Speaker: Benjamin Plumridge, UTK

Title: Neural Network-Based Adaptive Filtering of the Spherical Harmonic Method
Abstract: 

We study the filtered spherical harmonics method ($FP_N$) applied to the radiation transport equation (RTE).

The $FP_N$ method is a variant of the spherical harmonic method ($P_N$) which introduces regularization, via a filter operator, dampening oscillatory 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 minimization problem and train the model with backpropagation 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 dramatically improved the quality of the $P_N$ approximation.