Journal of Computational Physics最新文献

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Non-uniform random walk for adaptive sampling 自适应抽样的非均匀随机漫步
IF 3.8 2区 物理与天体物理
Journal of Computational Physics Pub Date : 2025-06-13 DOI: 10.1016/j.jcp.2025.114160
Rouhan Wang, Dan Hu
{"title":"Non-uniform random walk for adaptive sampling","authors":"Rouhan Wang,&nbsp;Dan Hu","doi":"10.1016/j.jcp.2025.114160","DOIUrl":"10.1016/j.jcp.2025.114160","url":null,"abstract":"<div><div>Physics-Informed Neural Networks (PINNs) have been widely used in solving partial differential equations. In order to accelerate the convergence and improve the numerical accuracy of solutions, especially for solutions with low regularity, adaptive sampling strategies have been incorporated into PINNs, which place more sample points in regions with large residual. We propose a Non-Uniform Random Walk process for Adaptive Sampling (Nurwas). In Nurwas, sample points undergo a non-uniform random walk, with step-sizes inversely proportional to the square root of the desired probability density. Since the desired distribution is usually given by the residual of the numerical solution, Nurwas is performed without an explicit expression of the probability density function and without additional computational cost. As a result, compared to previous adaptive sampling strategies, Nurwas simplifies the sampling process and avoids the difficulties and limitations introduced by explicitly expressing the probability density function.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"538 ","pages":"Article 114160"},"PeriodicalIF":3.8,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144306259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Joint state-parameter estimation for the reduced fracture model via the united filter 基于联合滤波的压缩裂缝模型联合状态参数估计
IF 3.8 2区 物理与天体物理
Journal of Computational Physics Pub Date : 2025-06-13 DOI: 10.1016/j.jcp.2025.114159
Phuoc Toan Huynh , Thi-Thao-Phuong Hoang , Guannan Zhang , Feng Bao
{"title":"Joint state-parameter estimation for the reduced fracture model via the united filter","authors":"Phuoc Toan Huynh ,&nbsp;Thi-Thao-Phuong Hoang ,&nbsp;Guannan Zhang ,&nbsp;Feng Bao","doi":"10.1016/j.jcp.2025.114159","DOIUrl":"10.1016/j.jcp.2025.114159","url":null,"abstract":"<div><div>In this paper, we introduce an effective United Filter method for jointly estimating the solution state and physical parameters in flow and transport problems within fractured porous media. Fluid flow and transport in fractured porous media are critical in subsurface hydrology, geophysics, and reservoir geomechanics. Reduced fracture models, which represent fractures as lower-dimensional interfaces, enable efficient multi-scale simulations. However, reduced fracture models also face accuracy challenges due to modeling errors and uncertainties in physical parameters such as permeability and fracture geometry. To address these challenges, we propose a United Filter method, which integrates the Ensemble Score Filter (EnSF) for state estimation with the Direct Filter for parameter estimation. EnSF, based on a score-based diffusion model framework, produces ensemble representations of the state distribution without deep learning. Meanwhile, the Direct Filter, a recursive Bayesian inference method, estimates parameters directly from state observations. The United Filter combines these methods iteratively: EnSF estimates are used to refine parameter values, which are then fed back to improve state estimation. Numerical experiments demonstrate that the United Filter method surpasses the state-of-the-art Augmented Ensemble Kalman Filter, delivering more accurate state and parameter estimation for reduced fracture models. This framework also provides a robust and efficient solution for PDE-constrained inverse problems with uncertainties and sparse observations.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"538 ","pages":"Article 114159"},"PeriodicalIF":3.8,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144297165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Shape-informed surrogate models based on signed distance function domain encoding 基于符号距离函数域编码的形状通知代理模型
IF 3.8 2区 物理与天体物理
Journal of Computational Physics Pub Date : 2025-06-11 DOI: 10.1016/j.jcp.2025.114178
Linying Zhang , Stefano Pagani , Jun Zhang , Francesco Regazzoni
