arXiv subjects

Zhiwen Zhang

Publications and source records attributed to Zhiwen Zhang.

At least 19 recordsLinked to original sources

arXiv2026

Convergence Analysis of a Stochastic Interacting Particle-Field Algorithm for 3D Parabolic-Parabolic Keller-Segel Systems

Chemotaxis models describe the movement of organisms in response to chemical gradients. In this paper, we present a stochastic interacting particle-field algorithm with a random batch approximation (SIPF-$r$) for the three-dimensional (3D) parabolic-parabolic Keller-Segel (KS) system, also referred to as the fully parabolic KS system. The SIPF-$r$ method approximates the KS system by coupling particle-based representations of the density with a smooth field variable computed using spectral methods. By incorporating the random batch method (RBM), we bypass the mean-field limit and significantly reduce computational complexity. Under mild assumptions on the regularity of the original KS system and the boundedness of numerical approximations, we prove that the empirical measure of the SIPF-$r$ particle system converges, with high probability, to the exact measure of the limiting McKean-Vlasov process in the $1$-Wasserstein distance. Finally, we present numerical experiments to validate the theoretical convergence rates, and demonstrate the performance and robustness of the SIPF-$r$ method as a diagnostic tool for intense focusing and potential finite-time singularity in 3D, subject to critical initial mass thresholds in the system.

math.NA↗

arXiv2026

A fast stochastic interacting particle-field method for 3D parabolic parabolic Chemotaxis systems: numerical algorithms and error analysis

In this paper, we develop a novel numerical framework, namely the stochastic interacting particle-field method with particle-in-cell acceleration (SIPF-PIC), for the efficient simulation of the three-dimensional (3D) parabolic-parabolic Keller-Segel (KS) systems. The SIPF-PIC method integrates Lagrangian particle dynamics with spectral field solvers by leveraging localized particle-grid interpolations and fast Fourier transform (FFT) techniques. For $P$ particles and $H$ Fourier modes per spatial dimension, the SIPF-PIC method achieves a computational complexity of $O(P + H^3 \log H)$ per time step, a significant improvement over the original SIPF method (proposed in \cite{SIPF1}), which has a computational complexity of $O(PH^3)$, while preserving numerical accuracy. Moreover, we carry out a rigorous error analysis for the proposed method and establish the corresponding error estimates. Finally, we present numerical experiments to validate the convergence order and demonstrate the computational efficiency of SIPF-PIC. Further numerical experiments show the method's capability of capturing complex blowup dynamics beyond single-point collapse, including ring-shaped singularities.

math.NA↗

arXiv2026

A Unified Kullback--Leibler Divergence Analysis of Generative Diffusion Models via Entropy Production Rate

We introduce a unified framework for the error analysis of generative models based on the entropy production rate of the forward-reverse diffusion process pair. For a pair of continuity equation flows, the rate admits a closed velocity form identity whose time integral decomposes the terminal Kullback--Leibler (KL) divergence into the sum of an initialization error, a score approximation error, and a time-discretization error. By analyzing the entropy production at the level of marginal distributions, rather than in path space, our framework yields a sharp convergence rate of $\mathcal{O}(h^2)$ for the Euler-Maruyama sampler, where $h$ is the step size. This improves upon the $\mathcal{O}(h)$ rates typically obtained from Girsanov's path-space analyses. Furthermore, our framework unifies the analysis of score-based SDEs, probability-flow ODEs, and stochastic interpolants by varying diffusion coefficients within a single inequality, revealing the trade-off between deterministic and stochastic sampling. Numerical experiments confirm the predicted scaling with step size and terminal time.

math.NA↗

arXiv2026

A Pseudo-time Data-Driven Framework for Model Reduction of Linear Operator Equations

