We explore an application of the Physics Informed Neural Networks (PINNs) in conjunction with Airy stress functions and Fourier series to find optimal solutions to a few reference biharmonic problems of elasticity and elastic plate theory. Biharmonic relations are fourth-order partial differential equations (PDEs) that are challenging to solve using classical numerical methods, and have not been addressed using PINNs. Our work highlights a novel application of classical analytical methods to guide the construction of efficient neural networks with the minimal number of parameters that are very accurate and fast to evaluate. In particular, we find that enriching feature space using Airy stress functions can significantly improve the accuracy of PINN solutions for biharmonic PDEs.

翻译：我们结合空气压力功能和Fourier系列探索应用物理-知情神经网络(PINNs),以找到一些弹性和弹性板理论的参考双调问题的最佳解决办法。双调关系是四级局部方程式,使用传统数字方法难以解决,没有使用PINNs加以解决。我们的工作突出介绍了古典分析方法的新应用,以指导高效神经网络的建设,其参数数量极少,而且非常准确和快速地进行评估。我们特别发现,利用空气压力功能丰富地物空间可以大大提高双调人命的PINN解决方案的准确性。