Efficiently Solving High-Order and Nonlinear ODEs with Rational Fraction Polynomial: the Ratio Net

TL;DR

This paper introduces Ratio Net, a neural network architecture inspired by rational fraction polynomials, which improves efficiency in solving complex high-order and nonlinear ODEs compared to existing neural methods.

Contribution

The study proposes a novel neural network architecture called Ratio Net, inspired by Pade approximants, to enhance the efficiency of solving high-order and nonlinear ODEs.

Findings

Ratio Net outperforms polynomial and MLP-based methods in efficiency

Empirical results show higher accuracy in solving complex ODEs

The approach advances neural network methods for differential equations

Abstract

Recent advances in solving ordinary differential equations (ODEs) with neural networks have been remarkable. Neural networks excel at serving as trial functions and approximating solutions within functional spaces, aided by gradient backpropagation algorithms. However, challenges remain in solving complex ODEs, including high-order and nonlinear cases, emphasizing the need for improved efficiency and effectiveness. Traditional methods have typically relied on established knowledge integration to improve problem-solving efficiency. In contrast, this study takes a different approach by introducing a new neural network architecture for constructing trial functions, known as ratio net. This architecture draws inspiration from rational fraction polynomial approximation functions, specifically the Pade approximant. Through empirical trials, it demonstrated that the proposed method exhibits higher efficiency compared to existing approaches, including polynomial-based and multilayer perceptron (MLP) neural network-based methods. The ratio net holds promise for advancing the efficiency and effectiveness of solving differential equations.