This paper proposes a novel Pseudo-time Proper Orthogonal Decomposition (POD) framework to enable model reduction for stationary problems lacking temporal snapshot data. By recasting static operator equations into a pseudo-dynamic evolution form, we artificially generate temporal data while preserving the system's intrinsic spectral properties. Mathematically, we rigorously prove the exponential convergence of the pseudo-time trajectory to the exact stationary solution. Furthermore, the approximation properties of the generated POD basis functions are rigorously established. Finally, the universality and accuracy of the framework are validated both theoretically and numerically across two representative settings: elliptic inverse source problems and Fredholm integral equations of the first kind.

math.NA↗

arXiv2026

Parametric Sensitivity of POD Reduced-Order Models for Semilinear Evolution Equations with Applications to G-Equations

Proper Orthogonal Decomposition (POD) provides low-dimensional surrogate models of evolution equations from solution snapshots. In parameterized problems, however, a basis computed at one parameter value need not remain accurate at another, while recomputing the basis for every parameter in a many-query study is costly. We develop a sensitivity analysis for POD reduced-order models of a class of parameterized semilinear evolution equations that includes the viscous G-equation and a viscous strain G-equation in their mean-free formulations. The analysis is based on a cross-space Lipschitz condition for the nonlinear operator, which accommodates the nonsmooth, gradient-dependent nonlinearities of these flame-propagation models, and on continuity moduli quantifying the parameter dependence of the bilinear form and of the nonlinearity. We prove that when a POD basis constructed at a reference parameter is applied to nearby query parameters, the resulting error variation is controlled by the corresponding parameter modulus, with constants independent of the POD dimension, the number of time steps, and the perturbation magnitude. Numerical experiments are consistent with the predicted modulus-dependent sensitivity behavior, and illustrate the practical robustness of basis reuse for nearby parameters.

math.NA↗

arXiv2026

A Bernoulli Phase-Fitted finite difference method with wavenumber-explicit analysis for the Helmholtz problem

A new Bernoulli phase-fitted finite difference method for the Helmholtz equation is introduced, obtained by applying a complexified Scharfetter--Gummel flux to the one-way factors of the operator. The rigorous analysis is developed for the one-dimensional Helmholtz problem with impedance boundary conditions. For the homogeneous problem, the scheme reproduces sampled plane-waves exactly, both in the interior and at the discrete impedance boundary closures. For the inhomogeneous problem, we prove wavenumber-explicit stability, consistency, and second-order convergence estimates for all nondegenerate mesh wavenumbers $kh\notinπ\mathbb Z$. Under the fixed-resolution condition $kh\le s_0<π$ and $kL\geπ$, the estimates yield a pollution-free convergence theory. Numerical experiments confirm the plane-wave exactness and the predicted convergence behavior, and show favorable fixed-resolution performance compared with standard and dispersion-corrected finite difference methods.

math.NA↗

arXiv2026

A Lie-algebraic approach to non-Markovian quantum dynamics

In this paper, we study the non-Markovian quantum dynamics in quantum computations from the perspective of a Lie algebraic approach based on numerical analysis. By vectorizing the density matrix of quantum states, the non-Markovian evolutions can be represented with high-dimensional linear time-varying equations, where the time-varying parameters arise from the non-Markovian interactions between the quantum system and environment. We study the Magnus expansion of such linear time-varying quantum dynamics and clarify how the truncation errors for the first- and second-order Magnus expansions are influenced by the non-Markovian properties. Besides, when the quantum states are measured for filtering, the dynamics can be modeled as time-varying stochastic differential equations due to the existence of measurement noise. The Magnus expansions based on quantum stochastic filtering are different when the quantum measurement noises are modeled in an {Itô} or Stratonovich approach, rendering different truncation errors. Based on this, numerical simulations further demonstrate the efficiency of Magnus expansions in simulating non-Markovian quantum dynamics without or with stochasticity, and how the truncation errors are influenced by the Lie algebras in the Liouville space.

quant-ph↗

arXiv2026

Quasi-Monte Carlo time-splitting methods for the Schrödinger equation with Gaussian random potential

