Efficiently Solving High-Order and Nonlinear ODEs with Rational Fraction Polynomial: the Ratio Net
This work addresses efficiency challenges in solving complex ODEs for computational mathematics and engineering applications, representing an incremental improvement over prior neural network methods.
The paper tackles solving high-order and nonlinear ordinary differential equations (ODEs) by introducing a new neural network architecture called ratio net, inspired by rational fraction polynomial approximation, which demonstrated higher efficiency compared to existing methods like polynomial-based and multilayer perceptron approaches.
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.