Kinetic descriptions of fusion plasmas are throttled by the curse of dimensionality: a distribution function on a six-dimensional phase space is intractable to store on a dense grid, let alone to advance in time. We show that the kinetic distribution function is strongly compressible in a low-rank tensor-network (matrix-product-state, or tensor-train) representation and that this yields a practical solver on present-day classical hardware, not merely a storage economy. We cast the problem formally: the discretized distribution is a high-order tensor, its matrix-product-state factorization is exact up to a truncation error bounded by the discarded Schmidt weight (Eckart–Young), and the number of retained modes needed for a target accuracy scales with the entanglement entropy of the state. On a one-dimensional-in-space, one-dimensional-in-velocity Bhatnagar–Gross–Krook (BGK) kinetic test problem, discretized on a periodic N_x=128× N_v=64 grid and advanced by operator-split upwind streaming and implicit relaxation, a matrix-product-state truncation reaches a relative-L^2 error of 2.9×10^-6 at bond dimension =8 while storing 0.19× the dense grid, and the error falls exponentially in the bond dimension — from 1.8×10^-1 at =1 to 2.2×10^-10 at =16 and to the double-precision floor (~10^-14) by =24. The velocity moments inherit the same convergence (density error 1.1×10^-6, temperature error 2.8×10^-6 at =8), and the converged distribution carries a Schmidt entanglement entropy below 0.1 nats — the area-law signature that makes it compressible. Every headline quantity is a frozen anchor in the Kronos de-risking register (gate KX-L1-A16). Beyond the anchor we give the general-dimensional machinery that carries the method to the five- and six-dimensional gyrokinetic problem — matrix-product-operator representations of the streaming and collision maps, dimension-linear tensor-train construction by cross interpolation, and a rank-adaptive, conservative dynamical-low-rank integrator that evolves the state on the low-rank manifold without inverting small singular values — together with the derived exponential-to-linear crossover in both storage and per-step arithmetic that motivates it. The result is a classical, deployable-today complement to the fault-tolerant-horizon quantum kinetic route, and we place it as a cross-cutting, verification-first module inside the Kronos autonomous control architecture: an advisory/supervisory reduced-kinetics solver, exercised twin-first beneath a deterministic safety clamp, that hands authority back to the deterministic floor on out-of-distribution inputs. The single honest gate is that the demonstration is a collision-dominated reduced problem, and the rank growth of the fully nonlinear, collisionless six-dimensional gyrokinetic distribution at reactor parameters remains the open question we scope explicitly. Key results (frozen anchors): v = 2 v; t = 3.27 ×10^-3; sigma_1 = 9.4 ×10^-2; sigma_1 = 2.5 ×10^-3; sigma_1 = 4.7 ×10^-5. Methods & codes: CGYRO, TGLF, GENE, Sauter, VQE, ADAPT-VQE. Live verification: 1 gate validator(s) with live-recompute cards (1 reproduced). See the Verification section and data/verification.csv. Related: Paper page · De-risking register · 3D model · Learn more about Kronos Part of the 2026 Kronos publication series; independently re-run and stamped in the Kronos de-risking register (DOI 10.5281/zenodo.22645689). All numerical values are frozen design-point anchors; see the register. Public research artifact. No proprietary, financial, or supply-chain information is included. Previous version: DOI 10.5281/zenodo.22132192. Please cite this version.
Kinetic descriptions of fusion plasmas are throttled by the curse of dimensionality: a distribution function on a six-dimensional phase space is intractable to store on a dense grid, let alone to advance in time. We show that the kinetic distribution function is strongly compressible in a low-rank tensor-network (matrix-product-state, or tensor-train) representation and that this yields a practical solver on present-day classical hardware, not merely a storage economy. We cast the problem formally: the discretized distribution is a high-order tensor, its matrix-product-state factorization is exact up to a truncation error bounded by the discarded Schmidt weight (Eckart–Young), and the number of retained modes needed for a target accuracy scales with the entanglement entropy of the state. On a one-dimensional-in-space, one-dimensional-in-velocity Bhatnagar–Gross–Krook (BGK) kinetic test problem, discretized on a periodic N_x=128× N_v=64 grid and advanced by operator-split upwind streaming and implicit relaxation, a matrix-product-state truncation reaches a relative-L^2 error of 2.9×10^-6 at bond dimension =8 while storing 0.19× the dense grid, and the error falls exponentially in the bond dimension — from 1.8×10^-1 at =1 to 2.2×10^-10 at =16 and to the double-precision floor (~10^-14) by =24. The velocity moments inherit the same convergence (density error 1.1×10^-6, temperature error 2.8×10^-6 at =8), and the converged distribution carries a Schmidt entanglement entropy below 0.1 nats — the area-law signature that makes it compressible. Every headline quantity is a frozen anchor in the Kronos de-risking register (gate KX-L1-A16). Beyond the anchor we give the general-dimensional machinery that carries the method to the five- and six-dimensional gyrokinetic problem — matrix-product-operator representations of the streaming