Physics-Informed Neural Operator for Learning Partial Differential Equations

• URL: http://arxiv.org/abs/2111.03794v4
• Date: Sat, 29 Jul 2023 07:58:37 GMT
• Title: Physics-Informed Neural Operator for Learning Partial Differential Equations
• Authors: Zongyi Li, Hongkai Zheng, Nikola Kovachki, David Jin, Haoxuan Chen, Burigede Liu, Kamyar Azizzadenesheli, Anima Anandkumar
• Abstract summary: PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
• Score: 55.406540167010014
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we propose physics-informed neural operators (PINO) that combine training data and physics constraints to learn the solution operator of a given family of parametric Partial Differential Equations (PDE). PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. Specifically, in PINO, we combine coarse-resolution training data with PDE constraints imposed at a higher resolution. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families and shows no degradation in accuracy even under zero-shot super-resolution, i.e., being able to predict beyond the resolution of training data. PINO uses the Fourier neural operator (FNO) framework that is guaranteed to be a universal approximator for any continuous operator and discretization-convergent in the limit of mesh refinement. By adding PDE constraints to FNO at a higher resolution, we obtain a high-fidelity reconstruction of the ground-truth operator. Moreover, PINO succeeds in settings where no training data is available and only PDE constraints are imposed, while previous approaches, such as the Physics-Informed Neural Network (PINN), fail due to optimization challenges, e.g., in multi-scale dynamic systems such as Kolmogorov flows.

Related papers

• Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations [3.524869467682149]
  Solving systems of partial differential equations (PDEs) is a fundamental task in computational science. Recent advancements have introduced neural operators and physics-informed neural networks (PINNs) to tackle PDEs. We propose a novel framework for constructing GP priors that satisfy both general systems of linear PDEs with constant coefficients and linear boundary conditions.
  arXiv  Detail & Related papers  (2024-11-25T18:48:15Z)
• 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)
• RoPINN: Region Optimized Physics-Informed Neural Networks [66.38369833561039]
  Physics-informed neural networks (PINNs) have been widely applied to solve partial differential equations (PDEs) This paper proposes and theoretically studies a new training paradigm as region optimization. A practical training algorithm, Region Optimized PINN (RoPINN), is seamlessly derived from this new paradigm.
  arXiv  Detail & Related papers  (2024-05-23T09:45:57Z)
• 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)
• 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)
• Generic bounds on the approximation error for physics-informed (and) operator learning [7.6146285961466]
  We propose a framework for deriving rigorous bounds on the approximation error for physics-informed neural networks (PINNs) and operator learning architectures such as DeepONets and FNOs. These bounds guarantee that PINNs and (physics-informed) DeepONets or FNOs will efficiently approximate the underlying solution or solution operator of generic partial differential equations (PDEs)
  arXiv  Detail & Related papers  (2022-05-23T15:40:33Z)
• 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)
• dNNsolve: an efficient NN-based PDE solver [62.997667081978825]
  We introduce dNNsolve, that makes use of dual Neural Networks to solve ODEs/PDEs. We show that dNNsolve is capable of solving a broad range of ODEs/PDEs in 1, 2 and 3 spacetime dimensions.
  arXiv  Detail & Related papers  (2021-03-15T19:14:41Z)
• A nonlocal physics-informed deep learning framework using the peridynamic differential operator [0.0]
  We develop a nonlocal PINN approach using the Peridynamic Differential Operator (PDDO)---a numerical method which incorporates long-range interactions and removes spatial derivatives in the governing equations. Because the PDDO functions can be readily incorporated in the neural network architecture, the nonlocality does not degrade the performance of modern deep-learning algorithms. We document the superior behavior of nonlocal PINN with respect to local PINN in both solution accuracy and parameter inference.
  arXiv  Detail & Related papers  (2020-05-31T06:26:21Z)

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.