arXiv subjects

Jiale Linghu

Publications and source records attributed to Jiale Linghu.

13 recordsLinked to original sources

arXiv2026

Efficient higher-order multi-scale method and its convergence estimate for dynamic nonlinear hygro-thermo-mechanical coupling problems of heterogeneous structures

This paper presents a novel higher-order multi-scale (HOMS) computational framework for efficient, high-accuracy, and low-cost simulation of nonlinear hygro-thermo-mechanical (H-T-M) coupling problems in heterogeneous structures. The inherent nonlinearity in the investigated model stems primarily from temperature- or moisture-dependent material properties, and this model also accounts for temperature-dependent internal heat source and moisture sink terms induced by exothermic, moisture-consuming chemical reactions (e.g., hydration). The main contributions of this work are as follows. First, a high-accuracy multi-scale asymptotic model incorporating higher-order correction terms is constructed for nonlinear H-T-M coupling problems in heterogeneous structures with highly spatial inhomogeneity, using the multi-scale asymptotic approach together with Taylor series expansions. Second, rigorous error estimates in both point-wise and integral senses are derived for the multi-scale asymptotic solutions, which theoretically demonstrate the necessity and superiority of the proposed HOMS method. Third, an efficient two-stage numerical algorithm with off-line and on-line stages is developed, based on finite difference and finite element methods, and its convergence is also proved rigorously. Finally, two- and three-dimensional numerical experiments are performed to assess the computational performance of the proposed HOMS approach, showing excellent numerical accuracy and robustness with low computational overhead.

math.NA↗

arXiv2026

A high-order multi-scale method and its convergence analysis for temperature-dependent nonlinear thermal radiation problems of composite structures

Accurate prediction of the nonlinear radiation thermal transfer in composite structures with temperature-dependent properties is significant in high-temperature applications of the materials. This study establishes a high-accuracy multi-scale computational model incorporating novel high-order correction terms for the high-fidelity simulation of nonlinear thermal radiation in composite structures, enabling local balance preserving of heat quantity. Moreover, an explicit convergence rate is also derived for the resulting high-order multi-scale solutions. Furthermore, an efficient multi-scale algorithm consisting of off-line and on-line computation stages is developed for high-accuracy simulation of nonlinear thermal radiation behavior in composite structures, and corresponding convergence analysis is also obtained. Two- and three-dimensional numerical examples are presented to validate the competitive advantages of the proposed multi-scale approach, not only exceptional numerical accuracy, but also reduced computational cost in both storage requirements and computational time.

math.NA↗

arXiv2026

A Structure-Adaptive Random Feature Method for High-Dimensional Elliptic PDEs

Random-feature methods reduce high-dimensional elliptic PDE collocation to linear coefficient problems, but full-dimensional trial spaces overlook lower-dimensional structure. We introduce the Hierarchical Analysis-of-Variance Random Feature Method (HA-RFM), which selects coordinate blocks using closed Sobol indices of the PDE residual, identifies oblique low-rank features from fitted-predictor gradients, and couples all retained features in one regularized least-squares solve. Under structural and stability hypotheses, we establish an $L^2$ error bound that links solution and residual truncation to finite-width approximation and regularized finite-sample fitting, and we derive guarantees for width and structure recovery. The resulting width is polynomial in the dimension at fixed interaction order, with dimension-independent higher-order contributions under uniform structural control. Residual screening achieves exact recovery of the prescribed three-pair support, while fitted-predictor gradients recover oblique directions through dimension $50$. In random-ridge tests, less than $1\%$ additional width reduces errors by factors of $14$-$39$ over coordinate blocks and $34$-$100$ over equal-width full-dimensional RFM. Semilinear computations extend HA-RFM through dimension $100$, while dense and distributed interactions delineate the coordinate families required for broader structure.

