Typical machine learning (ML) methods are difficult to apply to radiation transport due to the large computational cost associated with simulating problems to create training data. Physics-informed Neural Networks (PiNNs) are a ML method that train a neural network with the residual of a governing equation as the loss function. This allows PiNNs to be trained in a low-data regime in the absence of (experimental or synthetic) data. PiNNs also are trained on points sampled within the phase-space volume of the problem, which means they are not required to be evaluated on a mesh, providing a distinct advantage in solving the linear Boltzmann transport equation, which is difficult to discretize. We have applied PiNNs to solve the streaming and interaction terms of the linear Boltzmann transport equation to create an accurate ML model that is wrapped inside a traditional source iteration process. We present an application of Fourier Features to PiNNs that yields good performance on heterogeneous problems. We also introduce a sampling method based on heuristics that improves the performance of PiNN simulations. The results are presented in a suite of one-dimensional radiation transport problems where PiNNs show very good agreement when compared to fine-mesh answers from traditional discretization techniques.

In this paper, a proper orthogonal decomposition based reduced-order model is presented for parametrized multiphysics computations. Our application physics is Doppler feedback in a simplified model of the molten salt fast reactor concept. The reduced model is created using the method of snapshots where the offline training set is obtained by exercising a full-order model created with the OpenFOAM based multiphysics solver, GeN-Foam. The steady state models solve the multi-group diffusion k-eigenvalue equations with moving precursors together with the energy equation. A fixed velocity field is assumed throughout the computations, hence the momentum and continuity equations are not solved. The discrete empirical interpolation method is used for the efficient coupling of the ROM solvers, while the input parameter space is surveyed using the improved distributed latin hypercube sampling algorithm.

It is well known that the diffusion synthetic acceleration (DSA) methods for the S_n equations become ineffective in the Fokker-Planck forward-peaked scattering limit. In response to this deficiency, Morel and Manteuffel ( 1991 Morel, J. E. and Manteuffel, T. A. 1991. An angular multigrid acceleration technique for S_n equations with highly forward-peaked scattering. Nuclear Science and Engineering, 107: 330–342. [Taylor & Francis Online], [Web of Science ®] [Google Scholar] ) developed an angular multigrid method for the 1-D S_n equations. This method is very effective, costing roughly twice as much as DSA per source iteration, and yielding a maximum spectral radius of approximately 0.6 in the Fokker-Planck limit. Pautz, Adams, and Morel (PAM) (1999) later generalized the angular multigrid to 2-D, but it was found that the method was unstable with sufficiently forward-peaked mappings between the angular grids. The method was stabilized via a filtering technique based on diffusion operators, but this filtering also degraded the effectiveness of the overall scheme. The spectral radius was not bounded away from unity in the Fokker-Planck limit, although the method remained more effective than DSA. The purpose of this article is to recast the multidimensional PAM angular multigrid method without the filtering as an S_n preconditioner and use it in conjunction with the Generalized Minimal RESidual (GMRES) Krylov method. The approach ensures stability and our computational results demonstrate that it is also significantly more efficient than an analogous DSA-preconditioned Krylov method.