A Generalized Schwarz-type Non-overlapping Domain Decomposition Method using Physics-constrained Neural Networks

• URL: http://arxiv.org/abs/2307.12435v1
• Date: Sun, 23 Jul 2023 21:18:04 GMT
• Title: A Generalized Schwarz-type Non-overlapping Domain Decomposition Method using Physics-constrained Neural Networks
• Authors: Shamsulhaq Basir, Inanc Senocak
• Abstract summary: We present a meshless Schwarz-type non-overlapping domain decomposition based on artificial neural networks. Our method is applicable to both the Laplace's and Helmholtz equations.
• Score: 0.9137554315375919
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present a meshless Schwarz-type non-overlapping domain decomposition method based on artificial neural networks for solving forward and inverse problems involving partial differential equations (PDEs). To ensure the consistency of solutions across neighboring subdomains, we adopt a generalized Robin-type interface condition, assigning unique Robin parameters to each subdomain. These subdomain-specific Robin parameters are learned to minimize the mismatch on the Robin interface condition, facilitating efficient information exchange during training. Our method is applicable to both the Laplace's and Helmholtz equations. It represents local solutions by an independent neural network model which is trained to minimize the loss on the governing PDE while strictly enforcing boundary and interface conditions through an augmented Lagrangian formalism. A key strength of our method lies in its ability to learn a Robin parameter for each subdomain, thereby enhancing information exchange with its neighboring subdomains. We observe that the learned Robin parameters adapt to the local behavior of the solution, domain partitioning and subdomain location relative to the overall domain. Extensive experiments on forward and inverse problems, including one-way and two-way decompositions with crosspoints, demonstrate the versatility and performance of our proposed approach.

Related papers

• The role of interface boundary conditions and sampling strategies for Schwarz-based coupling of projection-based reduced order models [0.0]
  We present a framework for the coupling of subdomain-local projection-based reduced order models (PROMs) using the Schwarz alternating method. We show that it is possible to obtain a stable and accurate coupled model utilizing Dirichlet-Dirichlet (rather than Robin-Robin or alternating Dirichlet-Neumann) transmission BCs on the subdomain boundaries. Our numerical results suggest that the proposed methodology has the potential to improve PROM accuracy.
  arXiv  Detail & Related papers  (2024-10-07T00:44:22Z)
• Non-overlapping, Schwarz-type Domain Decomposition Method for Physics and Equality Constrained Artificial Neural Networks [0.24578723416255746]
  We present a non-overlapping, Schwarz-type domain decomposition method with a generalized interface condition. Our approach employs physics and equality-constrained artificial neural networks (PECANN) within each subdomain. A distinct advantage our domain decomposition method is its ability to learn solutions to both Poisson's and Helmholtz equations.
  arXiv  Detail & Related papers  (2024-09-20T16:48:55Z)
• Symmetry group based domain decomposition to enhance physics-informed neural networks for solving partial differential equations [3.3360424430642848]
  We propose a symmetry group based domain decomposition strategy to enhance the PINN for solving the forward and inverse problems of the PDEs possessing a Lie symmetry group. For the forward problem, we first deploy the symmetry group to generate the dividing-lines having known solution information which can be adjusted flexibly. We then utilize the PINN and the symmetry-enhanced PINN methods to learn the solutions in each sub-domain and finally stitch them to the overall solution of PDEs.
  arXiv  Detail & Related papers  (2024-04-29T09:27:17Z)
• Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEs [93.82811501035569]
  We introduce a new data efficient and highly parallelizable operator learning approach with reduced memory requirement and better generalization. MG-TFNO scales to large resolutions by leveraging local and global structures of full-scale, real-world phenomena. We demonstrate superior performance on the turbulent Navier-Stokes equations where we achieve less than half the error with over 150x compression.
  arXiv  Detail & Related papers  (2023-09-29T20:18:52Z)
• 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)
• A domain-decomposed VAE method for Bayesian inverse problems [0.0]
  This paper proposes a domain-decomposed variational auto-encoder Markov chain Monte Carlo (DD-VAE-MCMC) method to tackle these challenges simultaneously. The proposed method first constructs local deterministic generative models based on local historical data, which provide efficient local prior representations.
  arXiv  Detail & Related papers  (2023-01-09T07:35:43Z)
• Neural PDE Solvers for Irregular Domains [25.673617202478606]
  We present a framework to neurally solve partial differential equations over domains with irregularly shaped geometric boundaries. Our network takes in the shape of the domain as an input and is able to generalize to novel (unseen) irregular domains.
  arXiv  Detail & Related papers  (2022-11-07T00:00:30Z)
• Adversarial Bi-Regressor Network for Domain Adaptive Regression [52.5168835502987]
  It is essential to learn a cross-domain regressor to mitigate the domain shift. This paper proposes a novel method Adversarial Bi-Regressor Network (ABRNet) to seek more effective cross-domain regression model.
  arXiv  Detail & Related papers  (2022-09-20T18:38:28Z)
• Neural Basis Functions for Accelerating Solutions to High Mach Euler Equations [63.8376359764052]
  We propose an approach to solving partial differential equations (PDEs) using a set of neural networks. We regress a set of neural networks onto a reduced order Proper Orthogonal Decomposition (POD) basis. These networks are then used in combination with a branch network that ingests the parameters of the prescribed PDE to compute a reduced order approximation to the PDE.
  arXiv  Detail & Related papers  (2022-08-02T18:27:13Z)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Mean-field Analysis of Piecewise Linear Solutions for Wide ReLU Networks [83.58049517083138]
  We consider a two-layer ReLU network trained via gradient descent. We show that SGD is biased towards a simple solution. We also provide empirical evidence that knots at locations distinct from the data points might occur.
  arXiv  Detail & Related papers  (2021-11-03T15:14:20Z)

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.