We propose physics-informed holomorphic neural networks (PIHNNs) as a method to solve boundary value problems where the solution can be represented via holomorphic functions. Specifically, we consider the case of plane linear elasticity and, by leveraging the Kolosov-Muskhelishvili representation of the solution in terms of holomorphic potentials, we train a complex-valued neural network to fulfill stress and displacement boundary conditions while automatically satisfying the governing equations. This is achieved by designing the network to return only approximations that inherently satisfy the Cauchy-Riemann conditions through specific choices of layers and activation functions. To ensure generality, we provide a universal approximation theorem guaranteeing that, under basic assumptions, the proposed holomorphic neural networks can approximate any holomorphic function. Furthermore, we suggest a new tailored weight initialization technique to mitigate the issue of vanishing/exploding gradients. Compared to the standard PINN approach, noteworthy benefits of the proposed method for the linear elasticity problem include a more efficient training, as evaluations are needed solely on the boundary of the domain, lower memory requirements, due to the reduced number of training points, and $C^\infty$ regularity of the learned solution. Several benchmark examples are used to verify the correctness of the obtained PIHNN approximations, the substantial benefits over traditional PINNs, and the possibility to deal with non-trivial, multiply-connected geometries via a domain-decomposition strategy.

翻译：我们提出物理信息全纯神经网络（PIHNNs）作为一种求解边界值问题的方法，其中解可通过全纯函数表示。具体而言，我们考虑平面线弹性情形，并利用基于全纯势函数的 Kolosov-Muskhelishvili 解表示法，训练一个复值神经网络以满足应力和位移边界条件，同时自动满足控制方程。这是通过设计网络使其仅返回近似解来实现的，这些近似解通过特定的层和激活函数选择，本质上满足 Cauchy-Riemann 条件。为确保通用性，我们提供了一个通用逼近定理，保证在基本假设下，所提出的全纯神经网络能够逼近任意全纯函数。此外，我们提出一种新的定制权重初始化技术以缓解梯度消失/爆炸问题。与标准 PINN 方法相比，所提方法在线弹性问题上的显著优势包括：由于仅需在区域边界上进行评估，训练效率更高；训练点数量减少，内存需求更低；以及所学解具有 $C^\infty$ 正则性。我们使用多个基准算例验证了所得 PIHNN 近似的正确性、相对于传统 PINN 的显著优势，以及通过区域分解策略处理非平凡多连通几何形状的可能性。