{"title":"Shape-informed surrogate models based on signed distance function domain encoding","authors":"Linying Zhang ,&nbsp;Stefano Pagani ,&nbsp;Jun Zhang ,&nbsp;Francesco Regazzoni","doi":"10.1016/j.jcp.2025.114178","DOIUrl":"10.1016/j.jcp.2025.114178","url":null,"abstract":"<div><div>We propose a non-intrusive method to build surrogate models that approximate the solution of parameterized partial differential equations (PDEs), capable of taking into account the dependence of the solution on the shape of the computational domain. Our approach is based on the combination of two neural networks (NNs). The first NN, conditioned on a latent code, provides an implicit representation of geometry variability through signed distance functions. This automated shape encoding technique generates compact, low-dimensional representations of geometries within a latent space, without requiring the explicit construction of an encoder. The second NN reconstructs the output physical fields independently for each spatial point, thus avoiding the computational burden typically associated with high-dimensional discretizations like computational meshes. Furthermore, we show that accuracy in geometrical characterization can be further enhanced by employing Fourier feature mapping and the distance function field as input features of the NN. The meshless nature of the proposed method, combined with the dimensionality reduction achieved through automatic feature extraction in latent space, makes it highly flexible and computationally efficient. This strategy eliminates the need for manual intervention in extracting geometric parameters, and can even be applied in cases where geometries undergo changes in their topology. Numerical tests in the field of fluid dynamics and solid mechanics demonstrate the effectiveness of the proposed method in accurately predict the solution of PDEs in domains of arbitrary shape. Remarkably, the results show that it achieves accuracy comparable to the best-case scenarios where an explicit parametrization of the computational domain is available.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"538 ","pages":"Article 114178"},"PeriodicalIF":3.8,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144306376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Multi-group maximum entropy method: Modeling translational non-equilibrium 多群最大熵法:模拟平移非平衡态
IF 3.8 2区 物理与天体物理
Journal of Computational Physics Pub Date : 2025-06-11 DOI: 10.1016/j.jcp.2025.114176
Anthony Chang, Narendra Singh, Marco Panesi
{"title":"Multi-group maximum entropy method: Modeling translational non-equilibrium","authors":"Anthony Chang,&nbsp;Narendra Singh,&nbsp;Marco Panesi","doi":"10.1016/j.jcp.2025.114176","DOIUrl":"10.1016/j.jcp.2025.114176","url":null,"abstract":"<div><div>The most rigorous physical description of non-equilibrium gas dynamics is rooted in the numerical solution of the Boltzmann equation. Yet, the large number of degrees of freedom and the wide range of both spatial and temporal scales render these equations intractable for many relevant problems. Drawing inspiration from model reduction techniques in statistical physics, this study constructs a reduced-order model for the Boltzmann equation, by combining coarse-graining modeling framework with the maximum entropy principle. This is accomplished by projecting the high-dimensional Boltzmann equation onto a carefully chosen lower-dimensional subspace, resulting from the discretization of the velocity space into sub-volumes. Within each sub-volume, the distribution function is reconstructed through the maximum entropy principle, ensuring compliance with the detailed balance. The resulting set of conservation equations comprises mass, momentum, and energy for each sub-volume, allowing for flexibility in the description of the velocity distribution function. This new set of governing equations, while retaining many of the mathematical characteristics of the conventional Navier-Stokes equations far outperforms them in terms of applicability. The proposed methodology is applied to the Bhatnagar, Gross, and Krook (BGK) formulation of the Boltzmann equation. To validate the model’s accuracy, we simulate the non-equilibrium relaxation of a gas under spatially uniform conditions and compare it directly with the analytical solution. Additionally, the model is used to analyze the shock structure of a 1-D standing shockwave across an extensive range of Mach numbers. Notably, both the non-equilibrium velocity distribution functions and macroscopic metrics derived from our model align remarkably with the direct solutions of the Boltzmann equation. These results are further validated by comparing them with available experimental data and simulation outcomes from the direct simulation Monte Carlo method, underscoring the robustness and accuracy of the proposed approach.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"538 ","pages":"Article 114176"},"PeriodicalIF":3.8,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313084","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A coordinate transformation-based physics-informed neural networks for hyperbolic conservation laws 基于坐标变换的双曲守恒定律物理信息神经网络
