Pub Date : 2025-02-05DOI: 10.1016/j.cma.2025.117798
Georgios Grekas , Charalambos G. Makridakis
While much attention of neural network methods is devoted to high-dimensional PDE problems, in this work we consider methods designed to work for elliptic problems on domains , in association with more standard finite elements. We suggest to connect finite elements and neural network approximations through training, i.e., using finite element spaces to compute the integrals appearing in the loss functionals. This approach, retains the simplicity of classical neural network methods for PDEs, uses well established finite element tools (and software) to compute the integrals involved and it gains in efficiency and accuracy. We demonstrate that the proposed methods are stable and furthermore, we establish that the resulting approximations converge to the solutions of the PDE. Numerical results indicating the efficiency and robustness of the proposed algorithms are presented.
{"title":"Deep Ritz - Finite element methods: Neural network methods trained with finite elements","authors":"Georgios Grekas , Charalambos G. Makridakis","doi":"10.1016/j.cma.2025.117798","DOIUrl":"10.1016/j.cma.2025.117798","url":null,"abstract":"<div><div>While much attention of neural network methods is devoted to high-dimensional PDE problems, in this work we consider methods designed to work for elliptic problems on domains <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mi>d</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></math></span> in association with more standard finite elements. We suggest to connect finite elements and neural network approximations through <em>training</em>, i.e., using finite element spaces to compute the integrals appearing in the loss functionals. This approach, retains the simplicity of classical neural network methods for PDEs, uses well established finite element tools (and software) to compute the integrals involved and it gains in efficiency and accuracy. We demonstrate that the proposed methods are stable and furthermore, we establish that the resulting approximations converge to the solutions of the PDE. Numerical results indicating the efficiency and robustness of the proposed algorithms are presented.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117798"},"PeriodicalIF":6.9,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143100876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-05DOI: 10.1016/j.cma.2025.117781
M. Zani, E. Panettieri, M. Montemurro
The paper investigates the influence of adaptive sampling strategies on the generation of a metamodel based on Non-Uniform Rational Basis Spline (NURBS) entities, obtained from unstructured data, with the purpose of improving accuracy while minimising computational resources. The metamodel is defined as solution of a constrained non-linear programming problem and it is solved through a three-step optimisation process based on a gradient-based algorithm. Moreover, this paper introduces a generalised formulation of the NURBS-based metamodel capable of handling unstructured sampling data, enabling simultaneous optimisation of control points and weights. Sensitivity analyses are performed to evaluate the influence of various adaptive sampling techniques, including cross-validation-based and geometry-based strategies, on the resulting metamodel, in terms of accuracy and computational costs. Analytical benchmarks functions and a complex real-world engineering problem (dealing with the non-linear thermomechanical analysis of a part produced with the fused deposition modelling technology) are used to prove the effectiveness of the NURBS-based metamodel coupled with adaptive sampling strategies in achieving high accuracy and efficiency.
{"title":"On adaptive sampling techniques for metamodels based on NURBS entities from unstructured data","authors":"M. Zani, E. Panettieri, M. Montemurro","doi":"10.1016/j.cma.2025.117781","DOIUrl":"10.1016/j.cma.2025.117781","url":null,"abstract":"<div><div>The paper investigates the influence of adaptive sampling strategies on the generation of a metamodel based on Non-Uniform Rational Basis Spline (NURBS) entities, obtained from unstructured data, with the purpose of improving accuracy while minimising computational resources. The metamodel is defined as solution of a constrained non-linear programming problem and it is solved through a three-step optimisation process based on a gradient-based algorithm. Moreover, this paper introduces a generalised formulation of the NURBS-based metamodel capable of handling unstructured sampling data, enabling simultaneous optimisation of control points and weights. Sensitivity analyses are performed to evaluate the influence of various adaptive sampling techniques, including cross-validation-based and geometry-based strategies, on the resulting metamodel, in terms of accuracy and computational costs. Analytical benchmarks functions and a complex real-world engineering problem (dealing with the non-linear thermomechanical analysis of a part produced with the fused deposition modelling technology) are used to prove the effectiveness of the NURBS-based metamodel coupled with adaptive sampling strategies in achieving high accuracy and efficiency.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117781"},"PeriodicalIF":6.9,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143100941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-05DOI: 10.1016/j.cma.2025.117791
Bin Shu , Gang Hu , Mao Cheng , Cunxia Zhang
