arXiv subjects

Zhongjian Wang

Publications and source records attributed to Zhongjian Wang.

At least 19 recordsLinked to original sources

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

Posterior sampling by source-space MCMC via prior-based few-step transport maps

Bayesian inference increasingly uses informative but implicit priors represented only by samples, such as historical ensembles, simulator outputs, and pretrained generative models. The same computational problem appears in the test-time guidance task (generalized Bayes), where an explicit positive weight, e.g., an exponentiated reward, tilts an implicit prior. We develop a framework for source-space generalized Bayesian inference that combines inexpensive few-step prior transports with posterior stability guarantees. Specifically, we represent the prior using a one- or few-step improved MeanFlow (iMF) map and perform posterior sampling in its Gaussian source space. We establish Wasserstein error bounds between the exact and learned posteriors in terms of the joint population iMF and auxiliary-velocity loss, decomposed into training suboptimality and model-class approximation error. In the iMF source space, we adopt parallel tempering with preconditioned Crank-Nicolson updates and introduce a hybrid variant that incorporates split Hamiltonian Monte Carlo to improve sampling efficiency. Synthetic experiments show that the proposed framework can approximate posterior distributions accurately and efficiently, while CLIP-guided ImageNet experiments demonstrate its ability to steer a pretrained iMF image prior toward text-specified preferences.

stat.ML↗

arXiv2026

Inverse scattering for waveguides in topological insulators

This paper concerns the inverse scattering problem of a topologically non-trivial waveguide separating two-dimensional topological insulators. We consider the specific model of a Dirac system. For scalar short-range perturbations, we prove linearized uniqueness and stability and obtain local uniqueness in a finite-dimensional setting under a smallness constraint. For general Hermitian perturbations, the scattering data are invariant under a natural gauge transformation; at the linearized level, the data determine the potential modulo precisely this gauge. We then solve the problem numerically by means of a standard adjoint method and illustrate our theoretical findings with several numerical simulations.

math-ph↗

arXiv2026

Arbitrary-Accuracy Neural Approximation with Optimal Neuron Count and Near-Optimal Bit Complexity

We study the minimum number of hidden neurons required for arbitrary-accuracy approximation of multivariate Hölder-continuous functions on $[0,1]^d$ and the associated encoding complexity. For $d\geq 2$, we construct a fixed, explicitly defined activation function for which a closed-form network with two hidden layers of widths $d$ and $1$ achieves arbitrary accuracy in the uniform norm. We prove that $d+1$ is the exact minimum total number of hidden neurons among standard feedforward networks with locally integrable activations and affine outputs. We further give a simpler construction using a single elementary activation that combines the floor and exponential functions. This construction requires three hidden layers of widths $d$, $1$, and $2$, only two neurons above the minimum. If a skip connection is allowed, widths $d$, $1$, and $1$ suffice. These constructions use explicit grid addressing and integer encoding of quantized function values. For a bounded $α$-Hölder class, they require $O(\varepsilon^{-d/α}\log(1/\varepsilon))$ bits, matching the metric-entropy lower bound up to a logarithmic factor.

cs.LG↗

arXiv2026

A priori regularity of the reverse heat flow and dimension-dependent complexity of higher-order diffusion samplers

We establish arbitrary-order a priori regularity estimates for the reverse heat flow in the Ornstein--Uhlenbeck setting, that is, for the probability-flow ODE of score-based diffusion models. For compactly supported, possibly singular targets satisfying a dimension-uniform doubling condition on supporting-cap masses, we bound the iterated material derivatives of the flow and their first spatial derivatives pointwise, with constants independent of the dimension $d$ and explicit in time and support radius. The proof rests on two ingredients: a spatially uniform bound on normal-direction posterior fluctuations, which is the only point where the geometry of the target enters, and a centered posterior-moment normal form closed under material differentiation. This algebraic structure, together with a weighted regularity calculus, yields quantitative bounds at every order, including the mixed space--time directional derivatives required by Runge--Kutta stages. As an application, for exact-score sampling stopped at forward time $δ>0$, order-$p$ Taylor and explicit Runge--Kutta schemes reach accuracy $\varepsilon$ in $\widetilde O(d^{1/p}\varepsilon^{-1/p})$ steps in total variation, and the Taylor scheme in $\widetilde O(d^{1/(2p)}\varepsilon^{-1/p})$ steps in $W_2$.

