  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.