math.NA↗

arXiv2026

Residual-Christoffel Sampling for Random Feature Collocation of Linear PDEs

Random feature collocation fixes a randomly generated trial space and determines its coefficients from a linear least-squares system. Stability then depends on whether the sampled residual equations represent the geometry induced by the differential operator. We construct an operator-aware discretization in which the operator-applied features determine both the collocation measure and a coefficient whitening map. The randomized scheme combines a residual-Christoffel density with inverse-density weights, while a deterministic scalar-row alternative maximizes successive regularized log-determinant increments. Conditional on the realized trial space, the sampled whitened interior Gram is a spectral approximation to the reference Gram on the retained residual space, with sample complexity linear in the retained dimension up to a logarithmic factor. For uniformly analytic residual kernels, the associated operator has stretched-exponentially decaying eigenvalues and ridge effective dimension that is polylogarithmic in the inverse ridge scale. Experiments on scalar and vector equations, varied geometries, and one to three spatial dimensions show that residual-space sampling and whitening produce numerically full-rank transformed systems with substantially smaller condition numbers and iteration counts. The deterministic construction attains the lowest errors at the smallest scalar sample sizes. Residual-space geometry therefore yields a principled design for stable strong-form random feature collocation.

math.NA↗

arXiv2026

Trainable Photonic Measurement for Physics-Informed PDE Learning

Photonic quantum machine learning offers a route to trainable physical representations built from phase, interference and measurement. However, its role in scientific machine learning remains largely unexplored. Physics-informed neural fields provide a natural setting, because differential equations require trial spaces that preserve phase, frequency and derivative structure. Here we introduce a photonic quantum neural field in which coordinates become trainable optical phases, are mixed by multi-photon Fock-space interference and are decoded from photon-number measurements. The photonic circuit is optimized as the neural-field representation itself, not as a fixed feature map or hardware accelerator. Photonic measurement is therefore a trainable representation on which the physics-informed residual is minimized. Across seven elliptic, wave, nonlinear dispersive and inverse PDE benchmarks, we observe a phase-complexity transition: classical coordinate and Fourier-feature networks suffice in smooth regimes, whereas the photonic field is most accurate when residual derivatives amplify phase mismatch. In the hardest regimes it gives the lowest errors, with margins reaching an order of magnitude and about one quarter of the trainable parameters of classical baselines. Frozen and shuffled controls, together with noise stress tests, attribute this gain to learned interference and stable Fock-probability readout under compound perturbations. These results identify photonic quantum measurement as a representation-learning principle for scientific machine learning.

cs.LG↗

arXiv2026

Random-Feature Kalman Filtering for Linear PDE Data Assimilation

Data assimilation for time-dependent partial differential equations (PDEs) requires Bayesian updates of an evolving field from streaming, sparse, and noisy observations, while keeping the filtering state finite dimensional. We introduce a random-feature Kalman filtering framework for linear PDE data assimilation. Once the random features are frozen and the linear PDE is Galerkin discretized, the coefficient vector satisfies a finite-dimensional linear-Gaussian state-space model, so the Kalman recursion gives the exact posterior for the chosen coefficient model. For non-orthogonal random-feature draws, we construct a mass-whitened effective-rank coordinate system that removes near-null mass directions and identifies the posterior dimension $r$. For the heat equation with implicit-Euler time stepping, we prove a high-probability posterior-contraction and PDE-consistency theorem in these mass-whitened coordinates. The mean-square $L^2$ reconstruction error separates into an effective-rank feature approximation term, a deterministic time-consistency term, and a Bayesian estimation term. In the high-information regime, the leading posterior contribution scales as $r\sigma^2/N_o$, where $\sigma^2$ is the observation-noise variance and $N_o$ is the number of observations per analysis time. Thus the analysis distinguishes the exact coefficient-space posterior from deterministic PDE approximation errors, and gives a checkable uncertainty-quantification guarantee for random-feature filtering of a representative parabolic PDE.

math.NA↗

arXiv2026

Liquid Random Feature Methods for Time-Dependent Partial Differential Equations

A central challenge in mesh-free space--time approximation for time-dependent partial differential equations is to represent evolving temporal scales while keeping residual minimization computationally tractable. Random feature methods simplify this algebraic problem by freezing nonlinear trial functions and fitting only a linear readout, but standard static space--time activations provide no explicit relaxation-scale mechanism, making temporal-scale resolution a finite-dimensional bottleneck in stiff, dispersive, or multi-scale regimes. We introduce liquid random feature methods (L-RFM), which replace static temporal activations by closed-form liquid time-constant responses with sampled relaxation scales. The resulting frozen features form temporally structured local or global trial spaces with analytic space--time derivatives for residual least-squares assembly. A density theorem proves density of the deterministic trial spaces in the continuous space--time function class, and a temporal-rank calculation clarifies the role of sampled relaxation scales. Ablation and finite-feature tests identify the liquid temporal response as the primary source of the observed accuracy improvement. Across stiff reaction--diffusion, nonlinear transport, dispersive, complex-valued, and multidimensional benchmarks, L-RFM improves finite-feature accuracy in regimes where temporal-scale representation controls the approximation. By embedding relaxation scales directly into frozen trial functions, L-RFM provides a route to high-accuracy continuous space--time surrogates for evolutionary PDEs while preserving the simplicity of linear least-squares solvers.