Particle swarm optimization (PSO) is considered among the best seminal meta-heuristic algorithms,boasting merits of minimal parameter requirements, straightforward implementation, and highly accelerated convergence capacity, lower computational complexity, etc. Nevertheless, it also has drawbacks, for instance, it tends to converge prematurely at local optima, lack of diversity, and low accuracy. In order to effectively overcome these shortcomings, this paper presents a multi-strategy fusion enhanced PSO called MSFPSO algorithm. Firstly,It motivated by the black-winged kite algorithm, a migration mechanism based on Cauchy's variation is introduced. This mechanism contributes to the efficiency and effectiveness of the algorithm in exploiting the present search area. Also, it effectively balances the dynamics relationship between exploration and exploitation, boosting the algorithm's global and local search capabilities.Second, a joint-opposition selection strategy is introduced for expanding the solution search range. Our approach is designed to avoid getting stuck in local optima. Specifically, selective opposition obtains the proximity dimension of a candidate solution through a linearly decreasing threshold. Dynamic opposition further extends the process of investigating the solution space. The algorithm is fully incorporated with the differential creative search algorithm for dual-strategy scenarios to enhance the performance of the decision-making effectiveness, population diversity, exploitation capability of the PSO. Finally, an attraction-rejection optimization strategy is introduced to further obtain a good exploitation-exploration balance capability and avoid stagnation of the algorithm. In addition, the comparison results with eight advanced optimization algorithms and six improved particle swarm optimization algorithms on CEC2020 test sets, and the statistical analysis was conducted by Wilcoxon rank sum test. It illustrate the features of the MSFPSO developed within this research strong competitiveness. The convergence of the algorithm was verified at maximum iterations of 10000 on the CEC2017 test set. Meanwhile, the experimental outcomes of applying MSFPSO to 50 practical engineering design challenges prove its effectiveness and strong applicability. The test results and numerical computations manifest that the MSFPSO algorithm with strong competitiveness will become a preferred class of meta-heuristic algorithms to tackle issues within the realm of engineering optimization.
{"title":"MSFPSO: Multi-algorithm integrated particle swarm optimization with novel strategies for solving complex engineering design problems","authors":"Bin Shu , Gang Hu , Mao Cheng , Cunxia Zhang","doi":"10.1016/j.cma.2025.117791","DOIUrl":"10.1016/j.cma.2025.117791","url":null,"abstract":"<div><div>Particle swarm optimization (PSO) is considered among the best seminal meta-heuristic algorithms,boasting merits of minimal parameter requirements, straightforward implementation, and highly accelerated convergence capacity, lower computational complexity, etc. Nevertheless, it also has drawbacks, for instance, it tends to converge prematurely at local optima, lack of diversity, and low accuracy. In order to effectively overcome these shortcomings, this paper presents a multi-strategy fusion enhanced PSO called MSFPSO algorithm. Firstly,It motivated by the black-winged kite algorithm, a migration mechanism based on Cauchy's variation is introduced. This mechanism contributes to the efficiency and effectiveness of the algorithm in exploiting the present search area. Also, it effectively balances the dynamics relationship between exploration and exploitation, boosting the algorithm's global and local search capabilities.Second, a joint-opposition selection strategy is introduced for expanding the solution search range. Our approach is designed to avoid getting stuck in local optima. Specifically, selective opposition obtains the proximity dimension of a candidate solution through a linearly decreasing threshold. Dynamic opposition further extends the process of investigating the solution space. The algorithm is fully incorporated with the differential creative search algorithm for dual-strategy scenarios to enhance the performance of the decision-making effectiveness, population diversity, exploitation capability of the PSO. Finally, an attraction-rejection optimization strategy is introduced to further obtain a good exploitation-exploration balance capability and avoid stagnation of the algorithm. In addition, the comparison results with eight advanced optimization algorithms and six improved particle swarm optimization algorithms on CEC2020 test sets, and the statistical analysis was conducted by Wilcoxon rank sum test. It illustrate the features of the MSFPSO developed within this research strong competitiveness. The convergence of the algorithm was verified at maximum iterations of 10000 on the CEC2017 test set. Meanwhile, the experimental outcomes of applying MSFPSO to 50 practical engineering design challenges prove its effectiveness and strong applicability. The test results and numerical computations manifest that the MSFPSO algorithm with strong competitiveness will become a preferred class of meta-heuristic algorithms to tackle issues within the realm of engineering optimization.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117791"},"PeriodicalIF":6.9,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143276150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-05DOI: 10.1016/j.cma.2025.117799
R. Ayoub , M. Oulghelou , P.J. Schmid
