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.

翻译：近期，利用神经网络求解常微分方程（ODE）取得了显著进展。在梯度反向传播算法的辅助下，神经网络能有效充当试函数并在函数空间内近似解。然而，求解复杂ODE（包括高阶与非线性情况）仍面临挑战，亟需提升求解效率与效果。传统方法通常依赖整合已有知识来提高求解效率。与此不同，本研究另辟蹊径，引入一种用于构建试函数的新型神经网络架构——ratio net。该架构受有理分式多项式逼近函数（特别是Padé逼近）启发。实验表明，与基于多项式及多层感知机（MLP）神经网络等现有方法相比，所提方法具有更高的效率。Ratio net有望推动微分方程求解效率与效果的进一步突破。