Cristian Cárcamo, Alfonso Caiazzo, Felipe Galarce, Joaquín Mura
This work focuses on the numerical solution of the dynamics of a poroelastic material in the frequency domain. We provide a detailed stability analysis based on the application of the Fredholm alternative in the continuous case, considering a total pressure formulation of the Biot's equations. In the discrete setting, we propose a stabilized equal order finite element method complemented by an additional pressure stabilization to enhance the robustness of the numerical scheme with respect to the fluid permeability. Utilizing the Fredholm alternative, we extend the well-posedness results to the discrete setting, obtaining theoretical optimal convergence for the case of linear finite elements. We present different numerical experiments to validate the proposed method. First, we consider model problems with known analytic solutions in two and three dimensions. As next, we show that the method is robust for a wide range of permeabilities, including the case of discontinuous coefficients. Lastly, we show the application for the simulation of brain elastography on a realistic brain geometry obtained from medical imaging.
{"title":"A stabilized total pressure-formulation of the Biot's poroelasticity equations in frequency domain: numerical analysis and applications","authors":"Cristian Cárcamo, Alfonso Caiazzo, Felipe Galarce, Joaquín Mura","doi":"arxiv-2409.10465","DOIUrl":"https://doi.org/arxiv-2409.10465","url":null,"abstract":"This work focuses on the numerical solution of the dynamics of a poroelastic\u0000material in the frequency domain. We provide a detailed stability analysis\u0000based on the application of the Fredholm alternative in the continuous case,\u0000considering a total pressure formulation of the Biot's equations. In the\u0000discrete setting, we propose a stabilized equal order finite element method\u0000complemented by an additional pressure stabilization to enhance the robustness\u0000of the numerical scheme with respect to the fluid permeability. Utilizing the\u0000Fredholm alternative, we extend the well-posedness results to the discrete\u0000setting, obtaining theoretical optimal convergence for the case of linear\u0000finite elements. We present different numerical experiments to validate the\u0000proposed method. First, we consider model problems with known analytic\u0000solutions in two and three dimensions. As next, we show that the method is\u0000robust for a wide range of permeabilities, including the case of discontinuous\u0000coefficients. Lastly, we show the application for the simulation of brain\u0000elastography on a realistic brain geometry obtained from medical imaging.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"204 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ting Du, Xianliang Xu, Wang Kong, Ye Li, Zhongyi Huang
Learning operators for parametric partial differential equations (PDEs) using neural networks has gained significant attention in recent years. However, standard approaches like Deep Operator Networks (DeepONets) require extensive labeled data, and physics-informed DeepONets encounter training challenges. In this paper, we introduce a novel physics-informed tailored finite point operator network (PI-TFPONet) method to solve parametric interface problems without the need for labeled data. Our method fully leverages the prior physical information of the problem, eliminating the need to include the PDE residual in the loss function, thereby avoiding training challenges. The PI-TFPONet is specifically designed to address certain properties of the problem, allowing us to naturally obtain an approximate solution that closely matches the exact solution. Our method is theoretically proven to converge if the local mesh size is sufficiently small and the training loss is minimized. Notably, our approach is uniformly convergent for singularly perturbed interface problems. Extensive numerical studies show that our unsupervised PI-TFPONet is comparable to or outperforms existing state-of-the-art supervised deep operator networks in terms of accuracy and versatility.
