[Paper Review] MaxwellNet: Physics-driven deep neural network training based on Maxwell's equations

Joowon Lim, Demetri Psaltis|arXiv (Cornell University)|Jul 13, 2021

Neural Networks and Reservoir Computing34 references118 citations

TL;DR

This paper introduces MaxwellNet, a physics-driven deep neural network that solves Maxwell's equations for electromagnetic field prediction using the residual of the equations as a loss function, eliminating the need for ground-truth simulation data. The method enables fast, accurate simulation and inverse design of aspheric micro-lenses by training on permittivity distributions alone, achieving high consistency with COMSOL simulations and enabling optimization in a latent space for target focal intensity maximization.

ABSTRACT

Maxwell's equations govern light propagation and its interaction with matter. Therefore, the solution of Maxwell's equations using computational electromagnetic simulations plays a critical role in understanding light-matter interaction and designing optical elements. Such simulations are often time-consuming and recent activities have been described to replace or supplement them with trained deep neural networks (DNNs). Such DNNs typically require extensive, computationally demanding simulations using conventional electromagnetic solvers to compose the training dataset. In this paper, we present a novel scheme to train a DNN that solves Maxwell's equations speedily and accurately without relying on other computational electromagnetic solvers. Our approach is to train a DNN using the residual of Maxwell's equations as the physics-driven loss function for a network that finds the electric field given the spatial distribution of the material property. We demonstrate it by training a single network that simultaneously finds multiple solutions of various aspheric micro-lenses. Furthermore, we exploit the speed of this network in a novel inverse design scheme to design a micro-lens that maximizes a desired merit function. We believe that our approach opens up a novel way for light simulation and optical design of photonics devices.

Motivation & Objective

• To develop a deep neural network that solves Maxwell's equations for arbitrary inhomogeneous dielectric distributions without relying on precomputed simulation datasets.
• To enable fast and accurate electromagnetic field prediction for optical device design using a physics-informed loss function based on Maxwell's equations.
• To demonstrate inverse design of micro-lenses by optimizing shape in a latent space using the trained network as a forward simulator.
• To validate the method through comparison with high-fidelity COMSOL simulations for both field prediction and inverse design outcomes.

Proposed method

• Train a deep neural network (MaxwellNet) using the residual of Maxwell's equations as a physics-driven loss function, with relative permittivity distribution as input and electric field as output.
• Discretize the residual on a Yee grid using high-order finite difference approximations for gradient operators.
• Incorporate perfectly matched layers (PMLs) to enforce Sommerfeld radiation conditions and ensure physical consistency.
• Use symmetry to reduce computational domain and improve training efficiency.
• Train separate networks for TE and TM modes using a dataset of 2D aspheric micro-lens geometries with varying curvature and conic constants.
• Combine MaxwellNet with DeepSDF to represent lens shapes in a low-dimensional latent space, enabling inverse design by optimizing the latent vector to maximize a figure of merit (FOM).

Experimental results

Research questions

• RQ1 Can a deep neural network be trained to solve Maxwell’s equations directly using the equation residual as a loss function, without requiring ground-truth field data?
• RQ2 How accurately can such a physics-driven network predict electromagnetic field distributions for diverse aspheric micro-lens geometries compared to conventional solvers?
• RQ3 Can the trained network be used effectively as a forward simulator in an inverse design pipeline for photonic devices?
• RQ4 To what extent does the latent space representation via DeepSDF enable efficient and manufacturable inverse design of micro-lenses?

Key findings

• MaxwellNet predicts electric field distributions with high consistency to COMSOL simulations, as confirmed by direct pixel-wise comparison across multiple aspheric lens configurations.
• The physics-driven loss based on Maxwell’s equation residual effectively guides the network to converge to accurate solutions even without ground-truth labels.
• The inverse design process successfully optimized a spheric lens into an aspheric lens that focuses light at 8 μm, matching the target focal point.
• The intensity profile of the inverse-designed lens from MaxwellNet showed strong agreement with COMSOL simulation results, validating the accuracy of the forward model.
• The method enables rapid simulation and inverse design by replacing computationally expensive full-wave solvers with a single, trained neural network.
• The use of DeepSDF for latent space optimization allows for continuous, differentiable shape updates while maintaining manufacturability constraints.