Dynamic Gaussian Graph Operator: Learning parametric partial differential equations in arbitrary discrete mechanics problems

• URL: http://arxiv.org/abs/2403.02810v1
• Date: Tue, 5 Mar 2024 09:25:31 GMT
• Title: Dynamic Gaussian Graph Operator: Learning parametric partial differential equations in arbitrary discrete mechanics problems
• Authors: Chu Wang, Jinhong Wu, Yanzhi Wang, Zhijian Zha, Qi Zhou
• Abstract summary: We propose a novel operator learning algorithm that expands neural operators to learning parametric PDEs in arbitrary discrete mechanics problems. The efficiency and robustness of DGGO are validated by applying it to solve numerical arbitrary discrete mechanics problems. The proposed method is utilized to forecast stress field of hyper-elastic material with geometrically variable void as engineering application.
• Score: 33.32926047057572
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Deep learning methods have access to be employed for solving physical systems governed by parametric partial differential equations (PDEs) due to massive scientific data. It has been refined to operator learning that focuses on learning non-linear mapping between infinite-dimensional function spaces, offering interface from observations to solutions. However, state-of-the-art neural operators are limited to constant and uniform discretization, thereby leading to deficiency in generalization on arbitrary discretization schemes for computational domain. In this work, we propose a novel operator learning algorithm, referred to as Dynamic Gaussian Graph Operator (DGGO) that expands neural operators to learning parametric PDEs in arbitrary discrete mechanics problems. The Dynamic Gaussian Graph (DGG) kernel learns to map the observation vectors defined in general Euclidean space to metric vectors defined in high-dimensional uniform metric space. The DGG integral kernel is parameterized by Gaussian kernel weighted Riemann sum approximating and using dynamic message passing graph to depict the interrelation within the integral term. Fourier Neural Operator is selected to localize the metric vectors on spatial and frequency domains. Metric vectors are regarded as located on latent uniform domain, wherein spatial and spectral transformation offer highly regular constraints on solution space. The efficiency and robustness of DGGO are validated by applying it to solve numerical arbitrary discrete mechanics problems in comparison with mainstream neural operators. Ablation experiments are implemented to demonstrate the effectiveness of spatial transformation in the DGG kernel. The proposed method is utilized to forecast stress field of hyper-elastic material with geometrically variable void as engineering application.

Related papers

• Finite Operator Learning: Bridging Neural Operators and Numerical Methods for Efficient Parametric Solution and Optimization of PDEs [0.0]
  We introduce a method that combines neural operators, physics-informed machine learning, and standard numerical methods for solving PDEs. We can parametrically solve partial differential equations in a data-free manner and provide accurate sensitivities. Our study focuses on the steady-state heat equation within heterogeneous materials.
  arXiv  Detail & Related papers  (2024-07-04T21:23:12Z)
• Geometry-Informed Neural Operator for Large-Scale 3D PDEs [76.06115572844882]
  We propose the geometry-informed neural operator (GINO) to learn the solution operator of large-scale partial differential equations. We successfully trained GINO to predict the pressure on car surfaces using only five hundred data points.
  arXiv  Detail & Related papers  (2023-09-01T16:59:21Z)
• Score-based Diffusion Models in Function Space [140.792362459734]
  Diffusion models have recently emerged as a powerful framework for generative modeling. We introduce a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space. We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution.
  arXiv  Detail & Related papers  (2023-02-14T23:50:53Z)
• Solving High-Dimensional PDEs with Latent Spectral Models [74.1011309005488]
  We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space. LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks.
  arXiv  Detail & Related papers  (2023-01-30T04:58:40Z)
• Rigorous dynamical mean field theory for stochastic gradient descent methods [17.90683687731009]
  We prove closed-form equations for the exact high-dimensionals of a family of first order gradient-based methods. This includes widely used algorithms such as gradient descent (SGD) or Nesterov acceleration.
  arXiv  Detail & Related papers  (2022-10-12T21:10:55Z)
• Equivariant Graph Mechanics Networks with Constraints [83.38709956935095]
  We propose Graph Mechanics Network (GMN) which is efficient, equivariant and constraint-aware. GMN represents, by generalized coordinates, the forward kinematics information (positions and velocities) of a structural object. Extensive experiments support the advantages of GMN compared to the state-of-the-art GNNs in terms of prediction accuracy, constraint satisfaction and data efficiency.
  arXiv  Detail & Related papers  (2022-03-12T14:22:14Z)
• Adjoint-aided inference of Gaussian process driven differential equations [0.8257490175399691]
  We show how the adjoint of a linear system can be used to efficiently infer forcing functions modelled as GPs. We demonstrate the approach on systems of both ordinary and partial differential equations.
  arXiv  Detail & Related papers  (2022-02-09T17:35:14Z)
• An Operator-Splitting Method for the Gaussian Curvature Regularization Model with Applications in Surface Smoothing and Imaging [6.860238280163609]
  We propose an operator-splitting method for a general Gaussian curvature model. The proposed method is not sensitive to the choice of parameters, its efficiency and performances being demonstrated.
  arXiv  Detail & Related papers  (2021-08-04T08:59:41Z)
• Fourier Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  We formulate a new neural operator by parameterizing the integral kernel directly in Fourier space. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. It is up to three orders of magnitude faster compared to traditional PDE solvers.
  arXiv  Detail & Related papers  (2020-10-18T00:34:21Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)

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.