{"title":"Physics-Informed Tailored Finite Point Operator Network for Parametric Interface Problems","authors":"Ting Du, Xianliang Xu, Wang Kong, Ye Li, Zhongyi Huang","doi":"arxiv-2409.10284","DOIUrl":"https://doi.org/arxiv-2409.10284","url":null,"abstract":"Learning operators for parametric partial differential equations (PDEs) using\u0000neural networks has gained significant attention in recent years. However,\u0000standard approaches like Deep Operator Networks (DeepONets) require extensive\u0000labeled data, and physics-informed DeepONets encounter training challenges. In\u0000this paper, we introduce a novel physics-informed tailored finite point\u0000operator network (PI-TFPONet) method to solve parametric interface problems\u0000without the need for labeled data. Our method fully leverages the prior\u0000physical information of the problem, eliminating the need to include the PDE\u0000residual in the loss function, thereby avoiding training challenges. The\u0000PI-TFPONet is specifically designed to address certain properties of the\u0000problem, allowing us to naturally obtain an approximate solution that closely\u0000matches the exact solution. Our method is theoretically proven to converge if\u0000the local mesh size is sufficiently small and the training loss is minimized.\u0000Notably, our approach is uniformly convergent for singularly perturbed\u0000interface problems. Extensive numerical studies show that our unsupervised\u0000PI-TFPONet is comparable to or outperforms existing state-of-the-art supervised\u0000deep operator networks in terms of accuracy and versatility.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents an energy-preserving machine learning method for inferring reduced-order models (ROMs) by exploiting the multi-symplectic form of partial differential equations (PDEs). The vast majority of energy-preserving reduced-order methods use symplectic Galerkin projection to construct reduced-order Hamiltonian models by projecting the full models onto a symplectic subspace. However, symplectic projection requires the existence of fully discrete operators, and in many cases, such as black-box PDE solvers, these operators are inaccessible. In this work, we propose an energy-preserving machine learning method that can infer the dynamics of the given PDE using data only, so that the proposed framework does not depend on the fully discrete operators. In this context, the proposed method is non-intrusive. The proposed method is grey box in the sense that it requires only some basic knowledge of the multi-symplectic model at the partial differential equation level. We prove that the proposed method satisfies spatially discrete local energy conservation and preserves the multi-symplectic conservation laws. We test our method on the linear wave equation, the Korteweg-de Vries equation, and the Zakharov-Kuznetsov equation. We test the generalization of our learned models by testing them far outside the training time interval.
{"title":"Structure-preserving learning for multi-symplectic PDEs","authors":"Süleyman Yıldız, Pawan Goyal, Peter Benner","doi":"arxiv-2409.10432","DOIUrl":"https://doi.org/arxiv-2409.10432","url":null,"abstract":"This paper presents an energy-preserving machine learning method for\u0000inferring reduced-order models (ROMs) by exploiting the multi-symplectic form\u0000of partial differential equations (PDEs). The vast majority of\u0000energy-preserving reduced-order methods use symplectic Galerkin projection to\u0000construct reduced-order Hamiltonian models by projecting the full models onto a\u0000symplectic subspace. However, symplectic projection requires the existence of\u0000fully discrete operators, and in many cases, such as black-box PDE solvers,\u0000these operators are inaccessible. In this work, we propose an energy-preserving\u0000machine learning method that can infer the dynamics of the given PDE using data\u0000only, so that the proposed framework does not depend on the fully discrete\u0000operators. In this context, the proposed method is non-intrusive. The proposed\u0000method is grey box in the sense that it requires only some basic knowledge of\u0000the multi-symplectic model at the partial differential equation level. We prove\u0000that the proposed method satisfies spatially discrete local energy conservation\u0000and preserves the multi-symplectic conservation laws. We test our method on the\u0000linear wave equation, the Korteweg-de Vries equation, and the\u0000Zakharov-Kuznetsov equation. We test the generalization of our learned models\u0000by testing them far outside the training time interval.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259146","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nadhir Ben Rached, Abdul-Lateef Haji-Ali, Raúl Tempone, Leon Wilkosz
This work develops a particle system addressing the approximation of McKean-Vlasov stochastic differential equations (SDEs). The novelty of the approach lies in involving low discrepancy sequences nontrivially in the construction of a particle system with coupled noise and initial conditions. Weak convergence for SDEs with additive noise is proven. A numerical study demonstrates that the novel approach presented here doubles the respective convergence rates for weak and strong approximation of the mean-field limit, compared with the standard particle system. These rates are proven in the simplified setting of a mean-field ordinary differential equation in terms of appropriate bounds involving the star discrepancy for low discrepancy sequences with a group structure, such as Rank-1 lattice points. This construction nontrivially provides an antithetic multilevel quasi-Monte Carlo estimator. An asymptotic error analysis reveals that the proposed approach outperforms methods based on the classic particle system with independent initial conditions and noise.
