Neural Electrostatics: A 3D Physics-Informed Boundary Element Poisson Equation Solver

Samuel Audia, Matthias Zwicker, Dinesh Manocha

Keywords: PINN, physics informed, scientific machine learning, electrostatics, scientific computing

TL;DR: We propose a method for solving a 3D Poisson equation using a neural network basis.

Abstract:

Electrostatics solvers relate an imposed voltage to a
corresponding charge density. Current classical methods require fine
discretization and scale poorly due to the construction of a large linear system
of equations. We recast the problem using neural networks and introduce
neural electrostatics, a hybrid 3D boundary element method (BEM). By using the
boundary element form, we are able to overcome many shortcomings of previous
neural solvers, such as learning trivial solutions and balancing loss terms
between the domain and boundary, at the cost of introducing a large integral
containing a singular kernel. We handle this singularity by locally
transforming the integral into polar coordinates and applying a numerical
quadrature. We also show that previous neural solver sampling methods are unable
to minimize the PDE residual, and propose a variational adaptive sampling
method. This technique is able to reduce mean absolute error by 5 times, while
keeping training time constant. Extensive scaling and ablation studies are
performed to justify our method. Results show that our method learns a charge
distribution within 1.2 $pC/m^2$ of mean absolute error from a classical BEM
solver, while using 25 times fewer rectangular elements.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

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Submission Number: 11610