Physics-guided Data Augmentation for Learning the Solution Operator of Linear Differential Equations

• URL: http://arxiv.org/abs/2212.04100v1
• Date: Thu, 8 Dec 2022 06:29:15 GMT
• Title: Physics-guided Data Augmentation for Learning the Solution Operator of Linear Differential Equations
• Authors: Ye Li, Yiwen Pang, and Bin Shan
• Abstract summary: We propose a physics-guided data augmentation (PGDA) method to improve the accuracy and generalization of neural operator models. We demonstrate the advantage of PGDA on a variety of linear differential equations, showing that PGDA can improve the sample complexity and is robust to distributional shift.
• Score: 2.1850269949775663
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Neural networks, especially the recent proposed neural operator models, are increasingly being used to find the solution operator of differential equations. Compared to traditional numerical solvers, they are much faster and more efficient in practical applications. However, one critical issue is that training neural operator models require large amount of ground truth data, which usually comes from the slow numerical solvers. In this paper, we propose a physics-guided data augmentation (PGDA) method to improve the accuracy and generalization of neural operator models. Training data is augmented naturally through the physical properties of differential equations such as linearity and translation. We demonstrate the advantage of PGDA on a variety of linear differential equations, showing that PGDA can improve the sample complexity and is robust to distributional shift.

Related papers

• DeltaPhi: Learning Physical Trajectory Residual for PDE Solving [54.13671100638092]
  We propose and formulate the Physical Trajectory Residual Learning (DeltaPhi) We learn the surrogate model for the residual operator mapping based on existing neural operator networks. We conclude that, compared to direct learning, physical residual learning is preferred for PDE solving.
  arXiv  Detail & Related papers  (2024-06-14T07:45:07Z)
• Transformers as Neural Operators for Solutions of Differential Equations with Finite Regularity [1.6874375111244329]
  We first establish the theoretical groundwork that transformers possess the universal approximation property as operator learning models. In particular, we consider three examples: the Izhikevich neuron model, the tempered fractional-order Leaky Integrate-and-Fire (LIFLIF) model, and the one-dimensional equation Euler problem.
  arXiv  Detail & Related papers  (2024-05-29T15:10:24Z)
• PICL: Physics Informed Contrastive Learning for Partial Differential Equations [7.136205674624813]
  We develop a novel contrastive pretraining framework that improves neural operator generalization across multiple governing equations simultaneously. A combination of physics-informed system evolution and latent-space model output are anchored to input data and used in our distance function. We find that physics-informed contrastive pretraining improves accuracy for the Fourier Neural Operator in fixed-future and autoregressive rollout tasks for the 1D and 2D Heat, Burgers', and linear advection equations.
  arXiv  Detail & Related papers  (2024-01-29T17:32:22Z)
• Generalization Error Guaranteed Auto-Encoder-Based Nonlinear Model Reduction for Operator Learning [12.124206935054389]
  In this paper, we utilize low-dimensional nonlinear structures in model reduction by investigating Auto-Encoder-based Neural Network (AENet) Our numerical experiments validate the ability of AENet to accurately learn the solution operator of nonlinear partial differential equations. Our theoretical framework shows that the sample complexity of training AENet is intricately tied to the intrinsic dimension of the modeled process.
  arXiv  Detail & Related papers  (2024-01-19T05:01:43Z)
• Training Deep Surrogate Models with Large Scale Online Learning [48.7576911714538]
  Deep learning algorithms have emerged as a viable alternative for obtaining fast solutions for PDEs. Models are usually trained on synthetic data generated by solvers, stored on disk and read back for training. It proposes an open source online training framework for deep surrogate models.
  arXiv  Detail & Related papers  (2023-06-28T12:02:27Z)
• Variational operator learning: A unified paradigm marrying training neural operators and solving partial differential equations [9.148052787201797]
  We propose a novel paradigm that provides a unified framework of training neural operators and solving PDEs with the variational form. With a label-free training set and a 5-label-only shift set, VOL learns solution operators with its test errors decreasing in a power law with respect to the amount of unlabeled data.
  arXiv  Detail & Related papers  (2023-04-09T13:20:19Z)
• Incorporating NODE with Pre-trained Neural Differential Operator for Learning Dynamics [73.77459272878025]
  We propose to enhance the supervised signal in learning dynamics by pre-training a neural differential operator (NDO) NDO is pre-trained on a class of symbolic functions, and it learns the mapping between the trajectory samples of these functions to their derivatives. We provide theoretical guarantee on that the output of NDO can well approximate the ground truth derivatives by proper tuning the complexity of the library.
  arXiv  Detail & Related papers  (2021-06-08T08:04:47Z)
• 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)
• 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)
• 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.