{"title":"Forward Propagation of Low Discrepancy Through McKean-Vlasov Dynamics: From QMC to MLQMC","authors":"Nadhir Ben Rached, Abdul-Lateef Haji-Ali, Raúl Tempone, Leon Wilkosz","doi":"arxiv-2409.09821","DOIUrl":"https://doi.org/arxiv-2409.09821","url":null,"abstract":"This work develops a particle system addressing the approximation of\u0000McKean-Vlasov stochastic differential equations (SDEs). The novelty of the\u0000approach lies in involving low discrepancy sequences nontrivially in the\u0000construction of a particle system with coupled noise and initial conditions.\u0000Weak convergence for SDEs with additive noise is proven. A numerical study\u0000demonstrates that the novel approach presented here doubles the respective\u0000convergence rates for weak and strong approximation of the mean-field limit,\u0000compared with the standard particle system. These rates are proven in the\u0000simplified setting of a mean-field ordinary differential equation in terms of\u0000appropriate bounds involving the star discrepancy for low discrepancy sequences\u0000with a group structure, such as Rank-1 lattice points. This construction\u0000nontrivially provides an antithetic multilevel quasi-Monte Carlo estimator. An\u0000asymptotic error analysis reveals that the proposed approach outperforms\u0000methods based on the classic particle system with independent initial\u0000conditions and noise.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper proposes high-order accurate, oscillation-eliminating Hermite weighted essentially non-oscillatory (OE-HWENO) finite volume schemes for hyperbolic conservation laws. The OE-HWENO schemes apply an OE procedure after each Runge--Kutta stage, dampening the first-order moments of the HWENO solution to suppress spurious oscillations without any problem-dependent parameters. This OE procedure acts as a filter, derived from the solution operator of a novel damping equation, solved exactly without discretization. As a result, the OE-HWENO method remains stable with a normal CFL number, even for strong shocks producing highly stiff damping terms. To ensure the method's non-oscillatory property across varying scales and wave speeds, we design a scale- and evolution-invariant damping equation and propose a dimensionless transformation for HWENO reconstruction. The OE-HWENO method offers several advantages over existing HWENO methods: the OE procedure is efficient and easy to implement, requiring only simple multiplication of first-order moments; it preserves high-order accuracy, local compactness, and spectral properties. The non-intrusive OE procedure can be integrated seamlessly into existing HWENO codes. Finally, we analyze the bound-preserving (BP) property using optimal cell average decomposition, relaxing the BP time step-size constraint and reducing decomposition points, improving efficiency. Extensive benchmarks validate the method's accuracy, efficiency, resolution, and robustness.
{"title":"High-Order Oscillation-Eliminating Hermite WENO Method for Hyperbolic Conservation Laws","authors":"Chuan Fan, Kailiang Wu","doi":"arxiv-2409.09632","DOIUrl":"https://doi.org/arxiv-2409.09632","url":null,"abstract":"This paper proposes high-order accurate, oscillation-eliminating Hermite\u0000weighted essentially non-oscillatory (OE-HWENO) finite volume schemes for\u0000hyperbolic conservation laws. The OE-HWENO schemes apply an OE procedure after\u0000each Runge--Kutta stage, dampening the first-order moments of the HWENO\u0000solution to suppress spurious oscillations without any problem-dependent\u0000parameters. This OE procedure acts as a filter, derived from the solution\u0000operator of a novel damping equation, solved exactly without discretization. As\u0000a result, the OE-HWENO method remains stable with a normal CFL number, even for\u0000strong shocks producing highly stiff damping terms. To ensure the method's\u0000non-oscillatory property across varying scales and wave speeds, we design a\u0000scale- and evolution-invariant damping equation and propose a dimensionless\u0000transformation for HWENO reconstruction. The OE-HWENO method offers several\u0000advantages over existing HWENO methods: the OE procedure is efficient and easy\u0000to implement, requiring only simple multiplication of first-order moments; it\u0000preserves high-order accuracy, local compactness, and spectral properties. The\u0000non-intrusive OE procedure can be integrated seamlessly into existing HWENO\u0000codes. Finally, we analyze the bound-preserving (BP) property using optimal\u0000cell average decomposition, relaxing the BP time step-size constraint and\u0000reducing decomposition points, improving efficiency. Extensive benchmarks\u0000validate the method's accuracy, efficiency, resolution, and robustness.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Discontinuous Galerkin (DG) schemes on unstructured meshes offer the advantages of compactness and the ability to handle complex computational domains. However, their robustness and reliability in solving hyperbolic conservation laws depend on two critical abilities: suppressing spurious oscillations and preserving intrinsic bounds or constraints. This paper introduces two significant advancements in enhancing the robustness and efficiency of DG methods on unstructured meshes for general hyperbolic conservation laws, while maintaining their accuracy and compactness. First, we investigate the oscillation-eliminating (OE) DG methods on unstructured meshes. These methods not only maintain key features such as conservation, scale invariance, and evolution invariance but also achieve rotation invariance through a novel rotation-invariant OE (RIOE) procedure. Second, we propose, for the first time, the optimal convex decomposition for designing efficient bound-preserving (BP) DG schemes on unstructured meshes. Finding the optimal convex decomposition that maximizes the BP CFL number is an important yet challenging problem.While this challenge was addressed for rectangular meshes, it remains an open problem for triangular meshes. This paper successfully constructs the optimal convex decomposition for the widely used $P^1$ and $P^2$ spaces on triangular cells, significantly improving the efficiency of BP DG methods.The maximum BP CFL numbers are increased by 100%--200% for $P^1$ and 280.38%--350% for $P^2$, compared to classic decomposition. Furthermore, our RIOE procedure and optimal decomposition technique can be integrated into existing DG codes with little and localized modifications. These techniques require only edge-neighboring cell information, thereby retaining the compactness and high parallel efficiency of original DG methods.