math.NA↗

arXiv2026

FLaG: Frequency-Domain Latent-attention Gated Pooling for Token Aggregation

Token aggregation converts token-level representations into fixed-dimensional sample representations, but most pooling methods operate only in the original token space. We introduce Frequency-Domain Latent-attention Gated Pooling (FLaG), a plug-in aggregation module that re-expresses encoder outputs in the Fourier domain before final pooling. FLaG represents the nonredundant rFFT spectrum through concatenated real and imaginary components, summarizes spectral tokens with learnable latent queries, derives a sample-conditioned channel gate, and reconstructs modulated token representations for downstream aggregation. We evaluate the same architecture across ESM2-based antimicrobial peptide (AMP) activity prediction, ResNet18 image classification on CIFAR-10 and CIFAR-100, and three RoBERTa-based language tasks. FLaG achieves the best macro-averaged Spearman correlation coefficient, RMSE, and Recall@50 across four AMP backbone-species settings and the highest top-1 accuracy on CIFAR 10. It also achieves the best mean results on five of seven language metrics, although mean pooling remains strongest on STSBenchmark. AMP-side mechanistic analyses reveal low-frequency prediction sensitivity across most encoder layers, with increased relative high-frequency sensitivity in the final layer, and pronounced peptide-specific positional responses. The residual gate broadly amplifies spectral channels while preserving the low-frequency-dominated energy profile, whereas latent cross-attention exhibits sample- and species-specific spectral allocation. Overall, FLaG provides a transferable frequency-domain aggregation bias across protein, visual, and textual representations, with benefits that depend on the backbone and downstream task. Supplementary materials, source code, and data are available at https://www.healthinformaticslab.org/supp/ and https://github.com/Kewei2023/AMPCliff/tree/FLaG.

cs.AI↗

arXiv2026

CliffRank: A Dual-Branch Framework for Activity-Cliff Ranking Prediction

Activity-cliff ranking remains difficult because local structural changes can cause large activity differences, while high-quality data that resolve the underlying mechanisms remain limited. To use available activity labels more effectively, we combine absolute-activity regression with ranking-consistency learning. CliffRank trains two parallel predictors with mean squared error, a thresholded listwise loss, and Pairwise Preference Consistency (PPC), which aligns relative ordering in the preference-probability space. On three antimicrobial peptide datasets, CliffRank with ESM2-t12 achieved the highest mean Spearman correlation of 0.5393 and mean Recall@50 of 21.4, although the leading method varied across individual datasets. On three small-molecule datasets, CliffRank with PNA, where PPC was activated after 120 epochs, achieved the highest mean Spearman correlation of 0.6890, while its mean Recall@50 of 30.4 matched that of ACANet-PNA. The PPC results also define its practical limits. Asymmetric initialization improved the MolCLR-GIN averages but did not improve every target. For PNA without pretrained weights, delayed PPC improved selected metrics, but no schedule was best for both mean Spearman correlation and mean Recall@50. Future work should evaluate more targets and antimicrobial peptide systems, develop adaptive PPC schedules, and incorporate protein or membrane context when available.

cs.LG↗

arXiv2026

COFM: Consistent Optimal Transport Flow Matching via Partially Input Convex Neural Networks

Optimal transport (OT) provides a principled framework for learning mappings between probability distributions, and has found broad applications in generative modeling, inverse problems and scientific computing. Recently, flow matching methods have emerged as an efficient paradigm for learning continuous-time transport dynamics. However, existing OT-based flow matching methods often suffer from either high computational cost due to inner optimization or limited consistency. Moreover, it remains challenging to design neural architectures that can simultaneously guarantee convexity, stability, and efficient transport learning. In this paper, we propose a framework for consistent optimal transport flow matching. Specifically, we parameterize the transport potential using partially input convex neural networks (PICNN), and incorporate a Hamilton-Jacobi residual into the training objective to enforce dynamical consistency of the learned flow. This design enables a unified formulation that supports both one-step transport and multi-step ODE-based sampling, without requiring costly inner optimization. Extensive experiments on benchmark datasets demonstrate that the proposed method achieves competitive performance compared with existing OT-based and flow matching approaches, while maintaining favorable computational efficiency. In particular, under the D=256 benchmark, COFM achieves more than a 2x reduction in L^2-UVP compared with state-of-the-art (SOTA) models, while requiring approximately 9x less computational time. These results suggest that combining convex potential structures with HJ-based dynamical regularization provides an effective framework for scalable and geometrically consistent transport learning.

cs.LG↗

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

ML-based approach to classification and generation of structured light propagation in turbulent media

We study the classification task of structured-light beams after propagation through a random turbulent medium. The received speckle patterns are generated by numerical simulation of a stochastic paraxial propagation model, and the classification task is formulated over a finite alphabet of 15 OAM source classes. We benchmark intensity and autocorrelation inputs using SimpleCNN and ResNet-18 as classifiers. We also quantify the effect of training-set size and receiver-window misalignment. Since additional propagated samples may be costly to obtain, we develop a class-conditioned diffusion model for generative augmentation of turbulence-degraded intensity images. The main contribution is a spectrum-aware diffusion objective: a pixel-domain loss combined with a Fourier-domain Bregman regularizer designed to preserve high-frequency speckle statistics. We prove that this hybrid objective is consistent with the posterior-mean regression target of the diffusion model and show that generated samples substantially improve low-data classification.

