Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformations
- URL: http://arxiv.org/abs/2405.17509v1
- Date: Mon, 27 May 2024 06:50:17 GMT
- Title: Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformations
- Authors: Ze Cheng, Zhongkai Hao, Xiaoqiang Wang, Jianing Huang, Youjia Wu, Xudan Liu, Yiru Zhao, Songming Liu, Hang Su,
- Abstract summary: For partial differential equations on domains of arbitrary shapes, existing works of neural operators attempt to learn a mapping from geometries to solutions.
We propose reference neural operators (RNO) to learn the smooth dependence of solutions on geometric deformations.
RNO outperforms baseline models in accuracy by a large lead and achieves up to 80% error reduction.
- Score: 13.208548352092455
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: For partial differential equations on domains of arbitrary shapes, existing works of neural operators attempt to learn a mapping from geometries to solutions. It often requires a large dataset of geometry-solution pairs in order to obtain a sufficiently accurate neural operator. However, for many industrial applications, e.g., engineering design optimization, it can be prohibitive to satisfy the requirement since even a single simulation may take hours or days of computation. To address this issue, we propose reference neural operators (RNO), a novel way of implementing neural operators, i.e., to learn the smooth dependence of solutions on geometric deformations. Specifically, given a reference solution, RNO can predict solutions corresponding to arbitrary deformations of the referred geometry. This approach turns out to be much more data efficient. Through extensive experiments, we show that RNO can learn the dependence across various types and different numbers of geometry objects with relatively small datasets. RNO outperforms baseline models in accuracy by a large lead and achieves up to 80% error reduction.
Related papers
- Computation-Aware Gaussian Processes: Model Selection And Linear-Time Inference [55.150117654242706]
We show that model selection for computation-aware GPs trained on 1.8 million data points can be done within a few hours on a single GPU.
As a result of this work, Gaussian processes can be trained on large-scale datasets without significantly compromising their ability to quantify uncertainty.
arXiv Detail & Related papers (2024-11-01T21:11:48Z) - Differentiable Visual Computing for Inverse Problems and Machine
Learning [27.45555082573493]
Visual computing methods are used to analyze geometry, physically simulate solids, fluids, and other media, and render the world via optical techniques.
Deep learning (DL) allows for the construction of general algorithmic models, side stepping the need for a purely first principles-based approach to problem solving.
DL is powered by highly parameterized neural network architectures -- universal function approximators -- and gradient-based search algorithms.
arXiv Detail & Related papers (2023-11-21T23:02:58Z) - Geometry-Informed Neural Operator for Large-Scale 3D PDEs [76.06115572844882]
We propose the geometry-informed neural operator (GINO) to learn the solution operator of large-scale partial differential equations.
We successfully trained GINO to predict the pressure on car surfaces using only five hundred data points.
arXiv Detail & Related papers (2023-09-01T16:59:21Z) - Learning Only On Boundaries: a Physics-Informed Neural operator for
Solving Parametric Partial Differential Equations in Complex Geometries [10.250994619846416]
We present a novel physics-informed neural operator method to solve parametrized boundary value problems without labeled data.
Our numerical experiments show the effectiveness of parametrized complex geometries and unbounded problems.
arXiv Detail & Related papers (2023-08-24T17:29:57Z) - Sparse Deep Neural Network for Nonlinear Partial Differential Equations [3.0069322256338906]
This paper is devoted to a numerical study of adaptive approximation of solutions of nonlinear partial differential equations.
We develop deep neural networks (DNNs) with a sparse regularization with multiple parameters to represent functions having certain singularities.
Numerical examples confirm that solutions generated by the proposed SDNN are sparse and accurate.
arXiv Detail & Related papers (2022-07-27T03:12:16Z) - Learning the Solution Operator of Boundary Value Problems using Graph
Neural Networks [0.0]
We design a general solution operator for two different time-independent PDEs using graph neural networks (GNNs) and spectral graph convolutions.
We train the networks on simulated data from a finite elements solver on a variety of shapes and inhomogeneities.
We find that training on a diverse dataset with lots of variation in the finite element meshes is a key ingredient for achieving good generalization results.
arXiv Detail & Related papers (2022-06-28T15:39:06Z) - Improved Training of Physics-Informed Neural Networks with Model
Ensembles [81.38804205212425]
We propose to expand the solution interval gradually to make the PINN converge to the correct solution.
All ensemble members converge to the same solution in the vicinity of observed data.
We show experimentally that the proposed method can improve the accuracy of the found solution.
arXiv Detail & Related papers (2022-04-11T14:05:34Z) - Message Passing Neural PDE Solvers [60.77761603258397]
We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators.
We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes.
We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
arXiv Detail & Related papers (2022-02-07T17:47:46Z) - 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.