Several forms for constructing novel physics-informed neural-networks (PINN) for the solution of partial-differential-algebraic equations based on derivative operator splitting are proposed, using the nonlinear Kirchhoff rod as a prototype for demonstration. The open-source DeepXDE is likely the most well documented framework with many examples. Yet, we encountered some pathological problems and proposed novel methods to resolve them. Among these novel methods are the PDE forms, which evolve from the lower-level form with fewer unknown dependent variables to higher-level form with more dependent variables, in addition to those from lower-level forms. Traditionally, the highest-level form, the balance-of-momenta form, is the starting point for (hand) deriving the lowest-level form through a tedious (and error prone) process of successive substitutions. The next step in a finite element method is to discretize the lowest-level form upon forming a weak form and linearization with appropriate interpolation functions, followed by their implementation in a code and testing. The time-consuming tedium in all of these steps could be bypassed by applying the proposed novel PINN directly to the highest-level form. We developed a script based on JAX. While our JAX script did not show the pathological problems of DDE-T (DDE with TensorFlow backend), it is slower than DDE-T. That DDE-T itself being more efficient in higher-level form than in lower-level form makes working directly with higher-level form even more attractive in addition to the advantages mentioned further above. Since coming up with an appropriate learning-rate schedule for a good solution is more art than science, we systematically codified in detail our experience running optimization through a normalization/standardization of the network-training process so readers can reproduce our results.

翻译：本文提出了基于导数算子分裂构建新型物理信息神经网络(PINN)求解偏微分-代数方程的多种形式，并以非线性Kirchhoff杆作为演示原型。开源框架DeepXDE可能是文档最完善且包含众多示例的框架，但我们在使用中遇到了一些病态问题，并提出了新方法予以解决。这些新方法包括偏微分方程形式的演进：从含较少未知因变量的低阶形式，到包含更多因变量的高阶形式（除低阶形式外）。传统上，最高阶形式——动量平衡形式——是（人工）推导最低阶形式的起点，需通过繁琐（且易出错）的连续代换过程。有限元方法的下一步是在形成弱形式并采用适当插值函数线性化后，对最低阶形式进行离散化，随后进行代码实现与测试。通过将所提出的新型PINN直接应用于最高阶形式，可以绕过所有这些步骤中耗时的繁琐工作。我们开发了基于JAX的脚本。虽然我们的JAX脚本未出现DDE-T（基于TensorFlow后端的DDE）的病态问题，但其计算速度慢于DDE-T。值得注意的是，DDE-T本身在处理高阶形式时比处理低阶形式更高效，这使得直接使用高阶形式更具吸引力——除了前文提到的优势外。由于为获得良好解而设计合适的学习率调度方案更偏向艺术而非科学，我们通过神经网络训练过程的归一化/标准化，系统化地详细编码了优化运行经验，以便读者能够复现我们的结果。