and collision maps, dimension-linear tensor-train construction by cross interpolation, and a rank-adaptive, conservative dynamical-low-rank integrator that evolves the state on the low-rank manifold without inverting small singular values — together with the derived exponential-to-linear crossover in both storage and per-step arithmetic that motivates it. The result is a classical, deployable-today complement to the fault-tolerant-horizon quantum kinetic route, and we place it as a cross-cutting, verification-first module inside the Kronos autonomous control architecture: an advisory/supervisory reduced-kinetics solver, exercised twin-first beneath a deterministic safety clamp, that hands authority back to the deterministic floor on out-of-distribution inputs. The single honest gate is that the demonstration is a collision-dominated reduced problem, and the rank growth of the fully nonlinear, collisionless six-dimensional gyrokinetic distribution at reactor parameters remains the open question we scope explicitly. Key results (frozen anchors): v = 2 v; t = 3.27 ×10^-3; sigma_1 = 9.4 ×10^-2; sigma_1 = 2.5 ×10^-3; sigma_1 = 4.7 ×10^-5. Methods & codes: CGYRO, TGLF, GENE, Sauter, VQE, ADAPT-VQE. Live verification: 1 gate validator(s) with live-recompute cards (1 reproduced). See the Verification section and data/verification.csv. Related: Paper page · De-risking register · 3D model · Learn more about Kronos Part of the 2026 Kronos publication series; independently re-run and stamped in the Kronos de-risking register (DOI 10.5281/zenodo.22645689). All numerical values are frozen design-point anchors; see the register. Public research artifact. No proprietary, financial, or supply-chain information is included. Previous version: DOI 10.5281/zenodo.22132192. Please cite this version.
Kinetic descriptions of fusion plasmas are throttled by the curse of dimensionality: a distribution function on a six-dimensional phase space is intractable to store on a dense grid, let alone to advance in time. We show that the kinetic distribution function is strongly compressible in a low-rank tensor-network (matrix-product-state, or tensor-train) representation and that this yields a practical solver on present-day classical hardware, not merely a storage economy. We cast the problem formally: the discretized distribution is a high-order tensor, its matrix-product-state factorization is exact up to a truncation error bounded by the discarded Schmidt weight (Eckart–Young), and the number of retained modes needed for a target accuracy scales with the entanglement entropy of the state. On a one-dimensional-in-space, one-dimensional-in-velocity Bhatnagar–Gross–Krook (BGK) kinetic test problem, discretized on a periodic N_x=128× N_v=64 grid and advanced by operator-split upwind streaming and implicit relaxation, a matrix-product-state truncation reaches a relative-L^2 error of 2.9×10^-6 at bond dimension =8 while storing 0.19× the dense grid, and the error falls exponentially in the bond dimension — from 1.8×10^-1 at =1 to 2.2×10^-10 at =16 and to the double-precision floor (~10^-14) by =24. The velocity moments inherit the same convergence (density error 1.1×10^-6, temperature error 2.8×10^-6 at =8), and the converged distribution carries a Schmidt entanglement entropy below 0.1 nats — the area-law signature that makes it compressible. Every headline quantity is a frozen anchor in the Kronos de-risking register (gate KX-L1-A16). Beyond the anchor we give the general-dimensional machinery that carries the method to the five- and six-dimensional gyrokinetic problem — matrix-product-operator representations of the streaming and collision maps, dimension-linear tensor-train construction by cross interpolation, and a rank-adaptive, conservative dynamical-low-rank integrator that evolves the state on the low-rank manifold without inverting small singular values — together with the derived exponential-to-linear crossover in both storage and per-step arithmetic that motivates it. The result is a classical, deployable-today complement to the fault-tolerant-horizon quantum kinetic route, and we place it as a cross-cutting, verification-first module inside the Kronos autonomous control architecture: an advisory/supervisory reduced-kinetics solver, exercised twin-first beneath a deterministic safety clamp, that hands authority back to the deterministic floor on out-of-distribution inputs. The single honest gate is that the demonstration is a collision-dominated reduced problem, and the rank growth of the fully nonlinear, collisionless six-dimensional gyrokinetic distribution at reactor parameters remains the open question we scope explicitly.Key results (frozen anchors): v = 2 v; t = 3.27 ×10^-3; N_v = 128 ×64; sigma_1 = 9.4 ×10^-2; sigma_1 = 2.5 ×10^-3; sigma_1 = 4.7 ×10^-5.Methods & codes: CGYRO, TGLF, GENE, Sauter, VQE, ADAPT-VQE.Live verification: 1 gate validator(s) with live-recompute cards (1 reproduced). See the Verification section and data/verification.csv.Related: Paper page · De-risking register · 3D model · Learn more about KronosPart of the 2026 Kronos publication series; independently re-run and stamped in the Kronos de-risking register (DOI 10.5281/zenodo.22645689).All numerical values are frozen design-point anchors; see the register.Public research artifact. No proprietary, financial, or supply-chain information is included.Note (REPLACE): This record replaces and supersedes DOI 10.5281/zenodo.22132192; please cite this version.