Model reduction is a key technology for large-scale physical systems in science and engineering, as it brings behavior expressed in many degrees of freedom to a more manageable size that subsequently allows control, optimization, and analysis with multi-query algorithms. We introduce an enhanced regression technique tailored to uncover quadratic parametric reduced-order dynamical systems from data. Our method, termed Improved Greedy Identification of Latent Dynamics (I-GILD), refines the learning phase of the original GILD approach proposed in Oulghelou et al. (2024). This refinement is achieved by reorganizing the quadratic model coefficients, allowing the minimum-residual problem to be reformulated using the Frobenius norm. Consequently, the optimality conditions lead to a generalized Sylvester equation, which is efficiently solved using the conjugate gradient method. Analysis of the convergence shows that I-GILD achieves superior convergence for quadratic model coefficients compared to GILD’s steepest gradient descent, reducing both computational complexity and iteration count. Additionally, we derive an error bound for the model predictions, offering insights into error growth in time and ensuring controlled accuracy as long as the magnitudes of initial error is small and learning residuals are well minimized. The efficacy of I-GILD is demonstrated through its application to numerical and experimental tests, specifically the flow past Ahmed body with a variable rear slant angle, and the lid-driven cylindrical cavity problem with variable Reynolds numbers, utilizing particle-image velocimetry (PIV) data. These tests confirm I-GILD’s ability to treat real-world dynamical system challenges and produce effective reduced-order models.
{"title":"Improved Greedy Identification of latent dynamics with application to fluid flows","authors":"R. Ayoub , M. Oulghelou , P.J. Schmid","doi":"10.1016/j.cma.2025.117799","DOIUrl":"10.1016/j.cma.2025.117799","url":null,"abstract":"<div><div>Model reduction is a key technology for large-scale physical systems in science and engineering, as it brings behavior expressed in many degrees of freedom to a more manageable size that subsequently allows control, optimization, and analysis with multi-query algorithms. We introduce an enhanced regression technique tailored to uncover quadratic parametric reduced-order dynamical systems from data. Our method, termed Improved Greedy Identification of Latent Dynamics (I-GILD), refines the learning phase of the original GILD approach proposed in Oulghelou et al. (2024). This refinement is achieved by reorganizing the quadratic model coefficients, allowing the minimum-residual problem to be reformulated using the Frobenius norm. Consequently, the optimality conditions lead to a generalized Sylvester equation, which is efficiently solved using the conjugate gradient method. Analysis of the convergence shows that I-GILD achieves superior convergence for quadratic model coefficients compared to GILD’s steepest gradient descent, reducing both computational complexity and iteration count. Additionally, we derive an error bound for the model predictions, offering insights into error growth in time and ensuring controlled accuracy as long as the magnitudes of initial error is small and learning residuals are well minimized. The efficacy of I-GILD is demonstrated through its application to numerical and experimental tests, specifically the flow past Ahmed body with a variable rear slant angle, and the lid-driven cylindrical cavity problem with variable Reynolds numbers, utilizing particle-image velocimetry (PIV) data. These tests confirm I-GILD’s ability to treat real-world dynamical system challenges and produce effective reduced-order models.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117799"},"PeriodicalIF":6.9,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143276151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-04DOI: 10.1016/j.cma.2025.117796
Giovanni Ziarelli, Stefano Pagani, Nicola Parolini, Francesco Regazzoni, Marco Verani
In this work, we aim to formalize a novel scientific machine learning framework to reconstruct the hidden dynamics of the transmission rate, whose inaccurate extrapolation can significantly impair the quality of the epidemic forecasts, by incorporating the influence of exogenous variables (such as environmental conditions and strain-specific characteristics). We propose a hybrid model that blends a data-driven layer with a physics-based one. The data-driven layer is based on a neural ordinary differential equation that learns the dynamics of the transmission rate, conditioned on the meteorological data and wave-specific latent parameters. The physics-based layer, instead, consists of a standard SEIR compartmental model, wherein the transmission rate represents an input. The learning strategy follows an end-to-end approach: the loss function quantifies the mismatch between the actual numbers of infections and its numerical prediction obtained from the SEIR model incorporating as an input the transmission rate predicted by the neural ordinary differential equation. We apply this original approach to both a synthetic test case and a realistic test case based on meteorological data (temperature and humidity) and influenza data from Italy between 2010 and 2020. In both scenarios, we achieve low generalization error on the test set and observe strong alignment between the reconstructed model and established findings on the influence of meteorological factors on epidemic spread. Finally, we implement a data assimilation strategy to adapt the neural equation to the specific characteristics of an epidemic wave under investigation, and we conduct sensitivity tests on the network’s hyperparameters.
