Multifidelity domain decomposition-based physics-informed neural networks and operators for time-dependent problems

• URL: http://arxiv.org/abs/2401.07888v2
• Date: Thu, 6 Jun 2024 08:50:53 GMT
• Title: Multifidelity domain decomposition-based physics-informed neural networks and operators for time-dependent problems
• Authors: Alexander Heinlein, Amanda A. Howard, Damien Beecroft, Panos Stinis,
• Abstract summary: A combination of multifidelity stacking PINNs and domain decomposition-based finite basis PINNs is employed. It can be observed that the domain decomposition approach clearly improves the PINN and stacking PINN approaches. It is demonstrated that the FBPINN approach can be extended to multifidelity physics-informed deep operator networks.
• Score: 40.46280139210502
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Multiscale problems are challenging for neural network-based discretizations of differential equations, such as physics-informed neural networks (PINNs). This can be (partly) attributed to the so-called spectral bias of neural networks. To improve the performance of PINNs for time-dependent problems, a combination of multifidelity stacking PINNs and domain decomposition-based finite basis PINNs is employed. In particular, to learn the high-fidelity part of the multifidelity model, a domain decomposition in time is employed. The performance is investigated for a pendulum and a two-frequency problem as well as the Allen-Cahn equation. It can be observed that the domain decomposition approach clearly improves the PINN and stacking PINN approaches. Finally, it is demonstrated that the FBPINN approach can be extended to multifidelity physics-informed deep operator networks.

Related papers

• Spectral Informed Neural Network: An Efficient and Low-Memory PINN [3.8534287291074354]
  We propose a spectral-based neural network that substitutes the differential operator with a multiplication. Compared to the PINNs, our approach requires lower memory and shorter training time. We provide two strategies to train networks by their spectral information.
  arXiv  Detail & Related papers  (2024-08-29T10:21:00Z)
• Deeper or Wider: A Perspective from Optimal Generalization Error with Sobolev Loss [2.07180164747172]
  We compare deeper neural networks (DeNNs) with a flexible number of layers and wider neural networks (WeNNs) with limited hidden layers. We find that a higher number of parameters tends to favor WeNNs, while an increased number of sample points and greater regularity in the loss function lean towards the adoption of DeNNs.
  arXiv  Detail & Related papers  (2024-01-31T20:10:10Z)
• PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks [22.39904196850583]
  Physics-Informed Neural Networks (PINNs) have emerged as a promising deep learning framework for approximating numerical solutions to partial differential equations (PDEs) We introduce a novel Transformer-based framework, termed PINNsFormer, designed to address this limitation. PINNsFormer achieves superior generalization ability and accuracy across various scenarios, including PINNs failure modes and high-dimensional PDEs.
  arXiv  Detail & Related papers  (2023-07-21T18:06:27Z)
• Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations [64.78260098263489]
  In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs) We show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity increases. We identify several reasons why this may be the case, including insufficient network capacity, poor conditioning of the ODEs, and high local curvature, as measured by the Laplacian of the PINN loss.
  arXiv  Detail & Related papers  (2022-10-14T15:01:32Z)
• Auto-PINN: Understanding and Optimizing Physics-Informed Neural Architecture [77.59766598165551]
  Physics-informed neural networks (PINNs) are revolutionizing science and engineering practice by bringing together the power of deep learning to bear on scientific computation. Here, we propose Auto-PINN, which employs Neural Architecture Search (NAS) techniques to PINN design. A comprehensive set of pre-experiments using standard PDE benchmarks allows us to probe the structure-performance relationship in PINNs.
  arXiv  Detail & Related papers  (2022-05-27T03:24:31Z)
• Revisiting PINNs: Generative Adversarial Physics-informed Neural Networks and Point-weighting Method [70.19159220248805]
  Physics-informed neural networks (PINNs) provide a deep learning framework for numerically solving partial differential equations (PDEs) We propose the generative adversarial neural network (GA-PINN), which integrates the generative adversarial (GA) mechanism with the structure of PINNs. Inspired from the weighting strategy of the Adaboost method, we then introduce a point-weighting (PW) method to improve the training efficiency of PINNs.
  arXiv  Detail & Related papers  (2022-05-18T06:50:44Z)
• Characterizing possible failure modes in physics-informed neural networks [55.83255669840384]
  Recent work in scientific machine learning has developed so-called physics-informed neural network (PINN) models. We demonstrate that, while existing PINN methodologies can learn good models for relatively trivial problems, they can easily fail to learn relevant physical phenomena even for simple PDEs. We show that these possible failure modes are not due to the lack of expressivity in the NN architecture, but that the PINN's setup makes the loss landscape very hard to optimize.
  arXiv  Detail & Related papers  (2021-09-02T16:06:45Z)
• Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations [20.277873724720987]
  We propose a new, scalable approach for solving large problems relating to differential equations called Finite Basis PINNs (FBPINNs) FBPINNs are inspired by classical finite element methods, where the solution of the differential equation is expressed as the sum of a finite set of basis functions with compact support. In FBPINNs neural networks are used to learn these basis functions, which are defined over small, overlapping subdomain problems.
  arXiv  Detail & Related papers  (2021-07-16T13:03:47Z)
• On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks [0.0]
  We show that neural networks (PINNs) struggle in cases where the target functions to be approximated exhibit high-frequency or multi-scale features. We construct novel architectures that employ multi-scale random observational features and justify how such coordinate embedding layers can lead to robust and accurate PINN models.
  arXiv  Detail & Related papers  (2020-12-18T04:19:30Z)
• 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.