A Deep Double Ritz Method (D<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e3266" altimg="si244.svg"><mml:msup><mml:mrow/><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>RM) for solving Partial Differential Equations using Neural Networks

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

doi:10.1016/j.cma.2023.115892

Cited by 8 publications

References 26 publications

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

Uriarte

Taylor²,

Pardo

et al. 2023

Lecture Notes in Computer Science

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

Uriarte

Taylor²,

Pardo

et al. 2023

Lecture Notes in Computer Science

Self Cite

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Solving partial differential equations using large-data models: a literature review

Hafiz,

Faiq,

Hassaballah

2024

Artif Intell Rev

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Mathematics lies at the heart of engineering science and is very important for capturing and modeling of diverse processes. These processes may be naturally-occurring or man-made. One important engineering problem in this regard is the modeling of advanced mathematical problems and their analysis. Partial differential equations (PDEs) are important and useful tools to this end. However, solving complex PDEs for advanced problems requires extensive computational resources and complex techniques. Neural networks provide a way to solve complex PDEs reliably. In this regard, large-data models are new generation of techniques, which have large dependency capturing capabilities. Hence, they can richly model and accurately solve such complex PDEs. Some common large-data models include Convolutional neural networks (CNNs) and their derivatives, transformers, etc. In this literature survey, the mathematical background is introduced. A gentle introduction to the area of solving PDEs using large-data models is given. Various state-of-the-art large-data models for solving PDEs are discussed. Also, the major issues and future scope of the area are identified. Through this literature survey, it is hoped that readers will gain an insight into the area of solving PDEs using large-data models and pursue future research in this interesting area.

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Some elliptic second order problems and neural network solutions: Existence and error estimates

Pousin

2024

Journal of Computational and Applied Mathematics

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A Deep Double Ritz Method (D2RM) for solving Partial Differential Equations using Neural Networks

Cited by 8 publications

References 26 publications

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

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

Solving partial differential equations using large-data models: a literature review

Some elliptic second order problems and neural network solutions: Existence and error estimates

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