A Deep Double Ritz Method (D2rm) for Solving Partial Differential Equations Using Neural Networks

Uriarte, Carlos; Pardo, David; Muga, Ignacio; Muñoz‐Matute, Judit

doi:10.2139/ssrn.4281320

Cited by 3 publications

(2 citation statements)

References 40 publications

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“…In [52], the same authors extend this method to time-harmonic Maxwell equations in the context of symmetric H(curl)-formulations. Also, in [54], in the context of Petrov-Galerkin formulations, the authors minimize the dual norm of the residual based on the concept of optimal testing form [22]. They approximate both the solution of the variational problem and the corresponding optimal test functions employing NNs by solving two nested deep Ritz problems.…”

Section: Introductionmentioning

confidence: 99%

Robust Variational Physics-Informed Neural Networks

Rojas,

Maczuga,

Muñoz-Matute

et al. 2024

Computer Methods in Applied Mechanics and Engineering

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Section: Introductionmentioning

confidence: 99%

Robust Variational Physics-Informed Neural Networks

Rojas,

Maczuga,

Muñoz-Matute

et al. 2024

Computer Methods in Applied Mechanics and Engineering

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“…Over the past decade, neural networks have proven to be a powerful tool in the context of solving Partial Differential Equations (PDEs) [5,8,9,12,[18][19][20]. In such cases, the traditional approach is to reformulate the PDE as a minimization problem, where the loss function is often described as a definite integral and, therefore, approximated or discretized by a quadrature rule.…”

Section: Introductionmentioning

confidence: 99%

Memory-Based Monte Carlo Integration for Solving Partial Differential Equations Using Neural Networks

Uriarte

Taylor²,

Pardo

et al. 2023

Computational Science – ICCS 2023

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Monte Carlo integration is a widely used quadrature rule to solve Partial Differential Equations with neural networks due to its ability to guarantee overfitting-free solutions and high-dimensional scalability. However, this stochastic method produces noisy losses and gradients during training, which hinders a proper convergence diagnosis. Typically, this is overcome using an immense (disproportionate) amount of integration points, which deteriorates the training performance. This work proposes a memory-based Monte Carlo integration method that produces accurate integral approximations without requiring the high computational costs of processing large samples during training.

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