In this paper, we study the Schrödinger equation with a Gaussian random potential (SE-GP) and develop an efficient numerical method to approximate the expectation of physical observables. The unboundedness of Gaussian random variables poses significant difficulties in both sampling and error analysis. Under time-splitting discretizations of SE-GP, we establish the regularity of the semi-discrete solution in the random space. Then, we introduce a non-standard weighted Sobolev space with properly chosen weight functions, and obtain a randomly shifted lattice-based quasi-Monte Carlo (QMC) quadrature rule for efficient sampling. This approach leads to a QMC time-splitting (QMC-TS) scheme for solving the SE-GP. We prove that the proposed QMC-TS method achieves a dimension-independent convergence rate that is almost linear with respect to the number of QMC samples. Numerical experiments illustrate the sharpness of the error estimate.

math.NA↗

arXiv2026

A Novel Stochastic Particle-Field Algorithm for a Reaction-Diffusion-Advection Cancer Invasion Model

In this paper, we present a novel numerical framework for solving a specific biological reaction-diffusion-advection system of cancer growth in three dimensions (3D) using particles of variable mass. We adopt empirical particle measures to represent cell density and dynamically construct the concentration fields of multiple related chemical species throughout the 3D domain. Efficient interaction between the particles and the spatial grid is achieved through a Particle-in-Cell (PIC) algorithm, while diffusion in space is solved rapidly using a spectral method. We demonstrate that for this particular system, the rate of change of particle mass remains bounded over finite time intervals. Furthermore, in addition to the inherent positivity preservation of cell density guaranteed by the empirical particle measures, the concentrations constructed by the algorithm are also unconditionally positivity-preserving on the spatial grid. Moreover, we present a rigorous error analysis for the proposed method, and numerical experiments confirm the theoretical convergence rates. To the best of our knowledge, this is the first numerical work to solve this system in three dimensions, wherein a rapid spread of cells driven by haptotactic flux is observed, similar to the behavior documented in the two-dimensional case.

math.NA↗

arXiv2026

Tensor train methods for high-dimensional nonlinear filtering problems with correlated noise

Nonlinear filtering with correlated noise leads to a Duncan-Mortensen-Zakai (DMZ) equation in the form of a stochastic partial differential equation (SPDE). Unlike the independent noise case, the presence of correlation prevents the classical invertible transformation that reduces the DMZ equation to a deterministic partial differential equation, requiring a direct numerical treatment of the SPDE. This paper develops a tensor train (TT) based framework for solving medium- to high-dimensional DMZ equations with correlated noise. Spatial discretization transforms the SPDE into a high-dimensional stochastic differential system, which is efficiently compressed using TT approximation. A semi-implicit Milstein scheme is employed for temporal integration to ensure stability and accuracy. Under suitable regularity assumptions, we establish a convergence analysis of the proposed method. In particular, the spatial error is controlled by both the mesh size and the prescribed TT approximation accuracy. In the temporal direction, the convergence is proved by estimating stochastic integrals involving drifted observations, without invoking a change-of-measure argument. Numerical experiments demonstrate that the proposed method achieves stable and accurate performance for cubic sensor problems. In challenging multi-modal settings, where particle filter and extended Kalman filter deteriorate, the proposed method maintains accuracy and effectively captures the posterior distribution.

math.NA↗

arXiv2026

On the Regularity and Generalization of One-Step Wasserstein-guided Generative Models for PDE-Induced Measures

Despite the remarkable empirical success of generative models, the available theory on their statistical accuracy in scientific computing remains largely pessimistic. This paper develops a theoretical framework for understanding the regularity of transport maps and the generalization properties of one-step Wasserstein-guided generative models for PDE-induced probability measures. We consider normalized target densities associated with linear elliptic and parabolic equations on bounded domains, as well as diffusion and Fokker--Planck equations on the torus. Under standard structural assumptions, we prove that these target measures satisfy doubling conditions. By combining this fact with regularity theory for optimal transport between doubling measures, we show that the optimal transport map from a uniform source measure to the target measure is Hölder continuous. This regularity yields an approximation-theoretic justification for one-step generative models that learn PDE-induced distributions via a single pushforward map. As a representative instance, we study DeepParticle and derive excess-risk bounds characterizing the discrepancy between the learned map and the population-optimal map. We also establish a robustness estimate under target shift and illustrate the theory with experiments which support the derived rates.