IF 3.8 2区 物理与天体物理
Journal of Computational Physics Pub Date : 2025-06-10 DOI: 10.1016/j.jcp.2025.114161
Yuanhong Chen , Zhen Gao , Jan S. Hesthaven , Yifan Lin , Xiang Sun
{"title":"A coordinate transformation-based physics-informed neural networks for hyperbolic conservation laws","authors":"Yuanhong Chen ,&nbsp;Zhen Gao ,&nbsp;Jan S. Hesthaven ,&nbsp;Yifan Lin ,&nbsp;Xiang Sun","doi":"10.1016/j.jcp.2025.114161","DOIUrl":"10.1016/j.jcp.2025.114161","url":null,"abstract":"<div><div>Hyperbolic conservation laws play a critical role in various fields, including aerodynamics, physics, and oceanography. However, traditional physics-informed neural networks (PINNs), despite their remarkable capabilities in solving partial differential equations (PDEs), often struggle to accurately resolve these problems. To address this challenge, a coordinate transformation-based PINN (CT-PINN) algorithm for hyperbolic conservation laws is proposed, which uses coordinate transformations along characteristic curves to prevent the generation and propagation of discontinuities. The coordinate transformation transforms subdomains divided along characteristic curves into regular domains governed by the corresponding transformed PDEs. The CT-PINN framework simultaneously learns the characteristic curves and the transformed solutions by optimizing a loss function that integrates both the transformed PDEs and the characteristic equations. Due to the equivalence between solutions in the transformed and original domains, predictions in arbitrary coordinates can be obtained without the need for interpolation. Moreover, different PINN architectures can be applied for each subdomain, with hyperparameters flexibly adjusted to enhance accuracy. The proposed method has been evaluated on a range of hyperbolic conservation laws, including the convection equation, the Burgers equation, the shallow water wave equation, the traffic flow equation and the Euler equation. The results demonstrate that CT-PINN can accurately solve the characteristic equation and PDEs, and effectively capture shock waves without transition points, outperforming traditional numerical approaches.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"538 ","pages":"Article 114161"},"PeriodicalIF":3.8,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144243530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Adaptive mesh refinement based cellular automata - lattice Boltzmann 3D simulation for multiple dendrites moving under forced convection 基于自适应网格细化的元胞自动机-晶格Boltzmann三维强迫对流多树突运动模拟
IF 3.8 2区 物理与天体物理
Journal of Computational Physics Pub Date : 2025-06-10 DOI: 10.1016/j.jcp.2025.114177
Shijie Zhang, Baofeng Zhu, Yang Zhang, Yunbo Li, Ri Li
{"title":"Adaptive mesh refinement based cellular automata - lattice Boltzmann 3D simulation for multiple dendrites moving under forced convection","authors":"Shijie Zhang,&nbsp;Baofeng Zhu,&nbsp;Yang Zhang,&nbsp;Yunbo Li,&nbsp;Ri Li","doi":"10.1016/j.jcp.2025.114177","DOIUrl":"10.1016/j.jcp.2025.114177","url":null,"abstract":"<div><div>A three-dimensional model of the coupled cellular automata and lattice Boltzmann methods, based on an adaptive mesh refinement scheme with a block grid structure, has been developed to emulate the motional growth processes of the binary alloy multi-dendrites. The model introduces an adaptive mesh refinement scheme with a uniform time scale, which effectively improves the computational efficiency and reduces the memory requirement of the 3D computational program. Moreover, the precision of the model is validated through the calculation of classical problems, including the flow through a stationary sphere and the sphere settling. The model's computational efficiency is also examined by emulating the growth problem of a stationary dendrite. Finally, the constructed model was deployed to emulate the kinematic growth procedure of Al-Cu alloy multi-dendrites in laminar, shear, and rotating flows, respectively. The findings imply that the growth of free dendrites is correlated to the relative motion speed between them and the melt. The final morphology of the dendrites is determined by the joint influences of crystal anisotropy, melt convection, and dendrite motion.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"538 ","pages":"Article 114177"},"PeriodicalIF":3.8,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144320850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Predictions based on pixel data: Insights from PDEs and finite differences 基于像素数据的预测:来自偏微分方程和有限差分的见解
IF 3.8 2区 物理与天体物理
