Harmonic (Quantum) Neural Networks

Harmonic functions are abundant in nature, appearing in limiting cases of Maxwell's, Navier-Stokes equations, the heat and the wave equation. Consequently, there are many applications of harmonic functions, spanning applications from industrial process optimisation to robotic path planning and the calculation of first exit times of random walks. Despite their ubiquity and relevance, there have been few attempts to develop effective means of representing harmonic functions in the context of machine learning architectures, either in machine learning on classical computers, or in the nascent field of quantum machine learning. Architectures which impose or encourage an inductive bias towards harmonic functions would facilitate data-driven modelling and the solution of inverse problems in a range of applications. For classical neural networks, it has already been established how leveraging inductive biases can in general lead to improved performance of learning algorithms. The introduction of such inductive biases within a quantum machine learning setting is instead still in its nascent stages. In this work, we derive exactly-harmonic (conventional- and quantum-) neural networks in two dimensions for simply-connected domains by leveraging the characteristics of holomorphic complex functions. We then demonstrate how these can be approximately extended to multiply-connected two-dimensional domains using techniques inspired by domain decomposition in physics-informed neural networks. We further provide architectures and training protocols to effectively impose approximately harmonic constraints in three dimensions and higher, and as a corollary we report divergence-free network architectures in arbitrary dimensions. Our approaches are demonstrated with applications to heat transfer, electrostatics and robot navigation, with comparisons to physics-informed neural networks included.