非结构网格上的非连续伽勒金（DG）方案具有结构紧凑和能够处理复杂计算域的优点。然而，它们在求解双曲守恒定律时的鲁棒性和可靠性取决于两个关键能力：抑制虚假振荡和保留固有边界或约束。本文介绍了在非结构网格上增强 DG 方法对一般双曲守恒定律的鲁棒性和效率的两个重要进展，同时保持了它们的精度和紧凑性。首先，我们研究了非结构网格上的振荡消除（OE）DG 方法。这些方法不仅保持了守恒性、尺度不变性和演化不变性等关键特征，还通过新颖的旋转不变 OE（RIOE）过程实现了旋转不变性。其次，我们首次提出了在非结构网格上设计高效保界（BP）DG 方案的最优凸分解。寻找能使 BP CFL 数最大化的最优凸分解是一个重要而又具有挑战性的问题。本文成功地为三角形单元上广泛使用的 $P^1$ 和 $P^2$ 空间构建了最优凸分解，显著提高了 BP DG 方法的效率。与经典分解相比，$P^1$ 的最大 BP CFL 数提高了 100%-200%，$P^2$ 的最大 BP CFL 数提高了 280.38%-350%。此外，我们的 RIOE 程序和最优分解技术可以集成到现有的 DG 代码中，只需进行少量的局部修改。这些技术只需要边缘相邻单元的信息，因此保留了原始 DG 方法的紧凑性和高并行效率。
{"title":"Robust DG Schemes on Unstructured Triangular Meshes: Oscillation Elimination and Bound Preservation via Optimal Convex Decomposition","authors":"Shengrong Ding, Shumo Cui, Kailiang Wu","doi":"arxiv-2409.09620","DOIUrl":"https://doi.org/arxiv-2409.09620","url":null,"abstract":"Discontinuous Galerkin (DG) schemes on unstructured meshes offer the\u0000advantages of compactness and the ability to handle complex computational\u0000domains. However, their robustness and reliability in solving hyperbolic\u0000conservation laws depend on two critical abilities: suppressing spurious\u0000oscillations and preserving intrinsic bounds or constraints. This paper\u0000introduces two significant advancements in enhancing the robustness and\u0000efficiency of DG methods on unstructured meshes for general hyperbolic\u0000conservation laws, while maintaining their accuracy and compactness. First, we\u0000investigate the oscillation-eliminating (OE) DG methods on unstructured meshes.\u0000These methods not only maintain key features such as conservation, scale\u0000invariance, and evolution invariance but also achieve rotation invariance\u0000through a novel rotation-invariant OE (RIOE) procedure. Second, we propose, for\u0000the first time, the optimal convex decomposition for designing efficient\u0000bound-preserving (BP) DG schemes on unstructured meshes. Finding the optimal\u0000convex decomposition that maximizes the BP CFL number is an important yet\u0000challenging problem.While this challenge was addressed for rectangular meshes,\u0000it remains an open problem for triangular meshes. This paper successfully\u0000constructs the optimal convex decomposition for the widely used $P^1$ and $P^2$\u0000spaces on triangular cells, significantly improving the efficiency of BP DG\u0000methods.The maximum BP CFL numbers are increased by 100%--200% for $P^1$ and\u0000280.38%--350% for $P^2$, compared to classic decomposition. Furthermore, our\u0000RIOE procedure and optimal decomposition technique can be integrated into\u0000existing DG codes with little and localized modifications. These techniques\u0000require only edge-neighboring cell information, thereby retaining the\u0000compactness and high parallel efficiency of original DG methods.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rafael Flock, Shuigen Liu, Yiqiu Dong, Xin T. Tong
We consider Bayesian inference for image deblurring with total variation (TV) prior. Since the posterior is analytically intractable, we resort to Markov chain Monte Carlo (MCMC) methods. However, since most MCMC methods significantly deteriorate in high dimensions, they are not suitable to handle high resolution imaging problems. In this paper, we show how low-dimensional sampling can still be facilitated by exploiting the sparse conditional structure of the posterior. To this end, we make use of the local structures of the blurring operator and the TV prior by partitioning the image into rectangular blocks and employing a blocked Gibbs sampler with proposals stemming from the Metropolis-Hastings adjusted Langevin Algorithm (MALA). We prove that this MALA-within-Gibbs (MLwG) sampling algorithm has