arXiv subjects

Yunqing Huang

Publications and source records attributed to Yunqing Huang.

At least 19 recordsLinked to original sources

arXiv2026

Sparsely connected neural network representation of Lagrange finite element function

We construct a mesh-induced sparsely connected neural network framework that exactly reproduces arbitrary-order Lagrange finite element spaces over simplicial meshes. Unlike conventional black-box neural surrogates, the proposed network architecture is fully dictated by finite element discretization rules: local computations stem from simplex geometry and barycentric coordinate transformations, while global consistency is enforced through shared degrees of freedom. For linear Lagrange elements, local basis functions are directly implemented via affine barycentric layers, and high-order polynomial bases are explicitly decomposed into barycentric product compositions realized by specially designed $\mathrm{ReLU}^p$ modules. Equipped with element indicator branches and multiplication units, these modular local components are globally assembled into a sparsely connected neural network whose function space coincides exactly with the target finite element space, thereby inheriting the complete classical finite element approximation theory. By prescribing customized backward differentiation rules for piecewise activations, function values and their spatial gradients can be simultaneously extracted via automatic differentiation within a unified computational graph, eliminating the separate gradient calculation subroutines required in standard finite element implementations. Numerical experiments verify the accuracy of the neural network representation of Lagrange finite elements. Furthermore, by virtue of the intrinsic mesh-free nature of this neural network representation, finite element functions can be interpolated between non-matching meshes, and the proposed scheme can be applied to adaptive finite element methods for solving parabolic partial differential equations. An open-source code implementation of the proposed architecture is made publicly available.

math.NA↗

arXiv2026

History-Compatible Energy-Stable Finite Element Schemes for Variable-Density Cahn--Hilliard--Navier--Stokes Flows on Evolving Meshes

We consider variable-density Cahn--Hilliard--Navier--Stokes (CHNS) discretizations on finite element meshes that may change between accepted time levels through fixed-topology motion or topology-changing remeshing. When the discrete spaces vary in time, the phase, kinetic, and pressure histories entering a multistep scheme are measured in different discrete structures and cannot, in general, be transferred by a single operator. We develop decoupled backward Euler (BE) and second-order backward differentiation formula (BDF2) schemes by combining exact physical cross-mesh pairings with history representations compatible with the corresponding phase-energy, kinetic-energy, and pressure-gradient storages. The phase update also determines an Abels--Garcke--Grün-consistent mass flux used in the momentum transport. A scalar capillary-exchange equation separates the phase and fluid solves while retaining the discrete energy exchange. The resulting field subproblems are linear, the scalar equation has a unique positive solution, and the schemes satisfy modified energy balances without a time-step restriction under the stated admissibility assumptions. Numerical experiments confirm second-order temporal convergence under both mesh updates, phase-mass conservation, modified-energy decay in the unforced tests, and comparable Rayleigh--Taylor and rising-bubble dynamics.

math.NA↗

arXiv2026

A Trace--Logarithmic Variational Functional for Equidistribution and Alignment in Moving Mesh Adaptation

Existing variational mesh functionals often require an empirical equidistribution--alignment weight or use a strongly nonlinear density. We propose a calibrated trace--logarithmic functional in the inverse pullback tensor $ A= J^{-1}M^{-1}J^{-T}$,with the logarithmic coefficient fixed by the local target $A=θI$. We prove strict convexity in the symmetric positive-definite (SPD) tensor variable, inverse-Jacobian polyconvexity and coercivity, and weak minimizer existence under explicit admissibility assumptions. The density also satisfies the standard coercivity condition for semi-discrete physical-coordinate mesh nonsingularity; the geometric discretization yields a compact computational-coordinate residual for the reported direct-secant implementation. Numerical tests for metric-induced mesh adaptation, a Burgers benchmark, a Rayleigh--Taylor instability simulation, and a matched high-anisotropy stress test show robust mesh redistribution; in the stress test, the trace--logarithmic runs complete over a broader fixed-protocol range of the compression parameter than the scanned Huang benchmark.

math.NA↗

arXiv2026

Neural network-enhanced $hr$-adaptive finite element algorithm for parabolic equations

