Neural Network Solutions to Differential Equations in Non-Convex Domains: Solving the Electric Field in the Slit-Well Microfluidic Device
This is an incremental improvement for computational physics, enabling reliable neural network solutions in challenging geometries for applications like particle simulations.
The authors tackled solving the electric field in a non-convex slit-well microfluidic device using neural networks, achieving acceptable accuracy validated against finite element methods and outperforming shallow networks in metrics like symmetry and flux conservation.
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.