Long-time Integration of Nonlinear Wave Equations with Neural Operators

• URL: http://arxiv.org/abs/2410.15617v3
• Date: Mon, 17 Feb 2025 03:08:36 GMT
• Title: Long-time Integration of Nonlinear Wave Equations with Neural Operators
• Authors: Guanhang Lei, Zhen Lei, Lei Shi,
• Abstract summary: We focus on solving the long-time integration of nonlinear wave equations via neural operators. We utilize some intrinsic features of these nonlinear wave equations, such as conservation laws and well-posedness, to improve the algorithm design and reduce accumulated error. Our numerical experiments examine these improvements in the Korteweg-de Vries (KdV) equation, the sine-Gordon equation, and the Klein-Gordon wave equation on the irregular domain.
• Score: 13.357441268268758
• License:
• Abstract: Neural operators have shown promise in solving many types of Partial Differential Equations (PDEs). They are significantly faster compared to traditional numerical solvers once they have been trained with a certain amount of observed data. However, their numerical performance in solving time-dependent PDEs, particularly in long-time prediction of dynamic systems, still needs improvement. In this paper, we focus on solving the long-time integration of nonlinear wave equations via neural operators by replacing the initial condition with the prediction in a recurrent manner. Given limited observed temporal trajectory data, we utilize some intrinsic features of these nonlinear wave equations, such as conservation laws and well-posedness, to improve the algorithm design and reduce accumulated error. Our numerical experiments examine these improvements in the Korteweg-de Vries (KdV) equation, the sine-Gordon equation, and the Klein-Gordon wave equation on the irregular domain.

Related papers

• MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation [48.41289705783405]
  We propose a PDE-embedded network with multiscale time stepping (MultiPDENet) In particular, we design a convolutional filter based on the structure of finite difference with a small number of parameters to optimize. A Physics Block with a 4th-order Runge-Kutta integrator at the fine time scale is established that embeds the structure of PDEs to guide the prediction.
  arXiv  Detail & Related papers  (2025-01-27T12:15:51Z)
• An efficient wavelet-based physics-informed neural networks for singularly perturbed problems [0.0]
  Physics-informed neural networks (PINNs) are a class of deep learning models that utilize physics in the form of differential equations to address complex problems. We present a wavelet-based PINNs model to tackle solutions of differential equations with rapid oscillations, steep gradients, or singular behavior. The proposed model significantly improves with traditional PINNs, recently developed wavelet-based PINNs, and other state-of-the-art methods.
  arXiv  Detail & Related papers  (2024-09-18T10:01:37Z)
• Spectral-Refiner: Fine-Tuning of Accurate Spatiotemporal Neural Operator for Turbulent Flows [6.961408873053586]
  We propose a new Stemporal Neural Operator (SFNO) that learns maps between Bochner spaces, and a new learning framework to address these issues. This new paradigm leverages wisdom from traditional numerical PDE theory and techniques to refine the pipeline of commonly adopted end-to-end neural operator training and evaluations. Numerical experiments on commonly used benchmarks for the 2D NSE demonstrate significant improvements in both computational efficiency and accuracy, compared to end-to-end evaluation and traditional numerical PDE solvers.
  arXiv  Detail & Related papers  (2024-05-27T14:33:06Z)
• Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methods [0.0]
  We present a data-driven modeling framework called neural dynamical operator that is continuous in both space and time. A key feature of the neural dynamical operator is the resolution-invariance with respect to both spatial and temporal discretizations. We show that the proposed model can better predict long-term statistics via the hybrid optimization scheme.
  arXiv  Detail & Related papers  (2023-11-20T14:31:18Z)
• 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)
• Semi-supervised Learning of Partial Differential Operators and Dynamical Flows [68.77595310155365]
  We present a novel method that combines a hyper-network solver with a Fourier Neural Operator architecture. We test our method on various time evolution PDEs, including nonlinear fluid flows in one, two, and three spatial dimensions. The results show that the new method improves the learning accuracy at the time point of supervision point, and is able to interpolate and the solutions to any intermediate time.
  arXiv  Detail & Related papers  (2022-07-28T19:59:14Z)
• Physics Informed RNN-DCT Networks for Time-Dependent Partial Differential Equations [62.81701992551728]
  We present a physics-informed framework for solving time-dependent partial differential equations. Our model utilizes discrete cosine transforms to encode spatial and recurrent neural networks. We show experimental results on the Taylor-Green vortex solution to the Navier-Stokes equations.
  arXiv  Detail & Related papers  (2022-02-24T20:46:52Z)
• Neural ODE Processes [64.10282200111983]
  We introduce Neural ODE Processes (NDPs), a new class of processes determined by a distribution over Neural ODEs. We show that our model can successfully capture the dynamics of low-dimensional systems from just a few data-points.
  arXiv  Detail & Related papers  (2021-03-23T09:32:06Z)
• Liquid Time-constant Networks [117.57116214802504]
  We introduce a new class of time-continuous recurrent neural network models. Instead of declaring a learning system's dynamics by implicit nonlinearities, we construct networks of linear first-order dynamical systems. These neural networks exhibit stable and bounded behavior, yield superior expressivity within the family of neural ordinary differential equations.
  arXiv  Detail & Related papers  (2020-06-08T09:53:35Z)

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.