In this paper, we propose a novel $hr$-adaptive finite element method, enhanced by neural networks, for parabolic equations. The main challenge of the conventional $h$-adaptive finite element method is interpolating the finite element solution from the previous step in the updated mesh. The interpolation dependent on the new mesh must be recomputed at each adaptive iteration, resulting in high computational costs. The new approach addresses this challenge by introducing a neural network to construct a mesh-free surrogate of the previous step finite element solution. Since the neural network is mesh-free, it only requires training once per time step, with its parameters initialized using the minimizer of the previous time step. This approach effectively overcomes the interpolation challenges associated with non-nested meshes in computation, making node insertion and movement more convenient and efficient. The new algorithm also emphasizes SIZE and GENERATE, allowing each refinement to roughly double the number of mesh nodes of the previous iteration and then redistribute them to form a new mesh that effectively captures the singularities. It significantly reduces the time required for repeated refinement of the conventional methods and achieves the desired accuracy in no more than seven space-adaptive iterations per time step. Numerical experiments confirm the efficiency of the proposed algorithm in capturing dynamic changes of singularities. The code is made publicly available on GitHub.

math.NA↗

arXiv2026

Adaptive multiscale model reduction for linear elasticity equation in perforated domains

In this paper, we develop a Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for solving linear elasticity problems in heterogeneous perforated domains. The presence of numerous perforations introduces multiple scales into the computational domain, making direct fine-grid simulations computationally expensive. The proposed method follows the standard offline--online decomposition of CEM-GMsFEM. In the offline stage, local spectral problems are solved on coarse elements to construct auxiliary spaces, and localized energy-minimizing basis functions are then computed on oversampled regions to capture fine-scale geometric information induced by the perforations. In the online stage, residual-driven basis functions are constructed in enlarged coarse neighborhoods to incorporate source-term information and improve the accuracy of the multiscale approximation adaptively. We establish convergence results for both the offline and online stages. In particular, we derive error estimates for the localized multiscale approximation and prove the convergence of the adaptive online enrichment algorithm. Moreover, we show that the oversampling regions used in the online stage can be determined locally, leading to a reduction in computational cost while maintaining convergence properties. Numerical experiments on perforated media with different geometric configurations demonstrate the accuracy and efficiency of the proposed method.

math.NA↗

arXiv2026

DataTransfer: Neural network based interpolation across non-nested meshes

In mesh-based numerical simulations, the interpolation of mesh-defined functions across different meshes is a critical task, and achieving high-precision interpolation is of great significance for improving the computational efficiency and numerical stability of algorithms. This paper proposes neural network based function mapping model across meshes, wherein the interpolation process is reformulated as a data-driven regression problem over scattered function data. Conventional interpolation and projection-based approaches are highly dependent on mesh connectivity and corresponding geometric properties, which renders such methods computationally costly and sensitive to mismatches between source and target meshes. The proposed method constructs a neural network approximator using nodal function values on the source mesh to obtain a global representation of the function, which can then be interpolated onto any other meshes. To investigate the network architectural impacts on model performance, three representative feedforward network structures are numerically compared in this work: multi-layer perceptrons, extreme learning machines, and network incorporating radial basis function hidden units. The results reveal distinct trade-offs among accuracy, computational efficiency and model robustness, among which the radial basis function-based network achieves the most desirable overall performance balance, enabling fast and precise function calculation. Numerical experiments conducted on non-nested meshes validate the efficacy of the proposed model in both function interpolation and cross-mesh data transmission tasks.

