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Table of Contents
1. Introduction
2. Theory
2.1. Exact Harmonicity in Two Dimensions
2.1.1. Exact Harmonicity: Holomorphic Networks on Simply-Connected Domains
2.1.2. Exact Harmonicity: Multiholomorphic Networks on Multiply-Connected Domains
2.2. Approximate Harmonicity: Curl-Driven Harmonic Networks
3. Applications
Summary
Harmonic functions are abundant in nature, with applications ranging from industrial process optimization to robotic path planning. In this work, the authors demonstrate effective means of representing harmonic functions in neural networks and extend the results to quantum neural networks. The paper focuses on incorporating inductive biases for harmonic functions in machine learning models. Various techniques are discussed for exact and approximate harmonicity in different dimensions and domains. The applications of these techniques are exemplified in electrostatics, heat distribution, robot navigation, and fluid flow examples.