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Tensor network techniques, known for their low-rank approximation abilitythat breaks the curse of dimensionality, are emerging as a foundation of newmathematical methods for ultra-fast numerical solutions of high-dimensionalPartial Differential Equations (PDEs). Here, we present a mixed Tensor Train(TT)/Quantized Tensor Train (QTT) approach for the numerical solution oftime-independent Boltzmann Neutron Transport equations (BNTEs) in Cartesiangeometry. Discretizing a realistic three-dimensional (3D) BNTE by (i) diamonddifferencing, (ii) multigroup-in-energy, and (iii) discrete ordinatecollocation leads to huge generalized eigenvalue problems that generallyrequire a matrix-free approach and large computer clusters. Starting from thisdiscretization, we construct a TT representation of the PDE fields and discreteoperators, followed by a QTT representation of the TT cores and solving thetensorized generalized eigenvalue problem in a fixed-point scheme with tensornetwork optimization techniques. We validate our approach by applying it to tworealistic examples of 3D neutron transport problems, currently solved by thePARallel TIme-dependent SN (PARTISN) solver. We demonstrate that our TT/QTTmethod, executed on a standard desktop computer, leads to a yottabytecompression of the memory storage, and more than 7500 times speedup with adiscrepancy of less than 1e-5 when compared to the PARTISN solution.

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