AI 中文总结
该研究提出一种结合帕德激活单元（PAUs）的有理神经网络方法，可追踪非线性PDE的复杂奇点，通过三个典型PDE案例验证了其定位、追踪奇点及推断特征现象形成的性能。
AI 中文摘要
我们提出了一种基于神经网络的方法，用于非线性偏微分方程（PDE）解的数值解析延拓及其复杂奇点的检测。该框架采用“不安全”帕德激活单元（PAUs）作为激活函数，同时采用一种新颖的无反向传播方法来计算隐藏层的权重和偏置。训练过程专门设计用于学习具有极点型奇点的亚纯函数。与现有具有固定隐藏层参数的方法不同，所提方法将权重和偏置视为时间相关函数，使网络能够适应复杂奇点的演化。利用该方法，我们可以高效定位延拓解的复杂奇点，追踪其随时间的轨迹，并基于这种动力学推断非线性PDE解中特征现象的形成。为验证方法性能，我们分析了三个知名案例：（1）表现出有限时间爆破的非线性热方程；（2）形成激波的非线性伯格斯方程；（3）出现畸形波的非线性薛定谔方程。
英文摘要
We present a neural network-based method for numerical analytic continuation of solutions of nonlinear partial differential equations (PDEs) and detection of their complex singularities. The proposed framework employs "unsafe" Padé activation units (PAUs) as activation functions, together with a novel backpropagation-free method for computing the weights and biases of the hidden layers. The training procedure is designed specifically for learning meromorphic functions with pole-type singularities. Unlike existing methods with fixed hidden-layer parameters, the proposed approach treats the weights and biases as time-dependent functions, allowing the network to adapt to the evolution of the complex singularities. Using this method, we can efficiently locate the complex singularities of the extended solution, track their trajectories over time, and, based on this dynamics, infer the formation of characteristic phenomena in the solutions of nonlinear PDEs. To demonstrate the performance of our method, we analyze three well-known cases: (1) the nonlinear heat equation, which exhibits finite-time blow up; (2) the nonlinear Burgers equation, which develops a shock; and (3) the nonlinear Schrödinger equation, in which rogue waves form.
Comments26 pages, 18 figures
