Coupled Multiwavelet Neural Operator Learning for Coupled Partial Differential Equations

• URL: http://arxiv.org/abs/2303.02304v4
• Date: Fri, 8 Dec 2023 06:06:57 GMT
• Title: Coupled Multiwavelet Neural Operator Learning for Coupled Partial Differential Equations
• Authors: Xiongye Xiao, Defu Cao, Ruochen Yang, Gaurav Gupta, Gengshuo Liu, Chenzhong Yin, Radu Balan, Paul Bogdan
• Abstract summary: We propose a textitcoupled multiwavelets neural operator (CMWNO) learning scheme by decoupling the coupled integral kernels. The proposed model achieves significantly higher accuracy compared to previous learning-based solvers.
• Score: 13.337268390844745
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Coupled partial differential equations (PDEs) are key tasks in modeling the complex dynamics of many physical processes. Recently, neural operators have shown the ability to solve PDEs by learning the integral kernel directly in Fourier/Wavelet space, so the difficulty for solving the coupled PDEs depends on dealing with the coupled mappings between the functions. Towards this end, we propose a \textit{coupled multiwavelets neural operator} (CMWNO) learning scheme by decoupling the coupled integral kernels during the multiwavelet decomposition and reconstruction procedures in the Wavelet space. The proposed model achieves significantly higher accuracy compared to previous learning-based solvers in solving the coupled PDEs including Gray-Scott (GS) equations and the non-local mean field game (MFG) problem. According to our experimental results, the proposed model exhibits a $2\times \sim 4\times$ improvement relative $L$2 error compared to the best results from the state-of-the-art models.

Related papers

• DimOL: Dimensional Awareness as A New 'Dimension' in Operator Learning [63.5925701087252]
  We introduce DimOL (Dimension-aware Operator Learning), drawing insights from dimensional analysis. To implement DimOL, we propose the ProdLayer, which can be seamlessly integrated into FNO-based and Transformer-based PDE solvers. Empirically, DimOL models achieve up to 48% performance gain within the PDE datasets.
  arXiv  Detail & Related papers  (2024-10-08T10:48:50Z)
• Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs [85.40198664108624]
  We propose Codomain Attention Neural Operator (CoDA-NO) to solve multiphysics problems with PDEs. CoDA-NO tokenizes functions along the codomain or channel space, enabling self-supervised learning or pretraining of multiple PDE systems. We find CoDA-NO to outperform existing methods by over 36% on complex downstream tasks with limited data.
  arXiv  Detail & Related papers  (2024-03-19T08:56:20Z)
• Enhancing Solutions for Complex PDEs: Introducing Complementary Convolution and Equivariant Attention in Fourier Neural Operators [17.91230192726962]
  We propose a novel hierarchical Fourier neural operator along with convolution-residual layers and attention mechanisms to solve complex PDEs. We find that the proposed method achieves superior performance in these PDE benchmarks, especially for equations characterized by rapid coefficient variations.
  arXiv  Detail & Related papers  (2023-11-21T11:04:13Z)
• 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)
• Multi-scale Physical Representations for Approximating PDE Solutions with Graph Neural Operators [14.466945570499183]
  We study three multi-resolution schema with integral kernel operators approximated with emphMessage Passing Graph Neural Networks (MPGNNs) To validate our study, we make extensive MPGNNs experiments with well-chosen metrics considering steady and unsteady PDEs.
  arXiv  Detail & Related papers  (2022-06-29T14:42:03Z)
• Learning Physics-Informed Neural Networks without Stacked Back-propagation [82.26566759276105]
  We develop a novel approach that can significantly accelerate the training of Physics-Informed Neural Networks. In particular, we parameterize the PDE solution by the Gaussian smoothed model and show that, derived from Stein's Identity, the second-order derivatives can be efficiently calculated without back-propagation. Experimental results show that our proposed method can achieve competitive error compared to standard PINN training but is two orders of magnitude faster.
  arXiv  Detail & Related papers  (2022-02-18T18:07:54Z)
• Multiwavelet-based Operator Learning for Differential Equations [3.0824316066680484]
  We introduce a textitmultiwavelet-based neural operator learning scheme that compresses the associated operator's kernel. By explicitly embedding the inverse multiwavelet filters, we learn the projection of the kernel onto fixed multiwavelet bases. Compared with the existing neural operator approaches, our model shows significantly higher accuracy and state-of-the-art in a range of datasets.
  arXiv  Detail & Related papers  (2021-09-28T03:21:47Z)
• 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)
• Actor-Critic Algorithm for High-dimensional Partial Differential Equations [1.5644600570264835]
  We develop a deep learning model to solve high-dimensional nonlinear parabolic partial differential equations. The Markovian property of the BSDE is utilized in designing our neural network architecture. We demonstrate those improvements by solving a few well-known classes of PDEs.
  arXiv  Detail & Related papers  (2020-10-07T20:53:24Z)
• Large-scale Neural Solvers for Partial Differential Equations [48.7576911714538]
  Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. Recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. We examine the applicability of continuous, mesh-free neural solvers for partial differential equations, physics-informed neural networks (PINNs) We discuss the accuracy of GatedPINN with respect to analytical solutions -- as well as state-of-the-art numerical solvers, such as spectral solvers.
  arXiv  Detail & Related papers  (2020-09-08T13:26:51Z)

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.