arXiv subjects

Wenrui Hao

Publications and source records attributed to Wenrui Hao.

At least 19 recordsLinked to original sources

arXiv2026

Multiscale modeling of host-pathogen interactions and mucociliary clearance during non-tuberculous mycobacterial pulmonary infection

Non-tuberculous mycobacterial (NTM) infections are a clinical challenge in cystic fibrosis (CF), where impaired mucociliary clearance and altered mucus rheology promote bacterial colonization despite host immune responses. Understanding how bacterial growth, immune cell dynamics, and mucus transport regulate infection progression is difficult because these processes interact across spatial and temporal scales. We develop a computational framework bridging a mechanistic agent-based model (ABM) of NTM infection with a spatially resolved partial differential equation (PDE) model. The PDE model couples bacterial proliferation, macrophage chemotaxis, immune-mediated clearance, mucus degradation, and viscoelastic transport in a two-compartment geometry representing mucus and lung tissue. Parameters are calibrated using data from the established ABM, yielding an efficient continuum representation while preserving cellular mechanisms. The PDE model reproduces bacterial and macrophage dynamics and enables analyses of mucus-related mechanisms and therapies. Sensitivity analysis identifies mucus viscosity, bacterial diffusivity, and macrophage mobility as key regulators of bacterial persistence through mucociliary clearance and tissue colonization. Simulations reveal nonlinear effects of mucolytic therapies: enhanced clearance reduces bacterial burden in mucus, whereas excessive viscosity reduction may promote migration into lung tissue, supporting combination with antibacterial treatment. This framework provides a quantitative platform for studying pulmonary infections, evaluating therapies, and developing patient-specific digital twins.

q-bio.QM↗

arXiv2026

Data-Driven Modeling of Spatiotemporal Dynamics Using Multimodal Imaging Data

Understanding how biological systems evolve across space and time remains a fundamental challenge, particularly when dynamic processes vary substantially across individuals. We present a personalized graph-based dynamical modeling framework for characterizing spatiotemporal biological dynamics from longitudinal multimodal imaging data. The framework constructs individualized brain graphs from MRI and PET measurements and learns patient-specific dynamical parameters governing regional structural and molecular changes. Applied to 1,891 participants from the Alzheimer's Disease Neuroimaging Initiative, the model captures the coordinated evolution of amyloid-$β$, tau, neurodegeneration, and cognition and accurately predicts their future trajectories, outperforming established clinical and neuroimaging benchmarks. Patient-specific dynamical parameters reveal distinct patterns of biological progression and provide improved prediction of future cognitive decline compared with standard biomarkers. Sensitivity analysis further identifies regional network features associated with the propagation of pathological and structural changes, recovering known temporolimbic and frontal vulnerability patterns. These results demonstrate how data-driven dynamical modeling can integrate multimodal longitudinal measurements to uncover individualized spatiotemporal patterns and latent mechanisms of biological change. The framework provides a quantitative approach for studying complex biological dynamics across heterogeneous individuals and establishes a foundation for personalized modeling of progressive biological processes.

q-bio.NC↗

arXiv2026

PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems

Many nonlinear models across physics, chemistry, and biology exhibit multiple solutions for the same parameters, and capturing this entire solution set is essential for understanding pattern-forming systems. Yet existing learned surrogates are fundamentally single-valued: neural operators map each parameter to a single output, and physics-informed neural networks converge to one branch. We develop \textbf{PatternFormer} (PF), a large language model-based framework for learning the multiple solutions of nonlinear partial differential equations. By transforming unordered coexisting solutions into canonical sequences, PF produces structured solution sets in a single autoregressive pass, terminating automatically for finite families and enforcing physical residual constraints for unbounded ones. On nonlinear elliptic problems it recovers all solution branches in one inference step; on Gray--Scott it generates coexisting Turing patterns, including physically valid states absent from the reference data and beyond training. PF can also be sequentially fine-tuned across multistable systems, toward general foundation models for solution landscapes.

math-ph↗

arXiv2026

Learning the Energy Landscapes of Dynamical Systems via Energetic Variational Optimal Transport under Data Quantity--Quality Trade-offs

