Finite Element Operator Network for Solving Elliptic-type parametric PDEs

• URL: http://arxiv.org/abs/2308.04690v3
• Date: Wed, 19 Feb 2025 09:47:56 GMT
• Title: Finite Element Operator Network for Solving Elliptic-type parametric PDEs
• Authors: Jae Yong Lee, Seungchan Ko, Youngjoon Hong,
• Abstract summary: Partial differential equations (PDEs) underlie our understanding and prediction of natural phenomena. We propose a novel approach for solving parametric PDEs using a Finite Element Operator Network (FEONet)
• Score: 9.658853094888125
• License:
• Abstract: Partial differential equations (PDEs) underlie our understanding and prediction of natural phenomena across numerous fields, including physics, engineering, and finance. However, solving parametric PDEs is a complex task that necessitates efficient numerical methods. In this paper, we propose a novel approach for solving parametric PDEs using a Finite Element Operator Network (FEONet). Our proposed method leverages the power of deep learning in conjunction with traditional numerical methods, specifically the finite element method, to solve parametric PDEs in the absence of any paired input-output training data. We performed various experiments on several benchmark problems and confirmed that our approach has demonstrated excellent performance across various settings and environments, proving its versatility in terms of accuracy, generalization, and computational flexibility. While our method is not meshless, the FEONet framework shows potential for application in various fields where PDEs play a crucial role in modeling complex domains with diverse boundary conditions and singular behavior. Furthermore, we provide theoretical convergence analysis to support our approach, utilizing finite element approximation in numerical analysis.

Related papers

• Physics Meets Pixels: PDE Models in Image Processing [55.2480439325792]
  Partial Differential Equations (PDEs) have long been recognized as powerful tools for image processing and analysis. We introduce novel physical-based PDE models specifically designed for various image processing tasks.
  arXiv  Detail & Related papers  (2024-12-11T23:11:50Z)
• Advancing Generalization in PINNs through Latent-Space Representations [71.86401914779019]
  Physics-informed neural networks (PINNs) have made significant strides in modeling dynamical systems governed by partial differential equations (PDEs) We propose PIDO, a novel physics-informed neural PDE solver designed to generalize effectively across diverse PDE configurations. We validate PIDO on a range of benchmarks, including 1D combined equations and 2D Navier-Stokes equations.
  arXiv  Detail & Related papers  (2024-11-28T13:16:20Z)
• Base Models for Parabolic Partial Differential Equations [30.565534769404536]
  Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects. It is often necessary to compute the solutions or a function of the solutions to a parametric PDE in multiple scenarios corresponding to different parameters of this PDE. We propose a framework for finding solutions to parabolic PDEs across different scenarios by meta-learning an underlying base.
  arXiv  Detail & Related papers  (2024-07-17T01:04:28Z)
• Unisolver: PDE-Conditional Transformers Are Universal PDE Solvers [55.0876373185983]
  We present the Universal PDE solver (Unisolver) capable of solving a wide scope of PDEs. Our key finding is that a PDE solution is fundamentally under the control of a series of PDE components. Unisolver achieves consistent state-of-the-art results on three challenging large-scale benchmarks.
  arXiv  Detail & Related papers  (2024-05-27T15:34:35Z)
• Deep Equilibrium Based Neural Operators for Steady-State PDEs [100.88355782126098]
  We study the benefits of weight-tied neural network architectures for steady-state PDEs. We propose FNO-DEQ, a deep equilibrium variant of the FNO architecture that directly solves for the solution of a steady-state PDE.
  arXiv  Detail & Related papers  (2023-11-30T22:34:57Z)
• Spectral operator learning for parametric PDEs without data reliance [6.7083321695379885]
  We introduce a novel operator learning-based approach for solving parametric partial differential equations (PDEs) without the need for data harnessing. The proposed framework demonstrates superior performance compared to existing scientific machine learning techniques.
  arXiv  Detail & Related papers  (2023-10-03T12:37:15Z)
• Efficient Neural PDE-Solvers using Quantization Aware Training [71.0934372968972]
  We show that quantization can successfully lower the computational cost of inference while maintaining performance. Our results on four standard PDE datasets and three network architectures show that quantization-aware training works across settings and three orders of FLOPs magnitudes.
  arXiv  Detail & Related papers  (2023-08-14T09:21:19Z)
• Learning differentiable solvers for systems with hard constraints [48.54197776363251]
  We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs) We develop a differentiable PDE-constrained layer that can be incorporated into any NN architecture. Our results show that incorporating hard constraints directly into the NN architecture achieves much lower test error when compared to training on an unconstrained objective.
  arXiv  Detail & Related papers  (2022-07-18T15:11:43Z)
• Finite Expression Method for Solving High-Dimensional Partial Differential Equations [5.736353542430439]
  This paper introduces a new methodology that seeks an approximate PDE solution in the space of functions with finitely many analytic expressions. It is proved in approximation theory that FEX can avoid the curse of dimensionality. An approximate solution with finite analytic expressions also provides interpretable insights into the ground truth PDE solution.
  arXiv  Detail & Related papers  (2022-06-21T05:51:10Z)
• Spectrally Adapted Physics-Informed Neural Networks for Solving Unbounded Domain Problems [0.0]
  In this work, we combine two classes of numerical methods: (i) physics-informed neural networks (PINNs) and (ii) adaptive spectral methods. The numerical methods that we develop take advantage of the ability of physics-informed neural networks to easily implement high-order numerical schemes to efficiently solve PDEs. We then show how recently introduced adaptive techniques for spectral methods can be integrated into PINN-based PDE solvers to obtain numerical solutions of unbounded domain problems that cannot be efficiently approximated by standard PINNs.
  arXiv  Detail & Related papers  (2022-02-06T05:25:22Z)
• DiffNet: Neural Field Solutions of Parametric Partial Differential Equations [30.80582606420882]
  We consider a mesh-based approach for training a neural network to produce field predictions of solutions to PDEs. We use a weighted Galerkin loss function based on the Finite Element Method (FEM) on a parametric elliptic PDE. We prove theoretically, and illustrate with experiments, convergence results analogous to mesh convergence analysis deployed in finite element solutions to PDEs.
  arXiv  Detail & Related papers  (2021-10-04T17:59:18Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.