This research introduces an extended application of neural networks for solving nonlinear partial differential equations (PDEs). A neural network, combined with a pseudo-arclength continuation, is proposed to construct bifurcation diagrams from parameterized nonlinear PDEs. Additionally, a neural network approach is also presented for solving eigenvalue problems to analyze solution linear stability, focusing on identifying the largest eigenvalue. The effectiveness of the proposed neural network is examined through experiments on the Bratu equation and the Burgers equation. Results from a finite difference method are also presented as comparison. Varying numbers of grid points are employed in each case to assess the behavior and accuracy of both the neural network and the finite difference method. The experimental results demonstrate that the proposed neural network produces better solutions, generates more accurate bifurcation diagrams, has reasonable computational times, and proves effective for linear stability analysis.

翻译：本研究拓展了神经网络在求解非线性偏微分方程中的应用。提出了一种结合伪弧长延拓的神经网络方法，用于从参数化非线性偏微分方程构建分岔图。此外，还提出了一种神经网络方法用于求解特征值问题以分析解的线性稳定性，重点关注最大特征值的识别。通过Bratu方程和Burgers方程的实验检验了所提神经网络的有效性，并给出了有限差分法的结果作为对比。每种情况均采用不同数量的网格点，以评估神经网络和有限差分法的性能与精度。实验结果表明，所提出的神经网络能产生更优的解、生成更精确的分岔图、具有合理的计算时间，并在线性稳定性分析中证明有效。