Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control

Ainsworth, Mark; Dong, Justin

doi:10.1137/20m1366587

Cited by 17 publications

(7 citation statements)

References 29 publications

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“…For example, Fourier transforms, which are employed in the neural network and to represent derivatives in the loss function, are known to be highly sensitive to discontinuities. Thus, potential improvements may encompass the use of Galerkin neural networks [44] to better handle discontinuities, and loss functions that incorporate particle number conservation. These improvements should be considered in future work.…”

Section: Burgers Equation 2d Inviscid Resultsmentioning

confidence: 99%

Applications of physics informed neural operators

Rosofsky

Majed

Huerta

2023

Mach. Learn.: Sci. Technol.

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We present a critical analysis of physics-informed neural operators to solve partial differential equations that are ubiquitous in the study and modeling of physics phenomena using carefully curated datasets. Further, we provide a benchmarking suite which can be used to evaluate physics-informed neural operators in solving such problems. We first demonstrate that our methods reproduce the accuracy and performance of other neural operators published elsewhere in the literature to learn the 1D wave equation and the 1D Burgers equation. Thereafter, we apply our physics-informed neural operators to learn new types of equations, including the 2D Burgers equation in the scalar, inviscid and vector types. Finally, we show that our approach is also applicable to learn the physics of the 2D linear and nonlinear shallow water equations, which involve three coupled partial differential equations. We release our artificial intelligence surrogates and scientific software to produce initial data and boundary conditions to study a broad range of physically motivated scenarios. We provide the \href{https://github.com/shawnrosofsky/PINO_Applications/tree/main}{source code}, an interactive \href{https://shawnrosofsky.github.io/PINO_Applications/}{website} to visualize the predictions of our physics informed neural operators, and a tutorial for their use at the \href{https://www.dlhub.org}{Data and Learning Hub for Science}.

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Section: Burgers Equation 2d Inviscid Resultsmentioning

confidence: 99%

Applications of physics informed neural operators

Rosofsky

Majed

Huerta

2023

Mach. Learn.: Sci. Technol.

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“…Many approaches have been proposed to mitigate these failures, e.g. [49][50][51][52][53][54][55][56][57][58][59][60], and we note that many of these can be applied simultaneously with the continual learning approach proposed here. For this paper we focus on PINNs applied alone with continual learning to show the improvement that continual learning can offer.…”

Section: Physics-informed Trainingmentioning

confidence: 99%

A multifidelity approach to continual learning for physical systems

Howard,

Fu,

Stinis

2024

Mach. Learn.: Sci. Technol.

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We introduce a novel continual learning method based on multifidelity deep neural networks. This method learns the correlation between the output of previously trained models and the desired output of the model on the current training dataset, limiting catastrophic forgetting. On its own the multifidelity continual learning method shows robust results that limit forgetting across several datasets. Additionally, we show that the multifidelity method can be combined with existing continual learning methods, including replay and memory aware synapses, to further limit catastrophic forgetting. The proposed continual learning method is especially suited for physical problems where the data satisfy the same physical laws on each domain, or for physics-informed neural networks, because in these cases we expect there to be a strong correlation between the output of the previous model and the model on the current training domain.

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“…However, as in other ML-based approaches, they are not able to accurately predict numerical solutions to the stiff PDEs which exhibit a sharp transition near the boundary. Recently, Galerkin Neural Network (GNN) was introduced to learn a test function for a single instance in [55], [56] and solve a boundary layer issue arose from reaction-diffusion types of singular perturbation problems.…”

Section: Related Workmentioning

confidence: 99%

Unsupervised Legendre–Galerkin Neural Network for Solving Partial Differential Equations

2023

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In recent years, machine learning methods have been used to solve partial differential equations (PDEs) and dynamical systems, leading to the development of a new research field called scientific machine learning, which combines techniques such as deep neural networks and statistical learning with classical problems in applied mathematics. In this paper, we present a novel numerical algorithm that uses machine learning and artificial intelligence to solve PDEs. Based on the Legendre-Galerkin framework, we propose an unsupervised machine learning algorithm that learns multiple instances of the solutions for different types of PDEs. Our approach addresses the limitations of both data-driven and physics-based methods. We apply the proposed neural network to general 1D and 2D PDEs with various boundary conditions, as well as convection-dominated singularly perturbed PDEs that exhibit strong boundary layer behavior.

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Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control

Cited by 17 publications

References 29 publications

Applications of physics informed neural operators

Applications of physics informed neural operators

A multifidelity approach to continual learning for physical systems

Unsupervised Legendre–Galerkin Neural Network for Solving Partial Differential Equations

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