Tensor Networks for Solving Realistic Time-independent Boltzmann Neutron Transport Equation

TL;DR

This paper introduces a tensor network method combining TT and QTT techniques to efficiently solve high-dimensional Boltzmann Neutron Transport equations, achieving massive data compression and significant computational speedup on standard hardware.

Contribution

The paper presents a novel TT/QTT tensor network approach for 3D neutron transport equations, enabling ultra-fast solutions with minimal memory and high accuracy, outperforming traditional solvers.

Findings

Achieved yottabyte-level data compression.

Realized over 7500 times speedup compared to PARTISN.

Maintained solution accuracy within 1e-5 discrepancy.

Abstract

Tensor network techniques, known for their low-rank approximation ability that breaks the curse of dimensionality, are emerging as a foundation of new mathematical methods for ultra-fast numerical solutions of high-dimensional Partial Differential Equations (PDEs). Here, we present a mixed Tensor Train (TT)/Quantized Tensor Train (QTT) approach for the numerical solution of time-independent Boltzmann Neutron Transport equations (BNTEs) in Cartesian geometry. Discretizing a realistic three-dimensional (3D) BNTE by (i) diamond differencing, (ii) multigroup-in-energy, and (iii) discrete ordinate collocation leads to huge generalized eigenvalue problems that generally require a matrix-free approach and large computer clusters. Starting from this discretization, we construct a TT representation of the PDE fields and discrete operators, followed by a QTT representation of the TT cores and solving the tensorized generalized eigenvalue problem in a fixed-point scheme with tensor network optimization techniques. We validate our approach by applying it to two realistic examples of 3D neutron transport problems, currently solved by the PARallel TIme-dependent SN (PARTISN) solver. We demonstrate that our TT/QTT method, executed on a standard desktop computer, leads to a yottabyte compression of the memory storage, and more than 7500 times speedup with a discrepancy of less than 1e-5 when compared to the PARTISN solution.