Journal of Computational Physics Pub Date : 2025-06-09 DOI: 10.1016/j.jcp.2025.114166
Elena Celledoni , James Jackaman , Davide Murari , Brynjulf Owren
{"title":"Predictions based on pixel data: Insights from PDEs and finite differences","authors":"Elena Celledoni ,&nbsp;James Jackaman ,&nbsp;Davide Murari ,&nbsp;Brynjulf Owren","doi":"10.1016/j.jcp.2025.114166","DOIUrl":"10.1016/j.jcp.2025.114166","url":null,"abstract":"<div><div>As supported by abundant experimental evidence, neural networks are state-of-the-art for many approximation tasks in high-dimensional spaces. Still, there is a lack of a rigorous theoretical understanding of what they can approximate, at which cost, and at which accuracy. One network architecture of practical use, especially for approximation tasks involving images, is (residual) convolutional networks. However, due to the locality of the linear operators involved in these networks, their analysis is more complicated than that of fully connected neural networks. This paper deals with approximation of time sequences where each observation is a matrix. We show that with relatively small networks, we can represent exactly a class of numerical discretizations of PDEs based on the method of lines. We constructively derive these results by exploiting the connections between discrete convolution and finite difference operators. Our network architecture is inspired by those typically adopted in the approximation of time sequences. We support our theoretical results with numerical experiments simulating the linear advection, heat, and Fisher equations.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"538 ","pages":"Article 114166"},"PeriodicalIF":3.8,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144297164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Algorithms of very high space–time orders of accuracy for hyperbolic equations in the semidiscrete WENO–DeC framework 半离散WENO-DeC框架下双曲型方程的高时空阶精度算法
IF 3.8 2区 物理与天体物理
Journal of Computational Physics Pub Date : 2025-06-09 DOI: 10.1016/j.jcp.2025.114167
Lorenzo Micalizzi , Eleuterio F. Toro
{"title":"Algorithms of very high space–time orders of accuracy for hyperbolic equations in the semidiscrete WENO–DeC framework","authors":"Lorenzo Micalizzi ,&nbsp;Eleuterio F. Toro","doi":"10.1016/j.jcp.2025.114167","DOIUrl":"10.1016/j.jcp.2025.114167","url":null,"abstract":"<div><div>In this work, we provide a deep investigation of a family of arbitrary high order numerical methods for hyperbolic partial differential equations (PDEs), with particular emphasis on very high order versions, i.e., with order higher than 5. More in detail, within the context of a generic Finite Volume (FV) semidiscretization, we consider Weighted Essentially Non–Oscillatory (WENO) spatial reconstruction and Deferred Correction (DeC) time discretization. The goal of this paper is twofold. On the one hand, we want to demonstrate the possibility of utilizing very high order schemes in concrete situations and highlight the related advantages. On the other one, we want to debunk the myth according to which, in the context of numerical resolution of hyperbolic PDEs with very high order spatial discretizations, the adoption of lower order time discretizations, e.g., strong stability preserving (SSP) or linearly strong stability preserving (<span><math><mrow><mi>ℓ</mi><mtext>SSP</mtext></mrow></math></span>) Runge–Kutta (RK) schemes, does not affect the overall accuracy of the resulting approach and consequently its computational efficiency. Numerical results are reported for the linear advection equation (LAE) and for the Euler equations of fluid dynamics, showing the advantages and the critical aspects of the adoption of very high order numerical methods. Overall, the results indicate the potential for their use in real–life applications, offering advantages in terms of efficiency, such as requiring shorter computational times to achieve a prescribed error, even in problems involving discontinuities. Furthermore, the results confirm order degradation and efficiency loss when coupling very high order space discretizations with lower order SSPRK time discretizations.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"538 ","pages":"Article 114167"},"PeriodicalIF":3.8,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144280105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Physics-informed neural operators for efficient modeling of infiltration in porous media 基于物理信息的神经算子在多孔介质中渗透的有效建模
IF 3.8 2区 物理与天体物理
Journal of Computational Physics Pub Date : 2025-06-08 DOI: 10.1016/j.jcp.2025.114156
Hamza Kamil , Azzeddine Soulaïmani , Abdelaziz Beljadid
