用神经网络+辅助方程法求解非线性偏微分方程，发现新解析解。
A novel auxiliary equation neural networks method for exactly explicit solutions of nonlinear partial differential equations
- 将神经网络与辅助方程法结合，引入基于Riccati方程的新激活函数。
- 成功求得3个方程的精确解析解，形式包括双曲、三角和有理函数。
- 适合研究偏微分方程求解、深度学习与数学物理交叉领域的学者。
本文提出一种新型辅助方程神经网络方法（AENNM），将神经网络模型与辅助方程法融合，用于求解非线性偏微分方程（NLPDEs）的精确解。其核心创新在于引入源自Riccati方程解的新激活函数，建立微分方程理论与深度学习之间的新联系。通过结合神经网络的强大逼近能力与符号计算的高精度，AENNM显著提升计算效率与准确性。为验证有效性，研究了三个数值例子：非线性演化方程、Korteweg-de Vries-Burgers方程及(2+1)维Boussinesq方程。通过在“2-2-2-1”和“3-2-2-1”神经网络结构中设定特定激活函数，构建了新的试探函数，并导出此前未报道的解析解。解以双曲函数、三角函数和有理函数形式表达。最后，通过三维图、等高线图与密度图展示解的动态特性。该研究为求解NLPDEs提供新方法框架，适用于科学与工程广泛领域。
原文摘要 · Abstract (English)
In this study, we firstly propose an auxiliary equation neural networks method (AENNM), an innovative analytical method that integrates neural networks (NNs) models with the auxiliary equation method to obtain exact solutions of nonlinear partial differential equations (NLPDEs). A key novelty of this method is the introduction of a novel activation function derived from the solutions of the Riccati equation, establishing a new mathematical link between differential equations theory and deep learning. By combining the strong approximation capability of NNs with the high precision of symbolic computation, AENNM significantly enhances computational efficiency and accuracy. To demonstrate the effectiveness of the AENNM in solving NLPDEs, three numerical examples are investigated, including the nonlinear evolution equation, the Korteweg-de Vries-Burgers equation, and the (2+1)-dimensional Boussinesq equation. Furthermore, some new trial functions are constructed by setting specific activation functions within the "2-2-2-1" and "3-2-2-1" NNs models. By embedding the auxiliary equation method into the NNs framework, we derive previously unreported solutions. The exact analytical solutions are expressed in terms of hyperbolic functions, trigonometric functions, and rational functions. Finally, three-dimensional plots, contour plots, and density plots are presented to illustrate the dynamic characteristics of the obtained solutions. This research provides a novel methodological framework for addressing NLPDEs, with broad applicability across scientific and engineering fields.
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