Dynamic optimal transport unifies optimal transport, fluid mechanics, and gradient-flow theory within a continuous dynamical framework, offering a geometry-aware language for applications across physics, biology, and machine learning. However, conventional formulations cast it as a constrained optimization problem that must explicitly satisfy the continuity equation, hindering the reconstruction of the underlying dynamics directly from data. We propose the energetic variational method for dynamic optimal transport (EVMDOT), which reformulates it within an energetic variational framework by combining the flow map, the least action principle, and the maximum dissipation principle. The flow map recasts the constrained problem as an unconstrained one by automatically enforcing the continuity equation, while the balance between the conservative and dissipative forces determines the velocity field. Applied to the Fokker--Planck equation, the EVMDOT reconstructs both the energy landscape and the Waddington landscape directly from time-series density data. Through numerical experiments, we reveal that the EVMDOT achieves an intrinsic balance between data quantity and data quality: a sufficient data quantity compensates for limited data quality, making the reconstruction robust to the choice of the observation window. We further apply the EVMDOT to the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset to infer the potential landscape of amyloid-$β$ and tau, revealing two wells corresponding to the cognitively normal and Alzheimer's disease stages and the transition pathway between them.

math.DS↗

arXiv2026

A Geometric Local Parameterization Method for Generalized Hele-Shaw Free Boundary Problems with Source Terms

We develop a meshfree numerical framework for Hele--Shaw free boundary problems with surface tension and source terms based on geometric local parameterization and boundary integral methods. By decomposing the pressure into a particular solution and a harmonic component, the problem is reformulated into a boundary-only system, avoiding volumetric meshing of the evolving domain. For general source terms, we propose an eigenfunction-based approximation on a fixed domain and establish error estimates for both the truncation and coefficient approximation. Numerical experiments verify the accuracy and convergence of the proposed method, and an application to a tumor growth model demonstrates its effectiveness for coupled moving-boundary problems.

math.NA↗

arXiv2026

Pattern formation in a Reaction-Diffusion Model for Amyloid-$β$ and Tau Interactions in Alzheimer's Disease

Alzheimer's disease (AD) is characterized by the accumulation of Amyloid-$β$ ($Aβ$) plaques and hyperphosphorylated Tau proteins. However, many individuals exhibit substantial $Aβ$ and Tau pathology without developing dementia, suggesting that disease progression may depend not only on pathological burden but also on the spatial organization of these proteins. Motivated by this observation, we adapt Gray-Scott reaction-diffusion model to investigate pattern formation arising from the interactions between $Aβ$ and Tau. % To systematically identify stable spatial configurations, we employ a Companion-Based Multi-Level Finite Element Method (CBMFEM) on both two-dimensional domains and anatomically realistic cortical surface meshes. Numerical simulations reveal a rich landscape of multiple steady-state solutions, which are subsequently classified into representative pattern phenotypes using principal component analysis and clustering techniques. The results demonstrate that the coupled $Aβ$--Tau system admits numerous stable spatial patterns rather than a single pathological endpoint. % These findings provide a potential mathematical framework for understanding the heterogeneity of Alzheimer's disease and the existence of cognitively resilient individuals despite significant pathological burden. More broadly, the proposed framework suggests a pattern-based therapeutic paradigm in which disease dynamics are guided toward favorable stable states rather than solely targeting the elimination of pathological proteins.

q-bio.QM↗

arXiv2026

Neural operator-based digital twins for modeling amyloid-$β$ and tau propagation and treatment optimization in Alzheimer's disease

Accurately predicting the spatiotemporal evolution of amyloid-$β$ and tau proteins at the individual level is critical for improving the diagnosis and treatment of Alzheimer's disease. We consider the problem of constructing patient-specific digital twins that model the propagation of these biomarkers on the cortical surface using reaction--diffusion dynamics. A major challenge is that the underlying nonlinear aggregation mechanisms are unknown and must be inferred from sparse, noisy, and heterogeneous longitudinal PET imaging data. To address this, we develop a data-driven framework that learns biomarker dynamics directly from clinical observations. The approach combines operator learning with reduced-order representations to infer governing equations of disease progression from data. Using this framework, we achieve predictive accuracies of 87\% for amyloid-$β$ and 81\% for tau. Building on the learned dynamics, we further formulate a PDE-constrained optimal control problem to design personalized therapeutic strategies that regulate pathological protein propagation. By integrating data-driven dynamical modeling with treatment optimization, the proposed digital twin framework provides an interpretable and predictive platform for understanding disease progression and enabling precision interventions in neurodegenerative disorders.

