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The neural network method of solving differential equations is used toapproximate the electric potential and corresponding electric field in theslit-well microfluidic device. The device's geometry is non-convex, making thisa challenging problem to solve using the neural network method. To validate themethod, the neural network solutions are compared to a reference solutionobtained using the finite element method. Additional metrics are presented thatmeasure how well the neural networks recover important physical invariants thatare not explicitly enforced during training: spatial symmetries andconservation of electric flux. Finally, as an application-specific test ofvalidity, neural network electric fields are incorporated into particlesimulations. Conveniently, the same loss functional used to train the neuralnetworks also seems to provide a reliable estimator of the networks' trueerrors, as measured by any of the metrics considered here. In all metrics, deepneural networks significantly outperform shallow neural networks, even whennormalized by computational cost. Altogether, the results suggest that theneural network method can reliably produce solutions of acceptable accuracy foruse in subsequent physical computations, such as particle simulations.

Neural Network Solutions to Differential Equations in Non-Convex Domains: Solving the Electric Field in the Slit-Well Microfluidic Device