physics.comp-ph↗

arXiv2026

Local Surrogates for Harmonic Vibrational Entropy in Multilattices

Harmonic vibrational entropy is a key finite-temperature contribution to defect thermodynamics, but direct evaluation by dense Hessian diagonalization scales cubically with atom count and is too costly for supercell convergence, migration-path sampling, and high-throughput defect studies. We develop local surrogate models for harmonic entropy in multilattices, including semiconductors, ordered alloys, and multispecies crystals with multi-atom bases and internal degrees of freedom. Unlike Bravais lattices, multilattices contain internal-shift degrees of freedom and optical phonon modes coupled to acoustic strain; entropy models must therefore resolve sublattice and species labels. For finite-range or screened atomistic models, we prove sublattice-resolved locality and cutoff-error estimates that justify replacing the global entropy calculation by a local, symmetry-respecting regression problem with controlled truncation error. This turns vibrational entropy from a global spectral calculation into a reusable local site model with linear evaluation cost at fixed cutoff. Numerical tests confirm the predicted locality behavior and show that sublattice/species-resolved surrogates achieve accurate regression, transfer across supercell sizes, and linear-scaling evaluation on Stillinger--Weber Si and CdTe benchmarks. The resulting method enables repeated harmonic-entropy evaluations in multispecies defect calculations while retaining explicit stability, truncation, and surrogate-error controls.

physics.comp-ph↗

arXiv2025

Higher-order multi-scale computational method and its convergence analysis for hygro-thermo-mechanical coupling problems of quasi-periodic composite structures

This paper proposes a novel higher-order multi-scale (HOMS) computational method, which is highly targeted for efficient, high-accuracy and low-computational-cost simulation of hygro-thermo-mechanical (H-T-M) coupling problems in quasi-periodic composite structures. The first innovation of this work is that the establishment of the high-accuracy multi-scale model incorporating the higher-order correction terms for H-T-M coupling problems of quasi-periodic composite structures. The second innovation of this work is that the error analyses in the point-wise and integral senses are rigorously derived for multi-scale asymptotic solutions. Especially from the point-wise error analysis, the primary impetus for current study to develop the HOMS approach for quasi-periodic composite structures is illustrated. Furthermore, an high-accuracy multi-scale numerical algorithm is developed based on finite element method, while corresponding convergent analysis is also obtained. Finally, extensive numerical experiments are conducted to validate the computational performance of the proposed HOMS computational approach, demonstrating not only exceptional numerical accuracy, but also reduced computational cost.

math.NA↗

arXiv2025

A higher-order three-scale computational method for efficient nonlinear thermo-mechanical coupling simulation of heterogeneous structures with multiple spatial scales

Classical multi-scale methods involving two spatial scales face significant challenges when simulating heterogeneous structures with complicated three-scale spatial configurations. This study proposes an innovative higher-order three-scale (HOTS) computational method, aimed at accurately and efficiently computing the transient nonlinear thermo-mechanical coupling problems of heterogeneous structures with multiple spatial scales. In these heterogeneous structures, temperature-dependent material properties have an important impact on the thermo-mechanical coupling responses, which is the particular interest in this work. At first, the detailed macro-meso-micro correlative model with higher-order correction terms is established by recursively two-scale analysis between macro-meso and meso-micro scales, which enables high-accuracy analysis of temperature-dependent nonlinear thermo-mechanical behaviors of heterogeneous structures with complicated three-scale configurations. The local error analysis mathematically illustrates the well-balanced property of HOTS computational model, endowing it with high computational accuracy. In addition, a two-stage numerical algorithm with off-line and on-line stages is proposed in order to efficiently simulate the nonlinear thermo-mechanical responses of heterogeneous structures with three-level spatial scales and accurately capture their highly oscillatory information at micro-scale. Finally, the high computational efficiency, high numerical accuracy and low computational cost of the presented higher-order three-scale computational approach are substantiated via representative numerical experiments. It can be summarized that this scalable and robust HOTS computational approach offers a reliably numerical tool for nonlinear multiphysics simulation of large-scale heterogeneous structures in real-world applications.

