Physics-informed neural networks (PINNs) [1] represent one of the earliest attempts to embed physical laws directly into the training of neural networks. They have been successfully applied to the solution of partial differential equations in a meshless framework, demonstrating their effectiveness across a wide range of engineering fields. Nevertheless, their practical applicability is often constrained by high computational costs, slow training convergence, and potential numerical instabilities arising from the evaluation of higher-order derivatives. In this work, we address these limitations by strengthening the integration of physics into neural network architectures through the construction of models that satisfy the governing differential equations exactly. We begin by presenting holomorphic neural networks (HNNs) [2], a class of complex-valued neural networks whose outputs satisfy the Cauchy–Riemann conditions of holomorphicity a priori. We then show how HNNs can be used to efficiently solve boundary value problems whose solutions can be represented by holomorphic functions. Compared with the standard PINN approach, the key advantage is that the governing equations are inherently satisfied by construction. Consequently, only the boundary conditions must be learned, resulting in significantly improved training efficiency. To demonstrate the potential of the method, we present a series of numerical examples involving Laplace and linear elasticity problems. We first report results for the two-dimensional setting, where the formulation in terms of holomorphic functions is straightforward. Subsequently, we extend the methodology to the three-dimensional case using suitable potential formulations and present numerical results that highlight both the effectiveness and the limitations of the proposed approach. [1] Raissi M., Perdikaris, P., Karniadakis, G. E., Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics, 2019. [2] Calafà M., Hovad E., Engsig-Karup A. P., Andriollo T., Physics-Informed Holomorphic Neural Networks (PIHNNs): Solving 2D linear elasticity problems, Computer Methods in Applied Mechanics and Engineering, 2024.