cs.LG↗

arXiv2026

Geometric local parameterization for solving Hele-Shaw problems with surface tension

In this work, we introduce a novel computational framework for solving the two-dimensional Hele-Shaw free boundary problem with surface tension. The moving boundary is represented by point clouds, eliminating the need for a global parameterization. Our approach leverages Generalized Moving Least Squares (GMLS) to construct local geometric charts, enabling high-order approximations of geometric quantities such as curvature directly from the point cloud data. This local parameterization is systematically employed to discretize the governing boundary integral equation, including an analytical formula of the singular integrals. We provide a rigorous convergence analysis for the proposed spatial discretization, establishing consistency and stability under certain conditions. The resulting error bound is derived in terms of the size of the uniformly sampled point cloud data on the moving boundary, the smoothness of the boundary, and the order of the numerical quadrature rule. Numerical experiments confirm the theoretical findings, demonstrating high-order spatial convergence and the expected temporal convergence rates. The method's effectiveness is further illustrated through simulations of complex initial shapes, including interfaces driven by anisotropic surface tension, which correctly evolve towards circular equilibrium states under the influence of surface tension, highlighting the versatility of the method for complex geometry-dependent interface dynamics.

math.NA↗

arXiv2026

Numerical Identification of Stationary States and Their Stability in a Model of Quantum Droplets

In this work, we are motivated by a recent variant of the nonlinear Schrodinger (NLS) equation describing cold, dilute atomic condensates with quantum fluctuation effects. Our goal is to develop robust numerical methods capable of uncovering diverse stationary solutions in such NLS models. Specifically, and in line with recent theoretical and experimental interest, we focus on ultracold quantum droplets in Bose mixtures influenced by the Lee Huang Yang quantum fluctuation correction and study these systems in one and two dimensional settings. To this end, we deploy several numerical techniques. The homotopy grid method allows systematic refinement from coarse to fine spatial discretizations in one dimension, while the dimension by dimension homotopy approach extends one-dimensional solutions to two-dimensional domains. These methods effectively detect broad families of stationary states, many of which have not been previously reported, to the best of our knowledge. Furthermore, they enable the monitoring of solution continuation and bifurcation phenomena. During our investigation, we encounter unusual bifurcation events, including nonstandard pitchforks and saddle-center bifurcations, which exhibit novel stability transitions. For example, we identify continuous pathways connecting vortex and dark soliton stripe branches, absent in the standard cubic defocusing model. Overall, the presence of competing mean-field and quantum fluctuation interactions leads to a richer bifurcation structure than in traditional cubic NLS systems. These findings suggest that similar complex bifurcation and stability phenomena may appear in other settings, including higher-dimensional systems or models with competing nonlinearities such as cubic-quintic interactions, highlighting the importance of further theoretical and numerical exploration.

nlin.PS↗

arXiv2026

Unveiling Scaling Laws of Parameter Identifiability and Uncertainty Quantification in Data-Driven Biological Modeling

Integrating high-dimensional biological data into data-driven mechanistic modeling requires rigorous practical identifiability to ensure interpretability and generalizability. However, coordinate identifiability analysis often suffers from numerical instabilities near singular local minimizers. We present a computational framework that uncovers fundamental scaling laws governing practical identifiability through asymptotic analysis. By synthesizing Fisher information with perturbed Hessian matrices, we establish a hierarchical approach to quantify coordinate identifiability and inform uncertainty quantification within non-identifiable subspaces across different orders. Supported by rigorous mathematical analysis and validated on synthetic and real-world data, our framework was applied to HIV-host dynamics and spatiotemporal amyloid-beta propagation. These applications demonstrate the framework's efficiency in elucidating critical mechanisms underlying HIV diagnostics and Alzheimer's disease progression. In the era of large-scale mechanistic digital twins, our framework provides the scaling laws for data-driven modeling in terms of both parameter identifiability and uncertainty, ensuring that data-driven inferences are grounded in verifiable biological reality.

