Deep Learning in Physics: A Study of Dielectric Quasi-Cubic Particles in a Uniform Electric Field

Solving physics problems for which we know the equations, boundary conditions and symmetries can be done by deep learning. Here, we calculate the induced field inside and outside a dielectric quasi-cube placed in a uniform electric field, wherein the dielectric mismatch at edges and corners of the cube makes accurate calculations challenging. The electric potential is expressed as an ansatz incorporating neural networks with known leading order forms and symmetries, then Laplace's equation with boundary conditions at the dielectric interface is solved by minimizing a loss function inside a large solution domain. We study how the electric potential inside and outside the particle evolves through a sequence of shapes from a sphere to a cube. The neural network being differentiable, it is straightforward to calculate the electric field over the whole domain, the induced surface charge distribution and the polarizability. The neural network being retentive, one can efficiently follow how the field changes upon particle's shape or dielectric constant by iterating from previously converged solution. The present work's objective is two-fold, first to show how can a priori knowledge be incorporated into neural networks to achieve efficient learning and second to apply the method and study how the induced field and polarizability change when a dielectric particle progressively changes its shape from a sphere to a cube.