Quantum turbulence spans length scales from the system size L to the healing length ξ , making direct numerical simulations (DNS) of the Gross-Pitaevskii (GP) equation computationally expensive when L ≫ ξ . We present a matrix-product state (MPS) solver for the GP equation that efficiently compresses the wavefunction by truncating weak interlength-scale correlations. This approach reduces memory usage by factors ranging from 10 × to over 10 000 × compared to DNS. We benchmark our approach on nonlinear excitations, namely dark solitons (1D) and quantized vortices (2D, 3D), capturing key dynamics such as Kelvin-wave propagation and vortex-ring emission in the case of vortex-line reconnection. For turbulent states composed of multiple nonlinear excitations, we find that the memory compression of the MPS representation is directly proportional to the soliton or vortex densities. We also accurately reproduce established results from two-point correlation functions and energy spectra, where we recover the incompressible kinetic energy spectrum with little memory overhead. These results demonstrate the representative capabilities of the MPS ansatz for quantum turbulence and pave the way for studying this nonequilibrium state using previously prohibited system sizes. Moreover, the MPS ansatz can be adapted to existing quantum algorithms, providing a framework to implementing our time-evolution method on near-future quantum processing units.
Quantum turbulence spans length scales from the system size L to the healing length ξ , making direct numerical simulations (DNS) of the Gross-Pitaevskii (GP) equation computationally expensive when L ≫ ξ . We present a matrix-product state (MPS) solver for the GP equation that efficiently compresses the wavefunction by truncating weak interlength-scale correlations. This approach reduces memory usage by factors ranging from 10 × to over 10 000 × compared to DNS. We benchmark our approach on nonlinear excitations, namely dark solitons (1D) and quantized vortices (2D, 3D), capturing key dynamics such as Kelvin-wave propagation and vortex-ring emission in the case of vortex-line reconnection. For turbulent states composed of multiple nonlinear excitations, we find that the memory compression of the MPS representation is directly proportional to the soliton or vortex densities. We also accurately reproduce established results from two-point correlation functions and energy spectra, where we recover the incompressible kinetic energy spectrum with little memory overhead. These results demonstrate the representative capabilities of the MPS ansatz for quantum turbulence and pave the way for studying this nonequilibrium state using previously prohibited system sizes. Moreover, the MPS ansatz can be adapted to existing quantum algorithms, providing a framework to implementing our time-evolution method on near-future quantum processing units.
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.
Direct numerical simulation (DNS) of turbulent reactive flows has been the subject of significant research interest for several decades. Accurate prediction of the effects of turbulence on the rate of reactant conversion, and the subsequent influence of chemistry on hydrodynamics remain a challenge in combustion modeling. The key issue in DNS is to account for the wide range of temporal and spatial physical scales that are caused by complex interactions of turbulence and chemistry. In this work, a new computational methodology is developed that is shown to provide a viable alternative to DNS. The framework is the matrix product state (MPS), a form of tensor network (TN) as used in computational many body physics. The MPS is a well-established ansatz for efficiently representing many types of quantum states in condensed matter systems, allowing for an exponential compression of the required memory compared to exact diagonalization methods. Due to the success of MPS in quantum physics, the ansatz has been adapted to problems outside its historical domain, notably computational fluid dynamics. Here, the MPS is used for computational simulation of a shear flow under non-reacting and nonpremixed chemically reacting conditions. Reductions of 30% in memory are demonstrated for all transport variables, accompanied by excellent agreements with DNS. The anastaz accurately captures all pertinent flow physics such as reduced mixing due to exothermicity & compressibility, and the formation of eddy shocklets at high Mach numbers. A priori analysis of DNS data at higher Reynolds numbers shows compressions as large as 99.99% for some of the transport variables. This level of compression is encouraging and promotes the use of MPS for simulations of complex turbulent combustion systems.