{"title":"Physics-informed neural operators for efficient modeling of infiltration in porous media","authors":"Hamza Kamil ,&nbsp;Azzeddine Soulaïmani ,&nbsp;Abdelaziz Beljadid","doi":"10.1016/j.jcp.2025.114156","DOIUrl":"10.1016/j.jcp.2025.114156","url":null,"abstract":"<div><div>The development of efficient irrigation systems relies on accurately solving the water flow model described by the highly nonlinear Richards equation. Many forward simulations are required to evaluate various settings, such as irrigation duration, emitter flux rate, initial soil moisture distribution, and the type of irrigation system employed. This often leads to slow and cumbersome decision-making processes. In this study, we propose an efficient and accurate deep learning solver for modeling water flow in unsaturated soils. The model is a physics-informed deep operator network (DeepONet) with an improved architecture designed to accurately solve the highly nonlinear Richards equation. The aim is to train the DeepONet model to learn the mapping between different initial soil moisture conditions and emitter fluxes to the corresponding water content profile in the time-space domain. By incorporating physical constraints, we reduce the need for large amounts of labeled data while maintaining accuracy. Once trained, the DeepONet model provides soil moisture predictions within fractions of a second at any desired time or space location and with any desired resolution, given any initial conditions and fluxes. The reliability of the DeepONet model is evaluated through numerical experiments on two-dimensional soil geometries with different soil types. The results highlight the efficiency of DeepONet predictions, as they closely match reference solutions. Additionally, the model’s extrapolation capability was evaluated, showing that fine-tuning with physical constraints or available soil moisture data improved its prediction accuracy. This study represents a step towards developing an efficient and rapid physics-informed surrogate model for quick decision-making in irrigation strategies.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"538 ","pages":"Article 114156"},"PeriodicalIF":3.8,"publicationDate":"2025-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144290709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A hybrid iterative neural solver based on spectral analysis for parametric PDEs 基于谱分析的参数偏微分方程混合迭代神经求解器
IF 3.8 2区 物理与天体物理
Journal of Computational Physics Pub Date : 2025-06-07 DOI: 10.1016/j.jcp.2025.114165
Chen Cui, Kai Jiang, Yun Liu, Shi Shu
{"title":"A hybrid iterative neural solver based on spectral analysis for parametric PDEs","authors":"Chen Cui,&nbsp;Kai Jiang,&nbsp;Yun Liu,&nbsp;Shi Shu","doi":"10.1016/j.jcp.2025.114165","DOIUrl":"10.1016/j.jcp.2025.114165","url":null,"abstract":"<div><div>Deep learning-based hybrid iterative methods (DL-HIM) have emerged as a promising approach for designing fast neural solvers to tackle large-scale sparse linear systems. DL-HIM combine the smoothing effect of simple iterative methods with the spectral bias of neural networks, which allows them to effectively eliminate both high-frequency and low-frequency error components. However, their efficiency may decrease if simple iterative methods can not provide effective smoothing, making it difficult for the neural network to learn mid-frequency and high-frequency components. This paper first conducts a convergence analysis for general DL-HIM from a spectral viewpoint, concluding that under reasonable assumptions, DL-HIM exhibit a convergence rate independent of grid size <span><math><mi>h</mi></math></span> and physical parameters <span><math><mrow><mi>μ</mi></mrow></math></span>. To meet these assumptions, we design a neural network from an eigen perspective, focusing on learning the eigenvalues and eigenvectors corresponding to error components that simple iterative methods struggle to eliminate. Specifically, the eigenvalues are learned by a meta subnet, while the eigenvectors are approximated using Fourier modes with a transition matrix provided by another meta subnet. The resulting DL-HIM, termed the Fourier Neural Solver (FNS), can be trained to achieve a convergence rate independent of PDE parameters and grid size within a local neighborhood of the training scale by designing a loss function that ensures the neural network complements the smoothing effect of the damped Jacobi iterative methods. We verify the performance of FNS on five types of linear parametric PDEs.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"538 ","pages":"Article 114165"},"PeriodicalIF":3.8,"publicationDate":"2025-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144263085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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