{"title":"A model learning framework for inferring the dynamics of transmission rate depending on exogenous variables for epidemic forecasts","authors":"Giovanni Ziarelli, Stefano Pagani, Nicola Parolini, Francesco Regazzoni, Marco Verani","doi":"10.1016/j.cma.2025.117796","DOIUrl":"10.1016/j.cma.2025.117796","url":null,"abstract":"<div><div>In this work, we aim to formalize a novel scientific machine learning framework to reconstruct the hidden dynamics of the transmission rate, whose inaccurate extrapolation can significantly impair the quality of the epidemic forecasts, by incorporating the influence of exogenous variables (such as environmental conditions and strain-specific characteristics). We propose a hybrid model that blends a data-driven layer with a physics-based one. The data-driven layer is based on a neural ordinary differential equation that learns the dynamics of the transmission rate, conditioned on the meteorological data and wave-specific latent parameters. The physics-based layer, instead, consists of a standard <em>SEIR</em> compartmental model, wherein the transmission rate represents an input. The learning strategy follows an end-to-end approach: the loss function quantifies the mismatch between the actual numbers of infections and its numerical prediction obtained from the <em>SEIR</em> model incorporating as an input the transmission rate predicted by the neural ordinary differential equation. We apply this original approach to both a synthetic test case and a realistic test case based on meteorological data (temperature and humidity) and influenza data from Italy between 2010 and 2020. In both scenarios, we achieve low generalization error on the test set and observe strong alignment between the reconstructed model and established findings on the influence of meteorological factors on epidemic spread. Finally, we implement a data assimilation strategy to adapt the neural equation to the specific characteristics of an epidemic wave under investigation, and we conduct sensitivity tests on the network’s hyperparameters.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117796"},"PeriodicalIF":6.9,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143100940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a novel second-order fractional-step method for the numerical solution of incompressible Navier–Stokes equations. This fractional-step method consists of two splitting steps and it employs the second-order implicit backward differentiation formula for the time integration. Unlike most of the projection methods for solving incompressible Navier–Stokes equations, the proposed method is free from any numerical inconsistencies generated by the treatment of boundary conditions on the pressure solution. Two pressure-correction strategies including the scalar auxiliary variable approach are proposed to enhance the accuracy of the method. A rigorous stability analysis is also carried out in this study for the considered strategies. Numerical results are presented for three benchmark problems to validate the unconditional stability and to demonstrate the performance of the proposed fractional-step method for solving unsteady incompressible viscous flows. The obtained computational results support our theoretical expectations for an unconditionally stable second-order fractional-step method for the incompressible Navier–Stokes equations.
{"title":"A stable second-order splitting method for incompressible Navier–Stokes equations using the scalar auxiliary variable approach","authors":"Anouar Obbadi , Mofdi El-Amrani , Mohammed Seaid , Driss Yakoubi","doi":"10.1016/j.cma.2025.117801","DOIUrl":"10.1016/j.cma.2025.117801","url":null,"abstract":"<div><div>We propose a novel second-order fractional-step method for the numerical solution of incompressible Navier–Stokes equations. This fractional-step method consists of two splitting steps and it employs the second-order implicit backward differentiation formula for the time integration. Unlike most of the projection methods for solving incompressible Navier–Stokes equations, the proposed method is free from any numerical inconsistencies generated by the treatment of boundary conditions on the pressure solution. Two pressure-correction strategies including the scalar auxiliary variable approach are proposed to enhance the accuracy of the method. A rigorous stability analysis is also carried out in this study for the considered strategies. Numerical results are presented for three benchmark problems to validate the unconditional stability and to demonstrate the performance of the proposed fractional-step method for solving unsteady incompressible viscous flows. The obtained computational results support our theoretical expectations for an unconditionally stable second-order fractional-step method for the incompressible Navier–Stokes equations.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117801"},"PeriodicalIF":6.9,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143100937","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-03DOI: 10.1016/j.cma.2025.117790
Saurabh Deshpande , Hussein Rappel , Mark Hobbs , Stéphane P.A. Bordas , Jakub Lengiewicz
Many real-world applications demand accurate and fast predictions, as well as reliable uncertainty estimates. However, quantifying uncertainty on high-dimensional predictions is still a severely under-investigated problem, especially when input–output relationships are non-linear. To handle this problem, the present work introduces an innovative approach that combines autoencoder deep neural networks with the probabilistic regression capabilities of Gaussian processes. The autoencoder provides a low-dimensional representation of the solution space, while the Gaussian process is a Bayesian method that provides a probabilistic mapping between the low-dimensional inputs and outputs. We validate the proposed framework for its application to surrogate modeling of non-linear finite element simulations. Our findings highlight that the proposed framework is computationally efficient as well as accurate in predicting non-linear deformations of solid bodies subjected to external forces, all the while providing insightful uncertainty assessments.