dimension-independent block acceptance rates and dimension-independent convergence rate. In order to apply the MALA proposals, we approximate the TV by a smoothed version, and show that the introduced approximation error is evenly distributed and dimension-independent. Since the posterior is a Gibbs density, we can use the Hammersley-Clifford Theorem to identify the posterior conditionals which only depend locally on the neighboring blocks. We outline computational strategies to evaluate the conditionals, which are the target densities in the Gibbs updates, locally and in parallel. In two numerical experiments, we validate the dimension-independent properties of the MLwG algorithm and demonstrate its superior performance over MALA.
{"title":"Local MALA-within-Gibbs for Bayesian image deblurring with total variation prior","authors":"Rafael Flock, Shuigen Liu, Yiqiu Dong, Xin T. Tong","doi":"arxiv-2409.09810","DOIUrl":"https://doi.org/arxiv-2409.09810","url":null,"abstract":"We consider Bayesian inference for image deblurring with total variation (TV)\u0000prior. Since the posterior is analytically intractable, we resort to Markov\u0000chain Monte Carlo (MCMC) methods. However, since most MCMC methods\u0000significantly deteriorate in high dimensions, they are not suitable to handle\u0000high resolution imaging problems. In this paper, we show how low-dimensional\u0000sampling can still be facilitated by exploiting the sparse conditional\u0000structure of the posterior. To this end, we make use of the local structures of\u0000the blurring operator and the TV prior by partitioning the image into\u0000rectangular blocks and employing a blocked Gibbs sampler with proposals\u0000stemming from the Metropolis-Hastings adjusted Langevin Algorithm (MALA). We\u0000prove that this MALA-within-Gibbs (MLwG) sampling algorithm has\u0000dimension-independent block acceptance rates and dimension-independent\u0000convergence rate. In order to apply the MALA proposals, we approximate the TV\u0000by a smoothed version, and show that the introduced approximation error is\u0000evenly distributed and dimension-independent. Since the posterior is a Gibbs\u0000density, we can use the Hammersley-Clifford Theorem to identify the posterior\u0000conditionals which only depend locally on the neighboring blocks. We outline\u0000computational strategies to evaluate the conditionals, which are the target\u0000densities in the Gibbs updates, locally and in parallel. In two numerical\u0000experiments, we validate the dimension-independent properties of the MLwG\u0000algorithm and demonstrate its superior performance over MALA.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose PROSE-FD, a zero-shot multimodal PDE foundational model for simultaneous prediction of heterogeneous two-dimensional physical systems related to distinct fluid dynamics settings. These systems include shallow water equations and the Navier-Stokes equations with incompressible and compressible flow, regular and complex geometries, and different buoyancy settings. This work presents a new transformer-based multi-operator learning approach that fuses symbolic information to perform operator-based data prediction, i.e. non-autoregressive. By incorporating multiple modalities in the inputs, the PDE foundation model builds in a pathway for including mathematical descriptions of the physical behavior. We pre-train our foundation model on 6 parametric families of equations collected from 13 datasets, including over 60K trajectories. Our model outperforms popular operator learning, computer vision, and multi-physics models, in benchmark forward prediction tasks. We test our architecture choices with ablation studies.