physics.optics↗

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

Frequency-Domain Latent Attention Gating for Cross-Domain Token Aggregation

Token aggregation is a common bottleneck in models that map token representations to sample-level predictions, yet most pooling methods operate only in the original token domain. We propose FLaG, a plug-in aggregation module that transforms token representations with the real FFT, summarizes spectral components with learnable latent queries, applies a channel-wise gate, and reconstructs enhanced time-domain tokens for final pooling. We evaluate FLaG on antimicrobial peptide (AMP) activity prediction with ESM2, image classification with ResNet18 on CIFAR-10 and CIFAR-100, and text classification with RoBERTa on IMDB and GLUE. FLaG achieves its clearest gains on the ESM2-8M antimicrobial peptide tasks and on CIFAR-100, while remaining competitive with strong text baselines on IMDB and GLUE. Then we probe its behavior on the AMP setting with band knockouts, gate summaries, residue perturbations, latent-query readouts, and structure-proxy stratification. We find that low-frequency bands contribute the most overall, and the remaining higher-band pattern is more sample-specific. The gate acts as a broadly shared spectral reweighting stage and the cross-attention patterns are sample-specific with mild query-wise differentiation, and higher-helix peptides exhibit stronger average spectral sensitivity in both bacteria. The supplementary materials, source code and data are released at https://www.healthinformaticslab.org/supp/ and https://github.com/Kewei2023/AMPCliff/tree/FLaG.

cs.LG↗

arXiv2026

Preconditioned One-Step Generative Modeling for Bayesian Inverse Problems in Function Spaces

We propose a machine-learning algorithm for Bayesian inverse problems in the function-space regime. Based on one-step generative transport, the method learns an amortized neural operator whose pushforward of a Gaussian source approximates the posterior distribution conditioned on each new observation. We show that white-noise sources are incompatible with the function-space limit, and therefore adopt a prior-aligned GRF as the source. We justify this choice through the Lipschitz regularity of the resulting one-step conditional posterior transport and numerical experiments on linear inverse and PDE-based inverse problems. The method is not distilled from MCMC: it is trained only with prior samples and simulated partial noisy observations. Once trained, it generates a $64\times64$ posterior sample in $\sim 10^{-3}$s, avoiding repeated forward-model evaluations in MCMC and repeated network evaluations in multistep generative samplers while matching key posterior summaries.

stat.ML↗

arXiv2026

DC-LA: Difference-of-Convex Langevin Algorithm

We study a sampling problem whose target distribution is $π\propto \exp(-f-r)$ where the data fidelity term $f$ is Lipschitz smooth while the regularizer term $r=r_1-r_2$ is a non-smooth difference-of-convex (DC) function, i.e., $r_1,r_2$ are convex. By leveraging the DC structure of $r$, we can smooth out $r$ by applying Moreau envelopes to $r_1$ and $r_2$ separately. In line with DC programming, we then redistribute the concave part of the regularizer to the data fidelity and study its corresponding proximal Langevin algorithm (termed DC-LA). We establish convergence of DC-LA to the target distribution $π$, up to discretization and smoothing errors, in the $q$-Wasserstein distance for all $q \in \mathbb{N}^*$, under the assumption that $V$ is distant dissipative. Our results improve previous work on non-log-concave sampling in terms of a more general framework and assumptions. Numerical experiments show that DC-LA produces accurate distributions in synthetic settings and provides qualitatively reasonable uncertainty quantification in a real-world Computed Tomography application.

cs.LG↗

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

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↗

arXiv2025

Pathway to $O(\sqrt{d})$ Complexity bound under Wasserstein metric of flow-based models

We provide attainable analytical tools to estimate the error of flow-based generative models under the Wasserstein metric and to establish the optimal sampling iteration complexity bound with respect to dimension as $O(\sqrt{d})$. We show this error can be explicitly controlled by two parts: the Lipschitzness of the push-forward maps of the backward flow which scales independently of the dimension; and a local discretization error scales $O(\sqrt{d})$ in terms of dimension. The former one is related to the existence of Lipschitz changes of variables induced by the (heat) flow. The latter one consists of the regularity of the score function in both spatial and temporal directions. These assumptions are valid in the flow-based generative model associated with the Föllmer process and $1$-rectified flow under the Gaussian tail assumption. As a consequence, we show that the sampling iteration complexity grows linearly with the square root of the trace of the covariance operator, which is related to the invariant distribution of the forward process.

cs.LG↗