{"title":"Gaussian process regression + deep neural network autoencoder for probabilistic surrogate modeling in nonlinear mechanics of solids","authors":"Saurabh Deshpande , Hussein Rappel , Mark Hobbs , Stéphane P.A. Bordas , Jakub Lengiewicz","doi":"10.1016/j.cma.2025.117790","DOIUrl":"10.1016/j.cma.2025.117790","url":null,"abstract":"<div><div>Many real-world applications demand accurate and fast predictions, as well as reliable uncertainty estimates. However, quantifying uncertainty on high-dimensional predictions is still a severely under-investigated problem, especially when input–output relationships are non-linear. To handle this problem, the present work introduces an innovative approach that combines autoencoder deep neural networks with the probabilistic regression capabilities of Gaussian processes. The autoencoder provides a low-dimensional representation of the solution space, while the Gaussian process is a Bayesian method that provides a probabilistic mapping between the low-dimensional inputs and outputs. We validate the proposed framework for its application to surrogate modeling of non-linear finite element simulations. Our findings highlight that the proposed framework is computationally efficient as well as accurate in predicting non-linear deformations of solid bodies subjected to external forces, all the while providing insightful uncertainty assessments.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117790"},"PeriodicalIF":6.9,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143100938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-03DOI: 10.1016/j.cma.2025.117785
Mohammad Sadegh Eshaghi , Cosmin Anitescu , Manish Thombre , Yizheng Wang , Xiaoying Zhuang , Timon Rabczuk
Solving partial differential equations (PDEs) is a required step in the simulation of natural and engineering systems. The associated computational costs significantly increase when exploring various scenarios, such as changes in initial or boundary conditions or different input configurations. This study proposes the Variational Physics-Informed Neural Operator (VINO), a deep learning method designed for solving PDEs by minimizing the energy formulation of PDEs. This framework can be trained without any labeled data, resulting in improved performance and accuracy compared to existing deep learning methods and conventional PDE solvers. By discretizing the domain into elements, the variational format allows VINO to overcome the key challenge in physics-informed neural operators, namely the efficient evaluation of the governing equations for computing the loss. Comparative results demonstrate VINO’s superior performance, especially as the mesh resolution increases. As a result, this study suggests a better way to incorporate physical laws into neural operators, opening a new approach for modeling and simulating nonlinear and complex processes in science and engineering.
{"title":"Variational Physics-informed Neural Operator (VINO) for solving partial differential equations","authors":"Mohammad Sadegh Eshaghi , Cosmin Anitescu , Manish Thombre , Yizheng Wang , Xiaoying Zhuang , Timon Rabczuk","doi":"10.1016/j.cma.2025.117785","DOIUrl":"10.1016/j.cma.2025.117785","url":null,"abstract":"<div><div>Solving partial differential equations (PDEs) is a required step in the simulation of natural and engineering systems. The associated computational costs significantly increase when exploring various scenarios, such as changes in initial or boundary conditions or different input configurations. This study proposes the Variational Physics-Informed Neural Operator (VINO), a deep learning method designed for solving PDEs by minimizing the energy formulation of PDEs. This framework can be trained without any labeled data, resulting in improved performance and accuracy compared to existing deep learning methods and conventional PDE solvers. By discretizing the domain into elements, the variational format allows VINO to overcome the key challenge in physics-informed neural operators, namely the efficient evaluation of the governing equations for computing the loss. Comparative results demonstrate VINO’s superior performance, especially as the mesh resolution increases. As a result, this study suggests a better way to incorporate physical laws into neural operators, opening a new approach for modeling and simulating nonlinear and complex processes in science and engineering.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117785"},"PeriodicalIF":6.9,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143100939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-02DOI: 10.1016/j.cma.2025.117795
Chunlong Hai , Jiazhen Wang , Shimin Guo , Weiqi Qian , Liquan Mei
