The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems

Weinan, E; Yu, Bing

doi:10.1007/s40304-018-0127-z

Cited by 709 publications

(131 citation statements)

References 2 publications

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“…II A). In contrast to this work, the error analyses reported by E et al [14] and Anitescu et al [42] did not consider the physical realism of the NNM solutions (Sec. I D) nor the accuracy of the NNM solutions' gradients.…”

Section: Nnm With Complicated Geometriesmentioning

confidence: 66%

“…When Anitescu et al [42] revisited this needle problem using the original method of Dissanayake and Phan-Thien [2], they reported poorer convergence than obtained by E et al [14] with the Deep Ritz method. A similar observation was made during this work: reentrant corners significantly impair the convergence of the standard NNM (Sec.…”

Section: Nnm With Complicated Geometriesmentioning

confidence: 92%

“…For instance, the NNM is mesh free, and generally produces uniformly accurate solutions throughout the PDE domain [11,12]. Whereas earlier implementations used shallow neural networks (i.e., those having only one hidden layer), many authors have recently noted the significant benefits of using deep architectures [13][14][15][16][17][18][19][20][21][22][23][24][25][26]. In particular, it appears that the NNM with deep neural networks performs remarkably well in high-dimensional problems [13][14][15][17][18][19][21][22][23][24][25][26][27].…”

Section: A Neural Network Methodsmentioning

confidence: 99%

“…An exception to the above is given by E et al [14], who applied a variant of the NNM to a Poisson equation in a square domain with a reentrant needlelike boundary. This problem exhibits the same singular behavior as the slit-well problem with sharp corners (see Sec.…”

Section: Nnm With Complicated Geometriesmentioning

confidence: 99%

See 3 more Smart Citations

Neural network solutions to differential equations in nonconvex domains: Solving the electric field in the slit-well microfluidic device

Magill

Nagel

Haan

2020

Phys. Rev. Research

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The neural network method of solving differential equations is used to approximate the electric potential and corresponding electric field in the slit-well microfluidic device. The device's geometry is nonconvex, making this a challenging problem to solve using the neural network method. To validate the method, the neural network solutions are compared to a reference solution obtained using the finite-element method. Additional metrics are presented that measure how well the neural networks recover important physical invariants that are not explicitly enforced during training: spatial symmetries and conservation of electric flux. Finally, as an application-specific test of validity, neural network electric fields are incorporated into particle simulations. Conveniently, the same loss functional used to train the neural networks also seems to provide a reliable estimator of the networks' true errors, as measured by any of the metrics considered here. In all metrics, deep neural networks significantly outperform shallow neural networks, even when normalized by computational cost. Altogether, the results suggest that the neural network method can reliably produce solutions of acceptable accuracy for use in subsequent physical computations, such as particle simulations.

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Section: Nnm With Complicated Geometriesmentioning

confidence: 66%

Section: Nnm With Complicated Geometriesmentioning

confidence: 92%

Section: A Neural Network Methodsmentioning

confidence: 99%

Section: Nnm With Complicated Geometriesmentioning

confidence: 99%

See 2 more Smart Citations

Neural network solutions to differential equations in nonconvex domains: Solving the electric field in the slit-well microfluidic device

Magill

Nagel

Haan

2020

Phys. Rev. Research

View full text Add to dashboard Cite

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“…Additionally, Largaris et al [14] and more recently Berg and Nyström [16] showed fully connected networks can be used to learn PDE solutions on even complex domains. Recently, several investigators have examined the use of variational formulations of the governing equations as loss functions to solve various PDEs [3,17,18,19] which has been proven to be effective. Sirignano et al [20] show that the use of a fully connected network can be used for efficiently solving PDEs of high dimensionality where traditional discretization techniques become unfeasible.…”

Section: Introductionmentioning

confidence: 99%

Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks

Geneva

Zabaras

2020

Journal of Computational Physics

225

132

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In recent years, deep learning has proven to be a viable methodology for surrogate modeling and uncertainty quantification for a vast number of physical systems. However, in their traditional form, such models require a large amount of training data. This is of particular importance for various engineering and scientific applications where data may be extremely expensive to obtain. To overcome this shortcoming, physics-constrained deep learning provides a promising methodology as it only utilizes the governing equations. In this work, we propose a novel auto-regressive dense encoder-decoder convolutional neural network to solve and model transient systems with non-linear dynamics at a computational cost that is potentially magnitudes lower than standard numerical solvers. This model includes a Bayesian framework that allows for uncertainty quantification of the predicted quantities of interest at each time-step. We rigorously test this model on several non-linear transient partial differential equation systems including the turbulence of the Kuramoto-Sivashinsky equation, multi-shock formation and interaction with 1D Burgers' equation and 2D wave dynamics with coupled Burgers' equations. For each system, the predictive results and uncertainty are presented and discussed together with comparisons to the results obtained from traditional numerical analysis methods.

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Enhancing physics informed neural networks for solving Navier–Stokes equations

Farkane,

Ghogho,

Oudani

et al. 2023

Numerical Methods in Fluids

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SummaryFluid mechanics is a critical field in both engineering and science. Understanding the behavior of fluids requires solving the Navier–Stokes equation (NSE). However, the NSE is a complex partial differential equation that can be challenging to solve, and classical numerical methods can be computationally expensive. In this paper, we propose enhancing physics‐informed neural networks (PINNs) by modifying the residual loss functions and incorporating new computational deep learning techniques. We present two enhanced models for solving the NSE. The first model involves developing the classical PINN for solving the NSE, based on a stream function approach to the velocity components. We have added the pressure training loss function to this model and integrated the new computational training techniques. Furthermore, we propose a second, more flexible model that directly approximates the solution of the NSE without making any assumptions. This model significantly reduces the training duration while maintaining high accuracy. Moreover, we have successfully applied this model to solve the three‐dimensional NSE. The results demonstrate the effectiveness of our approaches, offering several advantages, including high trainability, flexibility, and efficiency.

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The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems

Cited by 709 publications

References 2 publications

Neural network solutions to differential equations in nonconvex domains: Solving the electric field in the slit-well microfluidic device

Neural network solutions to differential equations in nonconvex domains: Solving the electric field in the slit-well microfluidic device

Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks

Enhancing physics informed neural networks for solving Navier–Stokes equations

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