  The neural network method of solving differential equations is used to
approximate the electric potential and corresponding electric field in the
slit-well microfluidic device. The device's geometry is non-convex, making this
a challenging problem to solve using the neural network method. To validate the
method, the neural network solutions are compared to a reference solution
obtained using the finite element method. Additional metrics are presented that
measure how well the neural networks recover important physical invariants that
are not explicitly enforced during training: spatial symmetries and
conservation of electric flux. Finally, as an application-specific test of
validity, neural network electric fields are incorporated into particle
simulations. Conveniently, the same loss functional used to train the neural
networks also seems to provide a reliable estimator of the networks' true
errors, as measured by any of the metrics considered here. In all metrics, deep
neural networks significantly outperform shallow neural networks, even when
normalized by computational cost. Altogether, the results suggest that the
neural network method can reliably produce solutions of acceptable accuracy for
use in subsequent physical computations, such as particle simulations.
