neural Forms and Domain Fragmentation for solving Initial Value Problems

TL;DR

This paper introduces higher-order collocation neural forms combined with domain fragmentation to improve the accuracy and reliability of neural network solutions for large initial value problems in differential equations.

Contribution

It extends neural forms to higher polynomial orders and incorporates domain fragmentation, enhancing solution accuracy for large-scale initial value problems.

Findings

Higher-order neural forms improve solution accuracy.

Domain fragmentation enables solving over large domains.

The combined approach shows high reliability in experiments.

Abstract

Several neural network approaches for solving differential equations employ trial solutions with a feedforward neural network. There are different means to incorporate the trial solution in the construction, for instance one may include them directly in the cost function. Used within the corresponding neural network, the trial solutions define the so-called neural form. Such neural forms represent general, flexible tools by which one may solve various differential equations. In this article we consider time-dependent initial value problems, which require to set up the neural form framework adequately. The neural forms presented up to now in the literature for such a setting can be considered as first order polynomials. In this work we propose to extend the polynomial order of the neural forms. The novel collocation-type construction includes several feedforward neural networks, one for each order. Additionally, we propose the fragmentation of the computational domain into subdomains. The neural forms are solved on each subdomain, whereas the interfacing grid points overlap in order to provide initial values over the whole fragmentation. We illustrate in experiments that the combination of collocation neural forms of higher order and the domain fragmentation allows to solve initial value problems over large domains with high accuracy and reliability.