q-bio.QM↗

arXiv2026

SVD-Preconditioned Gradient Descent Method for Solving Nonlinear Least Squares Problems

This paper introduces a novel optimization algorithm designed for nonlinear least-squares problems. The method is derived by preconditioning the gradient descent direction using the Singular Value Decomposition (SVD) of the Jacobian. This SVD-based preconditioner is then integrated with the first- and second-moment adaptive learning rate mechanism of the Adam optimizer. We establish the local linear convergence of the proposed method under standard regularity assumptions and prove global convergence for a modified version of the algorithm under suitable conditions. The effectiveness of the approach is demonstrated experimentally across a range of tasks, including function approximation, partial differential equation (PDE) solving, and image classification on the CIFAR-10 dataset. Results show that the proposed method consistently outperforms standard Adam, achieving faster convergence and lower error in both regression and classification settings.

math.NA↗

arXiv2026

Fisher-Informed Parameterwise Aggregation for Federated Learning with Heterogeneous Data

Federated learning aggregates model updates from distributed clients, but standard first order methods such as FedAvg apply the same scalar weight to all parameters from each client. Under non-IID data, these uniformly weighted updates can be strongly misaligned across clients, causing client drift and degrading the global model. Here we propose Fisher-Informed Parameterwise Aggregation (FIPA), a second-order aggregation method that replaces client-level scalar weights with parameter-specific Fisher Information Matrix (FIM) weights, enabling true parameter-level scaling that captures how each client's data uniquely influences different parameters. With low-rank approximation, FIPA remains communication- and computation-efficient. Across nonlinear function regression, PDE learning, and image classification, FIPA consistently improves over averaging-based aggregation, and can be effectively combined with state-of-the-art client-side optimization algorithms to further improve image classification accuracy. These results highlight the benefits of FIPA for federated learning under heterogeneous data distributions.

cs.LG↗

arXiv2026

A Structure-Preserving Scheme for the Time-Dependent Ginzburg-Landau Model with BCS Gap Coupling

We propose a structure-preserving scheme for a hybrid model that couples the time-dependent Ginzburg-Landau (TDGL) equation of superconducting vortex dynamics and the nonlinear Bardeen-Cooper-Schrieffer (BCS) gap equation. This formulation is consistent with the classical TDGL equation in the near-critical temperature, while extending the applicability of the existing TDGL model to regimes beyond the critical temperature. The resulting system poses significant computational challenges due to its nonlinear and coupled structure. To achieve stable and reliable simulations of the vortex dynamics and accompanying morphological transitions, we develop a maximum bound preserving, energy-stable implicit-explicit (IMEX) scheme. The structure-preserving properties of the scheme are rigorously established, ensuring long-time stability and physical consistency. Through two- and three-dimensional simulations, the hybrid model successfully captures the temporal and spatial formation and alignment of vortices and the suppression of superconductivity under increasing magnetic fields, demonstrating both the accuracy and robustness of the proposed computational approach.

math.NA↗

arXiv2025

Multiscale Neural Networks for Approximating Green's Functions

Neural networks (NNs) have been widely used to solve partial differential equations (PDEs) in the applications of physics, biology, and engineering. One effective approach for solving PDEs with a fixed differential operator is learning Green's functions. However, Green's functions are notoriously difficult to learn due to their poor regularity, which typically requires larger NNs and longer training times. In this paper, we address these challenges by leveraging multiscale NNs to learn Green's functions. Through theoretical analysis using multiscale Barron space methods and experimental validation, we show that the multiscale approach significantly reduces the necessary NN size and accelerates training.

math.NA↗

arXiv2025

ZENN: A Thermodynamics-Inspired Computational Framework for Heterogeneous Data-Driven Modeling