math.NA↗

arXiv2026

A residual driven multiscale method for Darcy's flow in perforated domains

In this paper, we present a residual-driven multiscale method for simulating Darcy flow in perforated domains, where complex geometries and highly heterogeneous permeability make direct simulations computationally expensive. To address this, we introduce a velocity elimination technique that reformulates the mixed velocity-pressure system into a pressure-only formulation, significantly reducing complexity by focusing on the dominant pressure variable. Our method is developed within the Generalized Multiscale Finite Element Method (GMsFEM) framework. For each coarse block, we construct offline basis functions from local spectral problems that capture key geometric and physical features. Online basis functions are then adaptively enriched using residuals, allowing the method to incorporate global effects such as source terms and boundary conditions, thereby improving accuracy. We provide detailed error analysis demonstrating how the offline and online spaces contribute to the accuracy and efficiency of the solution. Numerical experiments confirm the method's effectiveness, showing substantial reductions in computational cost while maintaining high accuracy, particularly through adaptive online enrichment. These results highlight the method's potential for efficient and accurate simulation of Darcy flow in complex, heterogeneous perforated domains.

math.NA↗

arXiv2026

FEALPy: A Cross-platform Intelligent Numerical Simulation Engine

In resent years, the software ecosystem for numerical simulation still remains fragmented, with different algorithms and discretization methods often implemented in isolation, each with distinct data structures and programming conventions. This fragmentation is compounded by the growing divide between packages from different research fields and the lack of a unified, universal data structure, hindering the development of integrated, cross-platform solutions. In this work, we introduce FEALPy, a numerical simulation engine built around a unified tensor abstraction layer in a modular design. It enables seamless integration between diverse numerical methods along with deep learning workflows. By supporting multiple computational backends such as NumPy, PyTorch, and JAX, FEALPy ensures consistent adaptability across CPU and GPU hardware systems. Its modular architecture facilitates the entire simulation pipeline, from mesh handling and assembly to solver execution, with built-in support for automatic differentiation. In this paper, the versatility and efficacy of the framework are demonstrated through applications spanning linear elasticity, high-order PDEs, moving mesh methods, inverse problems and path planning.

math.NA↗

arXiv2026

The grad-div conforming virtual element method for the quad-div problem in three dimensions

We propose a new stable variational formulation for the quad-div problem in three dimensions and prove its well-posedness. Using this weak form, we develop and analyze the $\boldsymbol{H}(\operatorname{grad-div})$-conforming virtual element method of arbitrary approximation orders on polyhedral meshes. Three families of $\boldsymbol{H}(\operatorname{grad-div})$-conforming virtual elements are constructed based on the structure of a de Rham sub-complex with enhanced smoothness, resulting in an exact discrete virtual element complex. In the lowest-order case, the simplest element has only one degree of freedom at each vertex and face, respectively. We rigorously prove the interpolation error estimates, the stability of discrete bilinear forms, the well-posedness of discrete formulation and the optimal error estimates. Some numerical examples are shown to verify the theoretical results.

math.NA↗

arXiv2026

Data-integrated neural networks for solving partial differential equations

In this work, we propose data-integrated neural networks (DataInNet) for solving partial differential equations (PDEs), offering a novel approach to leveraging data (e.g., source terms, initial conditions, and boundary conditions). The core of this work lies in the integration of data into a unified network framework. DataInNet comprises two subnetworks: a data integration neural network responsible for accommodating and fusing various types of data, and a fully connected neural network dedicated to learning the residual physical information not captured by the data integration neural network. This network architecture inherently excludes function classes that violate known physical constraints, thereby substantially narrowing the solution search space. Numerical experiments demonstrate that the proposed DataInNet delivers superior performance on challenging problems, such as the Helmholtz equation (relative $L^2$ error: O($10^{-6}$)) and PDEs with high frequency solutions (relative $L^2$ error: O($10^{-5}$)).

math.NA↗

arXiv2025

A hierarchical approach for multicontinuum homogenization in high contrast media

A recently developed upscaling technique, the multicontinuum homogenization method, has gained significant attention for its effectiveness in modeling complex multiscale systems. This method defines multiple continua based on distinct physical properties and solves a series of constrained cell problems to capture localized information for each continuum. However, solving all these cell problems on very fine grids at every macroscopic point is computationally expensive, which is a common limitation of most homogenization approaches for non-periodic problems. To address this challenge, we propose a hierarchical multicontinuum homogenization framework. The core idea is to define hierarchical macroscopic points and solve the constrained problems on grids of varying resolutions. We assume that the local solutions can be represented as a combination of a linear interpolation of local solutions from preceding levels and an additional correction term. This combination is substituted into the original constrained problems, and the correction term is resolved using finite element (FE) grids of varying sizes, depending on the level of the macropoint. By normalizing the computational cost of fully resolving the local problem to $\mathcal{O}(1)$, we establish that our approach incurs a cost of $\mathcal{O}(L η^{(1-L)d})$, highlighting substantial computational savings across hierarchical layers $L$, coarsening factor $η$, and spatial dimension $d$. Numerical experiments validate the effectiveness of the proposed method in media with slowly varying properties, underscoring its potential for efficient multiscale modeling.