cs.LG↗

arXiv2026

A Recursive Polynomial Chaos Evolution Method for Stochastic Differential Equations

Numerical simulation of stochastic differential equations over long time intervals poses significant computational challenges. In this paper, we propose a novel recursive polynomial chaos evolution method that achieves model reduction without sampling by exploiting the Markov property to maintain a fixed low-dimensional representation throughout the time evolution. At each time step, we construct orthogonal polynomial bases adapted to the current probability measure, and project the one-step-ahead solution onto this new basis together with the new Brownian increments. This dynamic updating strategy effectively reduces the dimension of the random variables during long-time evolution. Under appropriate assumptions, we prove the convergence of the method, specifically that the distributions generated by the method preserve convergence in the Wasserstein-1 distance. We present numerical results demonstrating that the method can accurately capture complex dynamical behaviors with high accuracy and low computational cost.

math.NA↗

arXiv2026

Cell-induced densification and tether formation in fibrous extracellular matrices with biomimetic physics-informed neural networks

Nonconvex multi-well energies in cell-induced phase transitions give rise to fine-scale microstructures, low-regularity transition layers and sharp interfaces, all of which pose numerical challenges for physics-informed learning. Here we introduce biomimetic physics-informed neural networks (Bio-PINNs), which implement a near-to-far curriculum by progressively revealing the computational domain away from the cell boundary and combining this schedule with a deformation-uncertainty proxy that concentrates collocation points near evolving transition layers and tether-forming regions. Across single-cell and multicellular benchmarks, Bio-PINNs recover the densified phase more reliably near cell boundaries and in intercellular gaps, while capturing tether morphology more faithfully than representative ungated and residual-driven adaptive baselines.

cs.LG↗

arXiv2026

Two-Step Diffusion: Fast Sampling and Reliable Prediction for 3D Keller--Segel and KPP Equations in Fluid Flows

We study fast and reliable generative transport for the 3D KS (Keller-Segel) and KPP (Kolmogorov-Petrovsky-Piskunov) equations in the presence of fluid flows with the goal to approximate the map between initial and terminal distributions for a range of physical parameters $σ$ under the Wasserstein metric. To minimize the inaccuracy of direct Wasserstein solver, we propose a two-stage pipeline that retains one-step efficiency while reinstating an explicit $W_2$ objective where it is tractable. In Stage I, a Meanflow-style regressor yields a deterministic, one-step global transport that moves particles close to their terminal states. In Stage II, we freeze this initializer and train a near-identity corrector (Deep Particle, DP) that directly minimizes a mini-batch $W_2$ objective using warm-started optimal transport couplings computed on the Meanflow outputs. Crucially, after the one-step transport (from Stage I) concentrating mass on the approximated correct support, the induced geometry stabilizes high-dimensional $W_2$ computation of the direct Wasserstein solver. We validate our construction in the 3D KS and KPP equations subject to fluid flows with ordered and chaotic streamlines.

physics.comp-ph↗

arXiv2026

A Stochastic Genetic Interacting Particle Method for Reaction-Diffusion-Advection Equations

We develop and analyze a stochastic genetic interacting particle method (SGIP) for reaction-diffusion-advection (RDA) equations. The SGIP method employs operator splitting to approximate the advection-diffusion and reaction processes, treating the former using particle drift-diffusion and the latter via exact or implicit integration of reaction dynamics over bins, where particle density is estimated using a histogram. A key innovation is the incorporation of adaptive resampling to close the loop of particle and density field description of solutions, mimicking the selection mechanism in genetics. Resampling is also crucial for maintaining long-term stability by redistributing particles in accordance with the evolving density field. We provide a comprehensive error analysis and establish convergence bounds under appropriate regularity assumptions. Numerical experiments in one to three space dimensions demonstrate the method's effectiveness across various reaction types (Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP), cubic, Arrhenius) and flow configurations (shear, cellular, cat's eye, Arnold-Beltrami-Childress (ABC) flows), showing excellent agreement with the finite difference method (FDM) while offering computational advantages for complex flow geometries and higher-dimensional problems.

