A Kernel Approach for PDE Discovery and Operator Learning

• URL: http://arxiv.org/abs/2210.08140v2
• Date: Thu, 30 Mar 2023 18:04:22 GMT
• Title: A Kernel Approach for PDE Discovery and Operator Learning
• Authors: Da Long, Nicole Mrvaljevic, Shandian Zhe, and Bamdad Hosseini
• Abstract summary: kernel smoothing is utilized to denoise the data and approximate derivatives of the solution. The learned PDE is then used within a kernel based solver to approximate the solution of the PDE with a new source/boundary term.
• Score: 9.463496582811633
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This article presents a three-step framework for learning and solving partial differential equations (PDEs) using kernel methods. Given a training set consisting of pairs of noisy PDE solutions and source/boundary terms on a mesh, kernel smoothing is utilized to denoise the data and approximate derivatives of the solution. This information is then used in a kernel regression model to learn the algebraic form of the PDE. The learned PDE is then used within a kernel based solver to approximate the solution of the PDE with a new source/boundary term, thereby constituting an operator learning framework. Numerical experiments compare the method to state-of-the-art algorithms and demonstrate its competitive performance.

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)
• End-to-End Mesh Optimization of a Hybrid Deep Learning Black-Box PDE Solver [24.437884270729903]
  Recent research proposed a PDE correction framework that leverages deep learning to correct the solution obtained by a PDE solver on a coarse mesh. End-to-end training of such a PDE correction model requires the PDE solver to support automatic differentiation through the iterative numerical process. In this study, we explore the feasibility of end-to-end training of a hybrid model with a black-box PDE solver and a deep learning model for fluid flow prediction.
  arXiv  Detail & Related papers  (2024-04-17T21:49:45Z)
• 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)
• Gaussian Mixture Solvers for Diffusion Models [84.83349474361204]
  We introduce a novel class of SDE-based solvers called GMS for diffusion models. Our solver outperforms numerous SDE-based solvers in terms of sample quality in image generation and stroke-based synthesis.
  arXiv  Detail & Related papers  (2023-11-02T02:05:38Z)
• Elucidating the solution space of extended reverse-time SDE for diffusion models [54.23536653351234]
  Diffusion models (DMs) demonstrate potent image generation capabilities in various generative modeling tasks. Their primary limitation lies in slow sampling speed, requiring hundreds or thousands of sequential function evaluations to generate high-quality images. We formulate the sampling process as an extended reverse-time SDE, unifying prior explorations into ODEs and SDEs. We devise fast and training-free samplers, ER-SDE-rs, achieving state-of-the-art performance across all samplers.
  arXiv  Detail & Related papers  (2023-09-12T12:27:17Z)
• Weak-PDE-LEARN: A Weak Form Based Approach to Discovering PDEs From Noisy, Limited Data [0.0]
  We introduce Weak-PDE-LEARN, a discovery algorithm that can identify non-linear PDEs from noisy, limited measurements of their solutions. We demonstrate the efficacy of Weak-PDE-LEARN by learning several benchmark PDEs.
  arXiv  Detail & Related papers  (2023-09-09T06:45:15Z)
• Neural Partial Differential Equations with Functional Convolution [30.35306295442881]
  We present a lightweighted neural PDE representation to discover the hidden structure and predict the solution of different nonlinear PDEs. We leverage the prior of translational similarity'' of numerical PDE differential operators to drastically reduce the scale of learning model and training data.
  arXiv  Detail & Related papers  (2023-03-10T04:25:38Z)
• 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)
• Noise-aware Physics-informed Machine Learning for Robust PDE Discovery [5.746505534720594]
  This work is concerned with discovering the governing partial differential equation (PDE) of a physical system. Existing methods have demonstrated the PDE identification from finite observations but failed to maintain satisfying performance against noisy data. We introduce a noise-aware physics-informed machine learning framework to discover the governing PDE from data following arbitrary distributions.
  arXiv  Detail & Related papers  (2022-06-26T15:29:07Z)
• 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)
• Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers [26.444103444634994]
  We show that machine learning can improve the solution accuracy by correcting for effects not captured by the discretized PDE. We find that previously used learning approaches are significantly outperformed by methods that integrate the solver into the training loop. This provides the model with realistic input distributions that take previous corrections into account.
  arXiv  Detail & Related papers  (2020-06-30T18:00:03Z)

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.