LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Data

• URL: http://arxiv.org/abs/2206.09418v3
• Date: Tue, 7 May 2024 08:54:59 GMT
• Title: LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Data
• Authors: Xinquan Huang, Wenlei Shi, Xiaotian Gao, Xinran Wei, Jia Zhang, Jiang Bian, Mao Yang, Tie-Yan Liu,
• Abstract summary: We propose LordNet, a tunable and efficient neural network for modeling entanglements. The experiments on solving Poisson's equation and (2D and 3D) Navier-Stokes equation demonstrate that the long-range entanglements can be well modeled by the LordNet.
• Score: 47.49194807524502
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Neural operators, as a powerful approximation to the non-linear operators between infinite-dimensional function spaces, have proved to be promising in accelerating the solution of partial differential equations (PDE). However, it requires a large amount of simulated data, which can be costly to collect. This can be avoided by learning physics from the physics-constrained loss, which we refer to it as mean squared residual (MSR) loss constructed by the discretized PDE. We investigate the physical information in the MSR loss, which we called long-range entanglements, and identify the challenge that the neural network requires the capacity to model the long-range entanglements in the spatial domain of the PDE, whose patterns vary in different PDEs. To tackle the challenge, we propose LordNet, a tunable and efficient neural network for modeling various entanglements. Inspired by the traditional solvers, LordNet models the long-range entanglements with a series of matrix multiplications, which can be seen as the low-rank approximation to the general fully-connected layers and extracts the dominant pattern with reduced computational cost. The experiments on solving Poisson's equation and (2D and 3D) Navier-Stokes equation demonstrate that the long-range entanglements from the MSR loss can be well modeled by the LordNet, yielding better accuracy and generalization ability than other neural networks. The results show that the Lordnet can be $40\times$ faster than traditional PDE solvers. In addition, LordNet outperforms other modern neural network architectures in accuracy and efficiency with the smallest parameter size.

Related papers

• Latent Neural PDE Solver: a reduced-order modelling framework for partial differential equations [6.173339150997772]
  We propose to learn the dynamics of the system in the latent space with much coarser discretizations. A non-linear autoencoder is first trained to project the full-order representation of the system onto the mesh-reduced space. We showcase that it has competitive accuracy and efficiency compared to the neural PDE solver that operates on full-order space.
  arXiv  Detail & Related papers  (2024-02-27T19:36:27Z)
• Grad-Shafranov equilibria via data-free physics informed neural networks [0.0]
  We show that PINNs can accurately and effectively solve the Grad-Shafranov equation with several different boundary conditions. We introduce a parameterized PINN framework, expanding the input space to include variables such as pressure, aspect ratio, elongation, and triangularity.
  arXiv  Detail & Related papers  (2023-11-22T16:08:38Z)
• Deep Learning-based surrogate models for parametrized PDEs: handling geometric variability through graph neural networks [0.0]
  This work explores the potential usage of graph neural networks (GNNs) for the simulation of time-dependent PDEs. We propose a systematic strategy to build surrogate models based on a data-driven time-stepping scheme. We show that GNNs can provide a valid alternative to traditional surrogate models in terms of computational efficiency and generalization to new scenarios.
  arXiv  Detail & Related papers  (2023-08-03T08:14:28Z)
• A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networks [52.5899851000193]
  We develop an ODE based IVP solver which prevents the network from getting ill-conditioned and runs in time linear in the number of parameters. We show that current methods based on this approach suffer from two key issues. First, following the ODE produces an uncontrolled growth in the conditioning of the problem, ultimately leading to unacceptably large numerical errors.
  arXiv  Detail & Related papers  (2023-04-28T17:28:18Z)
• NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition [67.46012350241969]
  This paper proposes a general acceleration methodology called NeuralStagger. It decomposing the original learning tasks into several coarser-resolution subtasks. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations.
  arXiv  Detail & Related papers  (2023-02-20T19:36:52Z)
• 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)
• DOSnet as a Non-Black-Box PDE Solver: When Deep Learning Meets Operator Splitting [12.655884541938656]
  We develop a learning-based PDE solver, which we name Deep Operator-Splitting Network (DOSnet) DOSnet is constructed from the physical rules and operators governing the underlying dynamics contains learnable parameters. We train and validate it on several types of operator-decomposable differential equations.
  arXiv  Detail & Related papers  (2022-12-11T18:23:56Z)
• Neural Operator with Regularity Structure for Modeling Dynamics Driven by SPDEs [70.51212431290611]
  Partial differential equations (SPDEs) are significant tools for modeling dynamics in many areas including atmospheric sciences and physics. We propose the Neural Operator with Regularity Structure (NORS) which incorporates the feature vectors for modeling dynamics driven by SPDEs. We conduct experiments on various of SPDEs including the dynamic Phi41 model and the 2d Navier-Stokes equation.
  arXiv  Detail & Related papers  (2022-04-13T08:53:41Z)
• 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)
• PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs [8.220908558735884]
  Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep learning have shown the great potential of physics-informed neural networks (NNs) to solve PDEs as a basis for data-driven inverse analysis. We propose the novel physics-informed convolutional-recurrent learning architectures (PhyCRNet and PhCRyNet-s) for solving PDEs without any labeled data.
  arXiv  Detail & Related papers  (2021-06-26T22:22:19Z)

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.