Characteristic functions of weighted sums of independent random variables exhibit low-rank structure in the quantized tensor train (QTT) representation, also known as matrix product states (MPS), enabling up to exponential compression of their fully non-Gaussian probability distributions. Under variable independence, the global characteristic function factorizes into local terms. Its low-rank QTT structure arises from intrinsic spectral smoothness in continuous models, or from spectral energy concentration as the number of components D grows in discrete models. We demonstrate this on weighted sums of Bernoulli and lognormal random variables. In the former, despite an adversarial, incompressible small-D regime, the characteristic function undergoes a sharp bond-dimension collapse for D≳300 components, enabling polylogarithmic time and memory scaling. In the latter, the approach reaches high-resolution discretizations of N=230 frequency modes on standard hardware, far beyond the N=224 ceiling of dense implementations. These compressed representations enable efficient computation of Value at Risk (VaR) and Expected Shortfall (ES), supporting applications in quantitative finance and beyond.
This record has been superseded. The current, updated version is A Tensor-Network Kinetic Solver for Fusion Transport: Matrix-Product-State Compression of the Distribution Function with Exponential Error Decay (DOI 10.5281/zenodo.22645768). Please read and cite the current version. This page stays online so existing citations keep resolving. The kinetic distribution function is highly compressible in a low-rank tensor-network representation, and that yields a practical classical solver, not a storage trick. On a 1D1V BGK test, a matrix-product-state truncation reaches relative-L² error 2×10−⁴ at rank 8 using ~0.19× the dense storage, with the error falling exponentially in rank. Unlike the fault-tolerant-horizon quantum route, this runs today.
Tensor networks (TNs), originally developed for simulating many-body quantum systems, provide a systematic framework for approximating high-dimensional fields. This is achieved by factorizing the field into interconnected tensors with small bond dimensions, thereby restricting the correlations captured across field bipartitions. Belonging to the family of TNs, the matrix product state (MPS) ansatz is utilized here as a reduced-order modeling framework to construct truncated representations of isotropic turbulent flow data. Two direct numerical simulation (DNS) datasets are considered: the hydrodynamic field of an incompressible three-dimensional flow, and a conserved Fickian scalar in a similar flow. Each field is encoded as an MPS through a sequence of singular value decompositions (SVDs) in which small singular values are discarded. The truncated representation is contracted back to the full grid, and the resulting reconstructed field is compared against DNS. An interleaved ordering of the spatial tensor indices of the transport variables is applied prior to decomposition in order to localize the dominant inter-tensor correlations. Velocity reconstructions achieve 99.8% fidelity using only 5% of the original DNS memory, while the scalar field reaches the same fidelity at 15% memory usage. A wide range of lower- and higher-order statistics, including velocity gradients, dissipation, and structure functions, are systematically examined. At these compression levels, the total kinetic energy and the scalar energy are both recovered within 0.2% relative error, while the mean dissipation and mean scalar dissipation remain within approximately 10% of the DNS generated values. These findings support the suitability of MPS for scalable reduced-order analysis of complex turbulent datasets and motivate further exploration of TN-based methods in computational turbulence.
High performance computers produce extreme-scale data sets that require sampling or compression if they are to be used to their full potential. Existing data compression techniques typically exploit features such as sparsity in the data, homogeneity in the data, or {\it a priori} knowledge of what subsets of data are of most interest. Fluid dynamics data in general do not exhibit these features and so are attractive test beds for generic compression techniques that are objective, robust, and tuneable with respect to information lost due to compression. Presented here is a method based on tensor networks, specifically matrix product states or tensor trains, that meets these requirements. The method is demonstrated for compression in one-dimension and is extensible to higher dimensionality. Lossless compression is demonstrated for random Fourier series for sufficiently high bond dimension of the tensor network, with the memory required to store the tensor network scaling directly proportional to the bond dimension. The lossy compression exhibited at lower bond dimension can be well within the relative error of many fluid simulations. The compression algorithm is tested for the time evolution of Burger's equation with excellent results. We additionally demonstrate the capability to perform computations in the compressed form through a tensor network periodic convolution that can be orders of magnitude faster than using fast Fourier transforms and the convolution theorem. In addition to being an attractive method for working with data sets generated by existing computers, the tensor network methods utilised are directly translatable to the emerging paradigm of quantum computing.
Kinetic simulations of collisionless (or weakly collisional) plasmas using the Vlasov equation are often infeasible due to high-resolution requirements and the exponential scaling of computational cost with respect to dimension. Recently, it has been proposed that matrix product state (MPS) methods, a quantum-inspired but classical algorithm, can be used to solve partial differential equations with exponential speed-up, provided that the solution can be compressed and efficiently represented as a MPS within some tolerable error threshold. In this work, we explore the practicality of MPS methods for solving the Vlasov-Poisson equations for systems with one coordinate in space and one coordinate in velocity, and find that important features of linear and nonlinear dynamics, such as damping or growth rates and saturation amplitudes, can be captured while compressing the solution significantly. Furthermore, by comparing the performance of different mappings of the distribution functions onto the MPS, we develop an intuition of the MPS representation and its behavior in the context of solving the Vlasov-Poisson equations, which will be useful for extending these methods to higher-dimensional problems.