$PINN - a Domain Decomposition Method for Bayesian Physics-Informed Neural Networks

• URL: http://arxiv.org/abs/2504.19013v3
• Date: Thu, 01 May 2025 09:26:03 GMT
• Title: $PINN - a Domain Decomposition Method for Bayesian Physics-Informed Neural Networks
• Authors: Júlia Vicens Figueres, Juliette Vanderhaeghen, Federica Bragone, Kateryna Morozovska, Khemraj Shukla,
• Abstract summary: $PINN is a novel method of computing global uncertainty in PDEs using a Bayesian framework.<n>$PINN is verified by adding uncorrelated random noise to the training data up to 15% and testing for different domain sizes.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Physics-Informed Neural Networks (PINNs) are a novel computational approach for solving partial differential equations (PDEs) with noisy and sparse initial and boundary data. Although, efficient quantification of epistemic and aleatoric uncertainties in big multi-scale problems remains challenging. We propose \$PINN a novel method of computing global uncertainty in PDEs using a Bayesian framework, by combining local Bayesian Physics-Informed Neural Networks (BPINN) with domain decomposition. The solution continuity across subdomains is obtained by imposing the flux continuity across the interface of neighboring subdomains. To demonstrate the effectiveness of \$PINN, we conduct a series of computational experiments on PDEs in 1D and 2D spatial domains. Although we have adopted conservative PINNs (cPINNs), the method can be seamlessly extended to other domain decomposition techniques. The results infer that the proposed method recovers the global uncertainty by computing the local uncertainty exactly more efficiently as the uncertainty in each subdomain can be computed concurrently. The robustness of \$PINN is verified by adding uncorrelated random noise to the training data up to 15% and testing for different domain sizes.

Related papers

• PACMANN: Point Adaptive Collocation Method for Artificial Neural Networks [44.99833362998488]
  PINNs minimize a loss function which includes the PDE residual determined for a set of collocation points. Previous work has shown that the number and distribution of these collocation points have a significant influence on the accuracy of the PINN solution. We present the Point Adaptive Collocation Method for Artificial Neural Networks (PACMANN)
  arXiv  Detail & Related papers  (2024-11-29T11:31:11Z)
• SetPINNs: Set-based Physics-informed Neural Networks [31.193471532024407]
  We introduce SetPINNs, a framework that effectively captures local dependencies.<n>We partition a domain into sets to model local dependencies while simultaneously enforcing physical laws.
  arXiv  Detail & Related papers  (2024-09-30T11:41:58Z)
• RoPINN: Region Optimized Physics-Informed Neural Networks [66.38369833561039]
  Physics-informed neural networks (PINNs) have been widely applied to solve partial differential equations (PDEs) This paper proposes and theoretically studies a new training paradigm as region optimization. A practical training algorithm, Region Optimized PINN (RoPINN), is seamlessly derived from this new paradigm.
  arXiv  Detail & Related papers  (2024-05-23T09:45:57Z)
• Learning Only On Boundaries: a Physics-Informed Neural operator for Solving Parametric Partial Differential Equations in Complex Geometries [10.250994619846416]
  We present a novel physics-informed neural operator method to solve parametrized boundary value problems without labeled data. Our numerical experiments show the effectiveness of parametrized complex geometries and unbounded problems.
  arXiv  Detail & Related papers  (2023-08-24T17:29:57Z)
• Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks [51.92362217307946]
  Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems. PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
  arXiv  Detail & Related papers  (2023-03-03T08:17:47Z)
• Deep NURBS -- Admissible Physics-informed Neural Networks [0.0]
  We propose a new numerical scheme for physics-informed neural networks (PINNs) that enables precise and inexpensive solution for partial differential equations (PDEs) The proposed approach combines admissible NURBS parametrizations required to define the physical domain and the Dirichlet boundary conditions with a PINN solver.
  arXiv  Detail & Related papers  (2022-10-25T10:35:45Z)
• 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)
• Physics-Informed Neural Operator for Learning Partial Differential Equations [55.406540167010014]
  PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
  arXiv  Detail & Related papers  (2021-11-06T03:41:34Z)
• A Bit More Bayesian: Domain-Invariant Learning with Uncertainty [111.22588110362705]
  Domain generalization is challenging due to the domain shift and the uncertainty caused by the inaccessibility of target domain data. In this paper, we address both challenges with a probabilistic framework based on variational Bayesian inference. We derive domain-invariant representations and classifiers, which are jointly established in a two-layer Bayesian neural network.
  arXiv  Detail & Related papers  (2021-05-09T21:33:27Z)
• Train Once and Use Forever: Solving Boundary Value Problems in Unseen Domains with Pre-trained Deep Learning Models [0.20999222360659606]
  This paper introduces a transferable framework for solving boundary value problems (BVPs) via deep neural networks. First, we introduce emphgenomic flow network (GFNet), a neural network that can infer the solution of a BVP across arbitrary boundary conditions. Then, we propose emphmosaic flow (MF) predictor, a novel iterative algorithm that assembles or stitches the GFNet's inferences.
  arXiv  Detail & Related papers  (2021-04-22T05:20:27Z)
• Bayesian neural networks for weak solution of PDEs with uncertainty quantification [3.4773470589069473]
  A new physics-constrained neural network (NN) approach is proposed to solve PDEs without labels. We write the loss function of NNs based on the discretized residual of PDEs through an efficient, convolutional operator-based, and vectorized implementation. We demonstrate the capability and performance of the proposed framework by applying it to steady-state diffusion, linear elasticity, and nonlinear elasticity.
  arXiv  Detail & Related papers  (2021-01-13T04:57:51Z)
• A nonlocal physics-informed deep learning framework using the peridynamic differential operator [0.0]
  We develop a nonlocal PINN approach using the Peridynamic Differential Operator (PDDO)---a numerical method which incorporates long-range interactions and removes spatial derivatives in the governing equations. Because the PDDO functions can be readily incorporated in the neural network architecture, the nonlocality does not degrade the performance of modern deep-learning algorithms. We document the superior behavior of nonlocal PINN with respect to local PINN in both solution accuracy and parameter inference.
  arXiv  Detail & Related papers  (2020-05-31T06:26:21Z)

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.