In many industrial applications requiring computational modeling, the acquisition of high-fidelity data is often constrained by cost and technical limitations, while low-fidelity data, though cheaper and easier to obtain, lacks the same level of accuracy. Multi-fidelity data aggregation addresses this challenge by combining both types of data to construct surrogate models, balancing modeling accuracy with data cost. Optimizing the placement and distribution of high-fidelity samples is also essential to improving model performance. In this work, we propose online multi-fidelity data aggregation via hierarchical neural network (OMA-HNN). This method comprises two key components: multi-fidelity data aggregation via hierarchical neural network (MA-HNN) and an online progressive sampling framework. MA-HNN integrates data of varying fidelities within a hierarchical network structure, employing nonlinear components to capture the differences across multi-fidelity levels. The online progressive sampling framework manages high-fidelity data acquisition through two stages: initial sampling and incremental sampling. For these stages, we develop the low-fidelity-surrogate assisted sampling (LAS) strategy for the initial phase and the model divergence-based active learning (MDAL) strategy for incremental sampling. OMA-HNN was rigorously tested on 15 numerical examples across diverse multi-fidelity scenarios and further validated through three real-world applications. The results demonstrate its effectiveness and practicality, underscoring OMA-HNN’s potential to enhance the reliability and efficiency of multi-fidelity data aggregation in industrial contexts.
{"title":"Online multi-fidelity data aggregation via hierarchical neural network","authors":"Chunlong Hai , Jiazhen Wang , Shimin Guo , Weiqi Qian , Liquan Mei","doi":"10.1016/j.cma.2025.117795","DOIUrl":"10.1016/j.cma.2025.117795","url":null,"abstract":"<div><div>In many industrial applications requiring computational modeling, the acquisition of high-fidelity data is often constrained by cost and technical limitations, while low-fidelity data, though cheaper and easier to obtain, lacks the same level of accuracy. Multi-fidelity data aggregation addresses this challenge by combining both types of data to construct surrogate models, balancing modeling accuracy with data cost. Optimizing the placement and distribution of high-fidelity samples is also essential to improving model performance. In this work, we propose online multi-fidelity data aggregation via hierarchical neural network (OMA-HNN). This method comprises two key components: multi-fidelity data aggregation via hierarchical neural network (MA-HNN) and an online progressive sampling framework. MA-HNN integrates data of varying fidelities within a hierarchical network structure, employing nonlinear components to capture the differences across multi-fidelity levels. The online progressive sampling framework manages high-fidelity data acquisition through two stages: initial sampling and incremental sampling. For these stages, we develop the low-fidelity-surrogate assisted sampling (LAS) strategy for the initial phase and the model divergence-based active learning (MDAL) strategy for incremental sampling. OMA-HNN was rigorously tested on 15 numerical examples across diverse multi-fidelity scenarios and further validated through three real-world applications. The results demonstrate its effectiveness and practicality, underscoring OMA-HNN’s potential to enhance the reliability and efficiency of multi-fidelity data aggregation in industrial contexts.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117795"},"PeriodicalIF":6.9,"publicationDate":"2025-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143100934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.cma.2025.117793
Yongcheng Li , Changsheng Wang , Wenbin Hou
Nonlinear finite element analysis (FEA) is typically time-consuming, primarily due to its reliance on incremental solution schemes which require repeated stiffness matrix assembly and inversion at each step. In scenarios like structural optimization, where numerous FEA iterations are needed, deep learning-based surrogate models are usually employed as alternatives owing to their extremely high inference efficiency. However, they may exhibit weak generalization ability and produce predictions that violate established physical laws. Furthermore, their network types, such as multi-layer perceptron (MLP), limit the scalability of surrogate modeling methods, as a single model is restricted to a specific structural topology. To address these issues, we propose a universal surrogate modeling method based on heterogeneous graph neural network (HGNN) for nonlinear analysis, enhancing both scalability and generalization. Our method starts by decomposing an arbitrary engineering structure into components of different types and representing it as heterogeneous graph data, which establish a foundation for the method’s universality. Then, each increment step in the nonlinear FEA is used to extract a new sample, achieving significant data augmentation without additional computation. To further improve prediction accuracy, we leverage a physical loss derived from the nonlinear equations of each increment step to direct the model’s training process. Numerical experiments on the car body frame and car roof achieved prediction accuracies of 99.45% and 99.66%, respectively, demonstrating our method’s feasibility and efficacy.
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