math.NA↗

arXiv2025

IG-PINNs: Interface-gated physics-informed neural networks for solving elliptic interface problems

In this work, we develop interface-gated physics-informed neural networks (IG-PINNs) to solve elliptic interface equations. In IG-PINNs, we use a fully connected neural network to capture the smooth behavior across the entire domain. In each subdomain separated by the interface, an interface-gated network is utilized to provide corrections at the interface. In the architectural design of the interface-gated network, we introduce a gating mechanism and a level-set function derived from the interface. This design enables the interface-gated network to effectively handle discontinuous jumps across the interface. Some numerical experiments have confirmed the effectiveness of the IG-PINNs, demonstrating higher accuracy compared with PINNs, interface PINNs (I-PINNs) and multi-domain PINNs (M-PINNs).

math.NA↗

arXiv2025

A grad-curl conforming virtual element method for a grad-curl problem linking the 3D quad-curl problem and Stokes system

Based on the Stokes complex with vanishing boundary conditions and its dual complex, we reinterpret a grad-curl problem arising from the quad-curl problem as a new vector potential formulation of the three-dimensional Stokes system. By extending the analysis to the corresponding non-homogeneous problems and the accompanying trace complex, we construct a novel $\boldsymbol{H}(\operatorname{grad-curl})$-conforming virtual element space with arbitrary approximation order that satisfies the exactness of the associated discrete Stokes complex. In the lowest-order case, three degrees of freedom are assigned to each vertex and one to each edge. For the grad-curl problem, we rigorously establish the interpolation error estimates, the stability of discrete bilinear forms, and the convergence of the proposed element on polyhedral meshes. As a discrete vector potential formulation of the Stokes problem, the resulting system is pressure-decoupled and symmetric positive definite. Some numerical examples are presented to verify the theoretical results.

math.NA↗

arXiv2025

Transcending Sparse Measurement Limits: Operator-Learning-Driven Data Super-Resolution for Inverse Source Problem

Inverse source localization from Helmholtz boundary data collected over a narrow aperture is highly ill-posed and severely undersampled, undermining classical solvers (e.g., the Direct Sampling Method). We present a modular framework that significantly improves multi-source localization from extremely sparse single-frequency measurements. First, we extend a uniqueness theorem for the inverse source problem, proving that a unique solution is guaranteed under limited viewing apertures. Second, we employ a Deep Operator Network (DeepONet) with a branch-trunk architecture to interpolate the sparse measurements, lifting six to ten samples within the narrow aperture to a sufficiently dense synthetic aperture. Third, the super-resolved field is fed into the Direct Sampling Method (DSM). For a single source, we derive an error estimate showing that sparse data alone can achieve grid-level precision. In two- and three-source trials, localization from raw sparse measurements is unreliable, whereas DeepONet-reconstructed data reduce localization error by about an order of magnitude and remain effective with apertures as small as $π/4$. By decoupling interpolation from inversion, the framework allows the interpolation and inversion modules to be swapped with neural operators and classical algorithms, respectively, providing a practical and flexible design that improves localization accuracy compared with standard baselines.

math.NA↗

arXiv2025

An adaptive time-stepping strategy for the modified phase field crystal model with a strong nonlinear vacancy potential