math.NA↗

arXiv2026

Moment propagation of a Vlasov-Poisson system for ions flow in the quasi-neutral regime

In light of recent work in the global well-posedness of solutions for an ionic Vlasov-Poisson system, as demonstrated by Griffin-Pickering and Iacobelli, the current work focuses on the moment propagation of the corresponding system in quasi-neutral regime. Such moment propagation result relies on an estimate of $Q_*(t)=|V(t;0,x,v)-V(0;0,x,v)|$, where $V(s;t,x,v)$ represents the solution of the characteristic ordinary differential equation associated with the Vlasov-Poisson system.

math.AP↗

arXiv2025

Computing large deviation rate functions of entropy production for diffusion processes by an interacting particle method

We develop an interacting particle method (IPM) for computing the large deviation rate function of entropy production for diffusion processes, with emphasis on the vanishing-noise limit and high dimensions. The crucial ingredient to obtain the rate function is the computation of the principal eigenvalue $λ$ of elliptic, non-self-adjoint operators. We show that this principal eigenvalue can be approximated in terms of the spectral radius of a discretized evolution operator, which is obtained from an operator splitting scheme and an Euler--Maruyama scheme with a small time step size. We also show that this spectral radius can be accessed through a large number of iterations of this discretized semigroup, which is suitable for computation using the IPM. The IPM applies naturally to problems in unbounded domains and scales easily to high dimensions. We show numerical examples of dimensions up to 16, and the results show that our numerical approximation of $λ$ converges to the analytical vanishing-noise limit within visual tolerance with a fixed number of particles and a fixed time step size. It is numerically shown that the IPM can adapt to singular behaviors in the vanishing-noise limit. We also apply the IPM to explore situations with no explicit formulas of the vanishing-noise limit. Our paper appears to be the first one to obtain numerical results of principal eigenvalue problems for non-self-adjoint operators in such high dimensions.

math.NA↗

arXiv2025

A DeepLagrangian method for learning and generating aggregation patterns in multi-dimensional Keller-Segel chemotaxis systems

The Keller-Segel (KS) chemotaxis system is used to describe the overall behavior of a collection of cells under the influence of chemotaxis. However, solving the KS chemotaxis system and generating its aggregation patterns remain challenging due to the emergence of solutions exhibiting near-singular behavior, such as finite-time blow-up or concentration phenomena. Building on a Lagrangian framework of the KS system, we develop DeepLagrangian, a self-adaptive density estimation method that learns and generates aggregation patterns and near-singular solutions of the KS system in two- and three-dimensional (2D and 3D) space under different physical parameters. The main advantage of the Lagrangian framework is its inherent ability to adapt to near-singular solutions. To develop this framework, we normalize the KS solution into a probability density function (PDF), derive the corresponding normalized KS system, and utilize the property of the continuity equation to rewrite the system into a Lagrangian framework. We then define a physics-informed Lagrangian loss to enforce this framework and incorporate a flow-based generative model, called the time-dependent KRnet, to approximate the PDF by minimizing the loss. Furthermore, we integrate time-marching strategies with the time-dependent KRnet to enhance the accuracy of the PDF approximation. After obtaining the approximate PDF, we recover the original KS solution. We also prove that the Lagrangian loss effectively controls the Kullback-Leibler (KL) divergence between the approximate PDF and the exact PDF. In the numerical experiments, we demonstrate the accuracy of our DeepLagrangian method for the 2D and 3D KS chemotaxis system with/without advection.

math.NA↗