Learning time-dependent PDE solver using Message Passing Graph Neural Networks

• URL: http://arxiv.org/abs/2204.07651v1
• Date: Fri, 15 Apr 2022 21:10:32 GMT
• Title: Learning time-dependent PDE solver using Message Passing Graph Neural Networks
• Authors: Pourya Pilva and Ahmad Zareei
• Abstract summary: We introduce a graph neural network approach to finding efficient PDE solvers through learning using message-passing models. We use graphs to represent PDE-data on an unstructured mesh and show that message passing graph neural networks (MPGNN) can parameterize governing equations. We show that a recurrent graph neural network approach can find a temporal sequence of solutions to a PDE.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: One of the main challenges in solving time-dependent partial differential equations is to develop computationally efficient solvers that are accurate and stable. Here, we introduce a graph neural network approach to finding efficient PDE solvers through learning using message-passing models. We first introduce domain invariant features for PDE-data inspired by classical PDE solvers for an efficient physical representation. Next, we use graphs to represent PDE-data on an unstructured mesh and show that message passing graph neural networks (MPGNN) can parameterize governing equations, and as a result, efficiently learn accurate solver schemes for linear/nonlinear PDEs. We further show that the solvers are independent of the initial trained geometry, i.e. the trained solver can find PDE solution on different complex domains. Lastly, we show that a recurrent graph neural network approach can find a temporal sequence of solutions to a PDE.

Related papers

• 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)
• Meta-learning of Physics-informed Neural Networks for Efficiently Solving Newly Given PDEs [33.072056425485115]
  We propose a neural network-based meta-learning method to efficiently solve partial differential equation (PDE) problems. The proposed method is designed to meta-learn how to solve a wide variety of PDE problems, and uses the knowledge for solving newly given PDE problems. We demonstrate that our proposed method outperforms existing methods in predicting solutions of PDE problems.
  arXiv  Detail & Related papers  (2023-10-20T04:35:59Z)
• A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networks [52.5899851000193]
  We develop an ODE based IVP solver which prevents the network from getting ill-conditioned and runs in time linear in the number of parameters. We show that current methods based on this approach suffer from two key issues. First, following the ODE produces an uncontrolled growth in the conditioning of the problem, ultimately leading to unacceptably large numerical errors.
  arXiv  Detail & Related papers  (2023-04-28T17:28:18Z)
• Meta-PDE: Learning to Solve PDEs Quickly Without a Mesh [24.572840023107574]
  Partial differential equations (PDEs) are often computationally challenging to solve. We present a meta-learning based method which learns to rapidly solve problems from a distribution of related PDEs.
  arXiv  Detail & Related papers  (2022-11-03T06:17:52Z)
• Learning to Solve PDE-constrained Inverse Problems with Graph Networks [51.89325993156204]
  In many application domains across science and engineering, we are interested in solving inverse problems with constraints defined by a partial differential equation (PDE) Here we explore GNNs to solve such PDE-constrained inverse problems. We demonstrate computational speedups of up to 90x using GNNs compared to principled solvers.
  arXiv  Detail & Related papers  (2022-06-01T18:48:01Z)
• Physics-constrained Unsupervised Learning of Partial Differential Equations using Meshes [1.066048003460524]
  Graph neural networks show promise in accurately representing irregularly meshed objects and learning their dynamics. In this work, we represent meshes naturally as graphs, process these using Graph Networks, and formulate our physics-based loss to provide an unsupervised learning framework for partial differential equations (PDE) Our framework will enable the application of PDE solvers in interactive settings, such as model-based control of soft-body deformations.
  arXiv  Detail & Related papers  (2022-03-30T19:22:56Z)
• Lie Point Symmetry Data Augmentation for Neural PDE Solvers [69.72427135610106]
  We present a method, which can partially alleviate this problem, by improving neural PDE solver sample complexity. In the context of PDEs, it turns out that we are able to quantitatively derive an exhaustive list of data transformations. We show how it can easily be deployed to improve neural PDE solver sample complexity by an order of magnitude.
  arXiv  Detail & Related papers  (2022-02-15T18:43:17Z)
• NeuralPDE: Modelling Dynamical Systems from Data [0.44259821861543996]
  We propose NeuralPDE, a model which combines convolutional neural networks (CNNs) with differentiable ODE solvers to model dynamical systems. We show that the Method of Lines used in standard PDE solvers can be represented using convolutions which makes CNNs the natural choice to parametrize arbitrary PDE dynamics. Our model can be applied to any data without requiring any prior knowledge about the governing PDE.
  arXiv  Detail & Related papers  (2021-11-15T10:59:52Z)
• GrADE: A graph based data-driven solver for time-dependent nonlinear partial differential equations [0.0]
  We propose a novel framework referred to as the Graph Attention Differential Equation (GrADE) for solving time dependent nonlinear PDEs. The proposed approach couples FNN, graph neural network, and recently developed Neural ODE framework. Results obtained illustrate the capability of the proposed framework in modeling PDE and its scalability to larger domains without the need for retraining.
  arXiv  Detail & Related papers  (2021-08-24T10:49:03Z)
• dNNsolve: an efficient NN-based PDE solver [62.997667081978825]
  We introduce dNNsolve, that makes use of dual Neural Networks to solve ODEs/PDEs. We show that dNNsolve is capable of solving a broad range of ODEs/PDEs in 1, 2 and 3 spacetime dimensions.
  arXiv  Detail & Related papers  (2021-03-15T19:14:41Z)

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.