{"title":"PROSE-FD: A Multimodal PDE Foundation Model for Learning Multiple Operators for Forecasting Fluid Dynamics","authors":"Yuxuan Liu, Jingmin Sun, Xinjie He, Griffin Pinney, Zecheng Zhang, Hayden Schaeffer","doi":"arxiv-2409.09811","DOIUrl":"https://doi.org/arxiv-2409.09811","url":null,"abstract":"We propose PROSE-FD, a zero-shot multimodal PDE foundational model for\u0000simultaneous prediction of heterogeneous two-dimensional physical systems\u0000related to distinct fluid dynamics settings. These systems include shallow\u0000water equations and the Navier-Stokes equations with incompressible and\u0000compressible flow, regular and complex geometries, and different buoyancy\u0000settings. This work presents a new transformer-based multi-operator learning\u0000approach that fuses symbolic information to perform operator-based data\u0000prediction, i.e. non-autoregressive. By incorporating multiple modalities in\u0000the inputs, the PDE foundation model builds in a pathway for including\u0000mathematical descriptions of the physical behavior. We pre-train our foundation\u0000model on 6 parametric families of equations collected from 13 datasets,\u0000including over 60K trajectories. Our model outperforms popular operator\u0000learning, computer vision, and multi-physics models, in benchmark forward\u0000prediction tasks. We test our architecture choices with ablation studies.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In arXiv:1906.09232v2, Golovaty et al. present a $Q$-tensor model for liquid crystal dynamics which reduces to the well-known Oseen-Frank director field model in uniaxial states. We study a closely related model and present an energy stable scheme for the corresponding gradient flow. We prove the convergence of this scheme via fixed-point iteration and rigorously show the $Gamma$-convergence of discrete minimizers as the mesh size approaches zero. In the numerical experiments, we successfully simulate isotropic-to-nematic phase transitions as expected.
{"title":"Finite element analysis of a nematic liquid crystal Landau-de Gennes model with quartic elastic terms","authors":"Jacob Elafandi, Franziska Weber","doi":"arxiv-2409.09837","DOIUrl":"https://doi.org/arxiv-2409.09837","url":null,"abstract":"In arXiv:1906.09232v2, Golovaty et al. present a $Q$-tensor model for liquid\u0000crystal dynamics which reduces to the well-known Oseen-Frank director field\u0000model in uniaxial states. We study a closely related model and present an\u0000energy stable scheme for the corresponding gradient flow. We prove the\u0000convergence of this scheme via fixed-point iteration and rigorously show the\u0000$Gamma$-convergence of discrete minimizers as the mesh size approaches zero.\u0000In the numerical experiments, we successfully simulate isotropic-to-nematic\u0000phase transitions as expected.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259103","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents a comprehensive numerical analysis of centrifugal clutch systems integrated with a two-speed automatic transmission, a key component in automotive torque transfer. Centrifugal clutches enable torque transmission based on rotational speed without external controls. The study systematically examines various clutch configurations effects on transmission dynamics, focusing on torque transfer, upshifting, and downshifting behaviors under different conditions. A Deep Neural Network (DNN) model predicts clutch engagement using parameters such as spring preload and shoe mass, offering an efficient alternative to complex simulations. The integration of deep learning and numerical modeling provides critical insights for optimizing clutch designs, enhancing transmission performance and efficiency.
{"title":"Analysis of Centrifugal Clutches in Two-Speed Automatic Transmissions with Deep Learning-Based Engagement Prediction","authors":"Bo-Yi Lin, Kai Chun Lin","doi":"arxiv-2409.09755","DOIUrl":"https://doi.org/arxiv-2409.09755","url":null,"abstract":"This paper presents a comprehensive numerical analysis of centrifugal clutch\u0000systems integrated with a two-speed automatic transmission, a key component in\u0000automotive torque transfer. Centrifugal clutches enable torque transmission\u0000based on rotational speed without external controls. The study systematically\u0000examines various clutch configurations effects on transmission dynamics,\u0000focusing on torque transfer, upshifting, and downshifting behaviors under\u0000different conditions. A Deep Neural Network (DNN) model predicts clutch\u0000engagement using parameters such as spring preload and shoe mass, offering an\u0000efficient alternative to complex simulations. The integration of deep learning\u0000and numerical modeling provides critical insights for optimizing clutch\u0000designs, enhancing transmission performance and efficiency.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"157 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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