Turbulent thermal convection governs heat transport in systems ranging from stellar interiors to industrial heat exchangers. Two-dimensional Rayleigh-Bénard convection serves as a paradigm for these flows, reproducing key features such as thin boundary layers, large-scale circulation, and sustained plume dynamics. While Matrix Product State (MPS) methods have demonstrated significant compression of isothermal turbulent fields, their application to buoyancy-driven flows with active thermal coupling has remained unexplored. We apply MPS to two-dimensional Rayleigh-Bénard convection with dynamical simulations up to Ra=1010. An a priori decomposition of DNS snapshots up to Ra=1011 shows that the bond dimension χ required to represent the flow fields grows without saturation, in contrast to the plateauing of χ reported for velocity fields in isothermal 2D turbulence. Crucially, however, dynamical simulations solving the governing equations directly in the compressed MPS format at fixed χ show that the χ required to recover statistical observables, such as the Nusselt number, scales significantly more favorably with Ra than the a priori complexity suggests. At Ra=1010, a relative error of 1.8% in the mean Nusselt number is achieved with a nearly 9-fold reduction in degrees of freedom, using a χ comparable to that required at Ra=109. Spectral analysis confirms the progressive recovery of spatial and temporal scales with increasing χ. These findings establish MPS as a scalable tool for simulating thermally driven turbulence, suggesting the method may remain viable for investigations of the ultimate regime at substantially higher Ra.
We examine the energy cascades and quantum vortex structures in two-dimensional quantum turbulence through a special unitary time evolution algorithm. An early attempt at using the Lattice Boltzmann Method proved successful in correctly representing some features of the Nonlinear Schrodinger System (NLS), such as the phase shift following the one-dimensional soliton-soliton collision, as well as the two-dimentional modulation instability. However, to accurately evaluate NLS, the implicit Euler method is required to resolve the time evolution, which is computationally expensive. A more accurate and efficient method, the Quantum Lattice Gas model is employed to simulate the quantum turbulence governed by the Gross-Pitaevskii equation, an equaiton that describes the evolution of the ground state wave function for a Bose-Einstein condensate (BEC). It is discovered that when the ratio of the internal energy to the kinetic energy is below 0.05, an unexpected short Poincare recurrence occurs independent of the initial profile of the wave function. It is demonstrated that this short recurrence is destroyed as the internal energy is strengthened. to compare the two-dimensional quantum turbulence with its classical counterpart, the incompressible energy spectra of quantum turbulence is analyzed. However, the result reveals no sign of dual cascades which is a hallmark of the classical incompressible two-dimensional fluid (inverse energy cascade to large scales with a direct cascade of enstrophy to small scales). It is the spectra of the compressible energy that can exhibits multiple cascades, but this is strongly dependent on the initial condition.
Abstract Computational fluid dynamics is both a thriving research field and a key tool for advanced industry applications. However, the simulation of turbulent flows in complex geometries is a compute-power intensive task due to the vast vector dimensions required by discretized meshes. We present a complete and self-consistent full-stack method to solve incompressible fluids with memory and run time scaling logarithmically in the mesh size. Our framework is based on matrix-product states, a compressed representation of quantum states. It is complete in that it solves for flows around immersed objects of arbitrary geometries, with non-trivial boundary conditions, and self-consistent in that it can retrieve the solution directly from the compressed encoding, i.e. without passing through the expensive dense-vector representation. This framework lays the foundation for a generation of more efficient solvers of real-life fluid problems.
Tensor network algorithms can efficiently simulate complex quantum many-body systems by utilizing knowledge of their structure and entanglement. These methodologies have been adapted recently for solving the Navier-Stokes equations, which describe a spectrum of fluid phenomena, from the aerodynamics of vehicles to weather patterns. Within this quantum-inspired paradigm, velocity is encoded as matrix product states (MPS), effectively harnessing the analogy between interscale correlations of fluid dynamics and entanglement in quantum many-body physics. This particular tensor structure is also called quantics tensor train (QTT). By utilizing NVIDIA's cuQuantum library to perform parallel tensor computations on GPUs, our adaptation speeds up simulations by up to 12.1 times. This allows us to study the algorithm in terms of its applicability, scalability, and performance. By simulating two qualitatively different but commonly encountered 2D flow problems at high Reynolds numbers up to 1 × 10 7 using a fourth-order time stepping scheme, we find that the algorithm has a potential advantage over direct numerical simulations in the turbulent regime as the requirements for grid resolution increase drastically. In addition, we derive the scaling χ = O ( poly ( 1 / ε ) ) for the maximum bond dimension χ of MPS representing turbulent flow fields, with an error ε , based on the spectral distribution of turbulent kinetic energy. Our findings motivate further exploration of related quantum algorithms and other tensor network methods.