math.NA↗

arXiv2025

SO-PIFRNN: Self-optimization physics-informed Fourier-features randomized neural network for solving partial differential equations

This study proposes a self-optimization physics-informed Fourier-features randomized neural network (SO-PIFRNN) framework, which significantly improves the numerical solving accuracy of PDEs through hyperparameter optimization mechanism. The framework employs a bi-level optimization architecture: the outer-level optimization utilizes a multi-strategy collaborated particle swarm optimization (MSC-PSO) algorithm to search for optimal hyperparameters of physics-informed Fourier-features randomized neural network, while the inner-level optimization determines the output layer weights of the neural network via the least squares method. The core innovation of this study is embodied in the following three aspects: First, the Fourier basis function activation mechanism is introduced in the hidden layer of neural network, which significantly enhances the ability of the network to capture multi-frequency components of the solution. Secondly, a novel derivative neural network method is proposed, which improves the calculation accuracy and efficiency of PIFRNN method. Finally, the MSC-PSO algorithm of the hybrid optimization strategy is designed to improve the global search ability and convergence accuracy through the synergistic effect of dynamic parameter adjustment, elitist and mutation strategies. Through a series of numerical experiments, including multiscale equations in complex regions, high-order equations, high-dimensional equations and nonlinear equations, the validity of SO-PIFRNN is verified. The experimental results affirm that SO-PIFRNN exhibits superior approximation accuracy and frequency capture capability.

cs.NE↗

arXiv2023

Higher-order multi-scale deep Ritz method for multi-scale problems of authentic composite materials

The direct deep learning simulation for multi-scale problems remains a challenging issue. In this work, a novel higher-order multi-scale deep Ritz method (HOMS-DRM) is developed for thermal transfer equation of authentic composite materials with highly oscillatory and discontinuous coefficients. In this novel HOMS-DRM, higher-order multi-scale analysis and modeling are first employed to overcome limitations of prohibitive computation and Frequency Principle when direct deep learning simulation. Then, improved deep Ritz method are designed to high-accuracy and mesh-free simulation for macroscopic homogenized equation without multi-scale property and microscopic lower-order and higher-order cell problems with highly discontinuous coefficients. Moreover, the theoretical convergence of the proposed HOMS-DRM is rigorously demonstrated under appropriate assumptions. Finally, extensive numerical experiments are presented to show the computational accuracy of the proposed HOMS-DRM. This study offers a robust and high-accuracy multi-scale deep learning framework that enables the effective simulation and analysis of multi-scale problems of authentic composite materials.

math.NA↗

arXiv2023

Self-optimization wavelet-learning method for predicting nonlinear thermal conductivity of highly heterogeneous materials with randomly hierarchical configurations

In the present work, we propose a self-optimization wavelet-learning method (SO-W-LM) with high accuracy and efficiency to compute the equivalent nonlinear thermal conductivity of highly heterogeneous materials with randomly hierarchical configurations. The randomly structural heterogeneity, temperature-dependent nonlinearity and material property uncertainty of heterogeneous materials are considered within the proposed self-optimization wavelet-learning framework. Firstly, meso- and micro-structural modeling of random heterogeneous materials are achieved by the proposed computer representation method, whose simulated hierarchical configurations have relatively high volume ratio of material inclusions. Moreover, temperature-dependent nonlinearity and material property uncertainties of random heterogeneous materials are modeled by a polynomial nonlinear model and Weibull probabilistic model, which can closely resemble actual material properties of heterogeneous materials. Secondly, an innovative stochastic three-scale homogenized method (STSHM) is developed to compute the macroscopic nonlinear thermal conductivity of random heterogeneous materials. Background meshing and filling techniques are devised to extract geometry and material features of random heterogeneous materials for establishing material databases. Thirdly, high-dimensional and highly nonlinear material features of material databases are preprocessed and reduced by wavelet decomposition technique. The neural networks are further employed to excavate the predictive models from dimension-reduced low-dimensional data.

physics.comp-ph↗