Traditional entropy-based methods - such as cross-entropy loss in classification problems - have long been essential tools for representing the information uncertainty and physical disorder in data and for developing artificial intelligence algorithms. However, the rapid growth of data across various domains has introduced new challenges, particularly the integration of heterogeneous datasets with intrinsic disparities. To address this, we introduce a zentropy-enhanced neural network (ZENN), extending zentropy theory into the data science domain via intrinsic entropy, enabling more effective learning from heterogeneous data sources. ZENN simultaneously learns both energy and intrinsic entropy components, capturing the underlying structure of multi-source data. To support this, we redesign the neural network architecture to better reflect the intrinsic properties and variability inherent in diverse datasets. We demonstrate the effectiveness of ZENN on classification tasks and energy landscape reconstructions, showing its superior generalization capabilities and robustness-particularly in predicting high-order derivatives. ZENN demonstrates superior generalization by introducing a learnable temperature variable that models latent multi-source heterogeneity, allowing it to surpass state-of-the-art models on CIFAR-10/100, BBCNews, and AGNews. As a practical application in materials science, we employ ZENN to reconstruct the Helmholtz energy landscape of Fe$_3$Pt using data generated from density functional theory (DFT) and capture key material behaviors, including negative thermal expansion and the critical point in the temperature-pressure space. Overall, this work presents a zentropy-grounded framework for data-driven machine learning, positioning ZENN as a versatile and robust approach for scientific problems involving complex, heterogeneous datasets.

cs.LG↗

arXiv2025

Optimal Control For Anti-Abeta Treatment in Alzheimer's Disease using a Reaction-Diffusion Model

Alzheimer's disease (AD) is a progressive neurodegenerative disorder that severely impairs survival and quality of life. While anti-amyloid beta (Abeta) therapies can slow disease progression, their efficacy depends on personalized dosing that maximizes benefits and minimizes risks such as amyloid related imaging abnormalities (ARIA). Mathematical modeling offers a powerful tool for understanding AD dynamics and optimizing treatment, yet most models focus solely on temporal behavior, overlooking spatial heterogeneity within the brain. In this study, we propose a spatially explicit reaction-diffusion model to describe Abeta plaque dynamics. We formulate an optimal control problem to minimize plaque concentration while balancing therapeutic efficacy and treatment risk. Under reasonable assumptions, we establish well-posedness and uniqueness of the optimal solution. A Finite Element Method (FEM) based numerical framework is developed to compute personalized treatment strategies. Our model is calibrated using longitudinal Abeta positron emission tomography (PET) data from the Alzheimer's Disease Neuroimaging Initiative (ADNI), enabling estimation of patient-specific parameters such as growth rate and effective diffusivity. Results show that optimized treatment strategies consistently outperform constant dosing regimens across patient groups, achieving substantial reductions in cumulative amyloid burden while minimizing side effects. This integrated, data-driven framework advances personalized, spatially informed therapeutic optimization for Alzheimer's disease.

math.OC↗

arXiv2025

Energy Approach from $\varepsilon$-Graph to Continuum Diffusion Model with Connectivity Functional

We derive an energy-based continuum limit for $\varepsilon$-graphs endowed with a general connectivity functional. We prove that the discrete energy and its continuum counterpart differ by at most $O(\varepsilon)$; the prefactor involves only the $W^{1,1}$-norm of the connectivity density as $\varepsilon\to0$, so the error bound remains valid even when that density has strong local fluctuations. As an application, we introduce a neural-network procedure that reconstructs the connectivity density from edge-weight data and then embeds the resulting continuum model into a brain-dynamics framework. In this setting, the usual constant diffusion coefficient is replaced by the spatially varying coefficient produced by the learned density, yielding dynamics that differ significantly from those obtained with conventional constant-diffusion models.

math.NA↗

arXiv2025

Optimal error estimates of the diffuse domain method for parabolic equations

In this paper, we study the convergence behavior of the diffuse domain method (DDM) for solving a class of second-order parabolic partial differential equations with Neumann boundary condition posed on general irregular domains. The DDM employs a phase-field function to extend the original parabolic problem to a similar but slightly modified problem defined over a larger rectangular domain that contains the target physical domain. Based on the weighted Sobolev spaces, we rigorously establish the convergence of the diffuse domain solution to the original solution as the interface thickness parameter goes to zero, together with the corresponding optimal error estimates under the weighted $L^2$ and $H^1$ norms. Numerical experiments are also presented to validate the theoretical results.

math.NA↗