This paper develops three linear and energy-stable schemes for a modified phase field crystal model with a strong nonlinear vacancy potential (VMPFC model). This sixth-order phase-field model enables realistic crystal growth simulation. Starting from a Crank-Nicolson scheme based on the stabilized-SAV (S-SAV) method, we optimize it via the generalized positive auxiliary variable (GPAV) and modified exponential scalar auxiliary variable (ESAV) methods, thereby reducing computational complexity or eliminating the requirement for the nonlinear free energy potential to be bounded from below. The newly developed Energy-Variation Moving Average (EV-MA) adaptive time-stepping strategy resolves numerical instabilities and mitigates the high parameter sensitivity of the conventional adaptive time algorithm during rapid energy decay in the strongly nonlinear system. Unlike conventional instantaneous energy-derivative monitors, the EV-MA technique incorporates a moving average of the energy variation. Additionally, the rate of change between adjacent time steps is constrained by a maximum change factor. This design effectively dampens spurious oscillations and enhances the robustness of time step selection. Extensive numerical experiments are conducted to validate the accuracy and energy stability of the proposed schemes. The EV-MA strategy is also demonstrated to perform robustly across a wide range of parameters.

math.NA↗

arXiv2025

Weights initialization of neural networks for function approximation

Neural network-based function approximation plays a pivotal role in the advancement of scientific computing and machine learning. Yet, training such models faces several challenges: (i) each target function often requires training a new model from scratch; (ii) performance is highly sensitive to architectural and hyperparameter choices; and (iii) models frequently generalize poorly beyond the training domain. To overcome these challenges, we propose a reusable initialization framework based on basis function pretraining. In this approach, basis neural networks are first trained to approximate families of polynomials on a reference domain. Their learned parameters are then used to initialize networks for more complex target functions. To enhance adaptability across arbitrary domains, we further introduce a domain mapping mechanism that transforms inputs into the reference domain, thereby preserving structural correspondence with the pretrained models. Extensive numerical experiments in one- and two-dimensional settings demonstrate substantial improvements in training efficiency, generalization, and model transferability, highlighting the promise of initialization-based strategies for scalable and modular neural function approximation. The full code is made publicly available on Gitee.

cs.LG↗

arXiv2025

Inverse source problems for the stochastic wave equations

To address the ill-posedness of the inverse source problem for the one-dimensional stochastic Helmholtz equations without attenuation, this study develops a novel computational framework designed to mitigate this inherent challenge at the numerical implementation level. For the stochastic wave equation driven by a finite-jump Lévy process (assuming that its jump amplitude obeys a Gaussian distribution and the jump time interval obeys a Poisson distribution), this paper firstly establish the existence of a mild solution to its direct problem satisfying a particular stability estimate. Building upon these theoretical foundations, we further investigate the well-posedness of the inverse problem and develop a methodology to reconstruct the unknown source terms $f$ and $g$ using the data of the wave field at the final time point $u(x,T)$. This work not only provides rigorous theoretical analysis and effective numerical schemes for solving inverse source problems in these two specific classes of stochastic wave equations, but also offers new perspectives and methodological approaches for addressing a broader range of wave propagation inverse problems characterized by non-Gaussian stochastic properties. The proposed framework demonstrates significant relevance for characterizing physical phenomena influenced by jump-type stochastic perturbations, offering promising applications in diverse domains including but not limited to seismic wave propagation analysis and financial market volatility modeling.

math.NA↗

arXiv2025

Robust space-time multiscale upscaling via multicontinuum homogenization for evolving perforated media

Time-evolving perforated domains arise in many engineering and geoscientific applications, including reactive transport, particle deposition, and structural degradation in porous media. Accurately capturing the macroscopic behavior of such systems poses significant computational challenges due to the dynamic fine-scale geometries. In this paper, we develop a robust and generalizable multiscale modeling framework based on multicontinuum homogenization to derive effective macroscopic equations in shrinking domains. The method distinguishes multiple continua according to the physical characteristics (e.g., channel widths), and couples them via space-time local cell problems formulated on representative volume elements. These local problems incorporate temporal derivatives and domain evolution, ensuring consistency with underlying fine-scale dynamics. The resulting upscaled system yields computable macroscopic coefficients and is suitable for large-scale simulations. Several numerical experiments are presented to validate the accuracy, efficiency, and potential applicability of the method to complex time-dependent engineering problems.

math.NA↗