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.

中文翻译：

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用于偏微分方程稳态分岔和线性稳定性分析的神经网络


本研究介绍了神经网络在求解非线性偏微分方程（PDE）方面的扩展应用。提出了一种与伪弧长延拓相结合的神经网络来从参数化非线性偏微分方程构建分叉图。此外，还提出了一种神经网络方法来解决特征值问题，以分析解的线性稳定性，重点是识别最大特征值。通过对 Bratu 方程和 Burgers 方程的实验检验了所提出的神经网络的有效性。还提供了有限差分法的结果作为比较。在每种情况下都采用不同数量的网格点来评估神经网络和有限差分方法的行为和准确性。实验结果表明，所提出的神经网络可以产生更好的解，生成更准确的分叉图，具有合理的计算时间，并且证明对线性稳定性分析有效。