Tensor Network Methods for Advection–Diffusion–Reaction Systems Using Quantum-Inspired Representations

Efficient Pauli-decomposition and multistage state-refinement for tensor network based differential equation solver

A multistage state-refinement heuristic that accelerates ITE convergence, reducing convergence time by up to two orders of magnitude is proposed.

Tensor-Network Finite Elements for Analytic Operator Equations

Operator equations (OEs) underpin quantitative modeling across science and engineering. Finite-element (FE) methods discretize continuous OEs into finite-dimensional algebraic systems, whereas tensor networks (TNs) provide flexible variational representations of correlated discrete systems. Here, we develop a framework that connects FE with TN for analytic OEs. The power of this method comes from its ability to convert highly non-linear partial differential equations into linear matrix equations. In particular, we show that FE discretization induces a hierarchy of multilinear interaction tensors, through which differential, integral, nonlinear, memory, and delay equations can be expressed within a common algebraic structure. The resulting systems are reformulated as weighted-residual optimization problems over TN degrees of freedom. Matrix-product-state calculations for one-dimensional linear and nonlinear diffusion reproduce conventional solutions with controlled error while preserving continuity and Neumann boundary conditions. The framework provides a common variational language for analytic OEs and establishes a direct connection between FE numerical formalism and TN variational algorithms, offering a general foundation for TN-based and quantum-inspired approaches to solving OEs.

Solving the Fokker–Planck Equation of Discretized Dean–Kawasaki Models with Functional Hierarchical Tensor

Abstract. We propose a particle-based workflow for approximating the time-dependent law of finite-volume discretizations of the Dean–Kawasaki model. After discretization, the state is a nonnegative vector whose total mass is conserved by the finite-volume scheme, and it is therefore supported on a probability simplex. To enable tensor-network density estimation, we map the simplex to an unconstrained Euclidean space using a centered logarithmic transform and then apply a wavelet transform that organizes degrees of freedom by spatial scale. On the transformed variables, we fit the probability density with a functional hierarchical tensor over a wavelet basis, i.e., a hierarchical-Tucker/tree-tensor-network representation of the coefficient tensor of a fixed univariate basis expansion. We illustrate the method on 1D and 2D examples with [Formula: see text] degrees of freedom, including cases with external potentials and pairwise interactions. The method accurately captures the site-wise correlations and other observables of the true model. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/Xun-Tang123/FHT_for_deans_equation and in the supplementary materials ( FHT_for_deans_equation-main.zip [27.5MB]). [Formula: see text]

Multi-scale physics-informed neural networks with Fourier features for approximating time-fractional PDEs

Efficient Treatment of Non-Linearity in Quantum Computational Fluid Dynamics Using Hybrid Tensor Networks

Nonlinear terms present a fundamental challenge for quantum computational fluid dynamics, as their implementation on inherently linear quantum hardware typically requires resource-intensive workarounds that limit scalability to large-scale simulations. We present a hybrid quantum-classical tensor network algorithm that addresses this bottleneck by combining variational time-stepping with quantum tensor programming to efficiently compile operators and time-dependent fields into quantum circuits. Within a probabilistic framework, we replace prior state-based nonlinear implementations with tensor-based block encodings, stabilizing success probabilities that otherwise decay exponentially with system size. Benchmarking on turbulent flow fields demonstrates that the algorithm maintains high success probabilities and moderate measurement overhead across increasing Reynolds numbers and grid resolutions. Compared to fully classical tensor network solvers, our hybrid approach yields substantial reductions in both memory footprint and computational cost, establishing a scalable pathway toward practical quantum advantage in scale-resolving CFD simulations.

Mass-Conserving Physics-Informed Neural Networks For The One-Dimensional Advection-Diffusion Equation

The advection-diffusion equation is a fundamental model of transport phenomena in which mass conservation is an essential physical constraint. While classical schemes such as Crank-Nicolson preserve this property by construction, Physics-Informed Neural Networks (PINNs) enforce only the local residual of the governing PDE and are therefore not guaranteed to conserve global quantities such as mass over long integration horizons. In this work, we examine the extent of this limitation for the periodic one-dimensional advection-diffusion equation and evaluate a Mass-Penalty PINN that augments the standard PINN loss with a soft mass-conservation constraint. We compare the performance of Vanilla PINN, Mass-Penalty PINN, and the Crank-Nicolson scheme across a range of Peclet numbers spanning diffusion-dominated to advection-dominated regimes, and over two simulation horizons representing short-term and long-term dynamics. The results show that, for short-term simulations, the Mass-Penalty PINN does not always provide a consistent improvement in accuracy. However, for long-term simulations, the Mass-Penalty PINN reduces the relative L2 error and mass conservation error by factors of approximately 9-67 and 15-215, respectively, compared with the Vanilla PINN, across the tested Peclet numbers. Further analysis reveals that the accuracy degradation observed in Vanilla PINN is predominantly caused by the accumulation of mass drift over time. These results demonstrate that incorporating a soft mass-conservation constraint substantially improves the long-term reliability of PINN for conservative transport problems, particularly in mitigating mass drift over extended simulation horizons.