Speaker: Benjamin Plumridge, UTK

Title: Filtered Spherical Harmonics with Neural Networks

Abstract: We study the filtered spherical harmonics method (FP_N) applied to the radiation transport equation (RTE) in 1-D slab geometries.  The FP_N method is a variant of the spherical harmonic method (P_N) which introduces regularization, via a filter operator, dampening ``unphysical" behavior associated with Gibb's phenomenon. The filter operator includes a tuning parameter called the filter strength, which determines the amount of smoothing that is introduced into the approximation. Selecting an optimal filter strength is challenging and often requires unattainable information (e.g. the true solution).   To address this, we model the filter strength as a neural network where the features include local state information and the material cross-sections. We write the optimal filter strength as the solution to a PDE-constrained optimization problem, where the true solution is closely estimated, training the model with back-propagation in PyTorch. These models are applied to a set of test problems to assess performance.  In all of our tests, the neural network filter strength reduced the error of the P_N approximation dramatically.