Beyond the Courant-Friedrichs-Lewy condition: Numerical methods for the wave problem using deep learning

Ovadia, Oded; Kahana, Adar; Turkel, Eli; Dekel, Shai

doi:10.1016/j.jcp.2021.110493

Cited by 12 publications

(10 citation statements)

References 10 publications

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“…The above-mentioned wavefields were simulated with the 2D acoustic wave equation (i.e. with 1-component waveforms), that saw several Neural Operators applications (Li et al, 2022;Ovadia et al, 2023;Yang et al, 2021). Table 1 summarizes the characteristics of those datasets.…”

Section: Geophysical Datasetsmentioning

confidence: 99%

Synthetic ground motions in heterogeneous geologies: the HEMEW-3D dataset for scientific machine learning

Lehmann,

Gatti,

Bertin

et al. 2024

Preprint

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Abstract. The ever-improving performances of physics-based simulations and the rapid developments of deep learning are offering new perspectives to study earthquake-induced ground motion. Due to the large amount of data required to train deep neural networks, applications have so far been limited to recorded data or two-dimensional simulations. To bridge the gap between deep learning and high-fidelity numerical simulations, this work introduces a new database of physics-based earthquake simulations. The HEMEW-3D database comprises 30,000 simulations of elastic wave propagation in three-dimensional (3D) geological domains. Each domain is parametrized by a different geological model built from a random arrangement of layers augmented by random fields that represent heterogeneities. For each simulation, ground motion is synthetized at the surface by a grid of virtual sensors. The high frequency of waveforms (fmax = 5 Hz) allows extensive analyses of surface ground motion. Existing and foreseen applications range from statistic analyses of the ground motion variability and machine learning methods on geological models, to deep learning-based predictions of ground motion depending on 3D heterogeneous geologies.

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Section: Geophysical Datasetsmentioning

confidence: 99%

Synthetic ground motions in heterogeneous geologies: the HEMEW-3D dataset for scientific machine learning

Lehmann,

Gatti,

Bertin

et al. 2024

Preprint

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“…This technique allows to obtain relatively high quality approximation with relatively small datasets. There are various neural network architectures that have been developed for this purpose, with different settings and strategies such as automation differentiation [18], numerical schemes [5], grid-free [3,4] or grid-dependent approaches [5], and the ability to handle different geometries [19]. In this paper, we present a generalization of spectral based deep learning methods for PDEs [23], [24], [25], [26]:…”

Section: Introductionmentioning

confidence: 99%

“…Allen-Cahn equation over S 2 -stability test results using the normalized metric(5) In this setting, the Laplace-Beltrami operator is[10] …”

mentioning

confidence: 99%

Numerical Methods for Pdes Over Manifolds Using Spectral Physics Informed Neural Networks

Dekel

Yuval

2023

Preprint

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“…Here, the neural network surrogate advances the state in time, conditioned on the provided parameters. Recent work has also explored alternative neural network architectures for surrogate models, such as transformers, graph neural networks, and Fourier neural operators [43,58,91,111].…”

Section: Deep Learning-based Surrogate Modelsmentioning

confidence: 99%

“…These transformers divide images into patches and apply the attention mechanism between each set of patches. Yet, since there is no natural way of reducing or expanding the dimensionality of the data, the ViT has been used to dimensionality reduction tasks only to a limited degree [64,90,111,126]. Importantly, current ViTs have not been integrated with increasing numbers of channels in convolutional layers to represent increasingly complicated features.…”

Section: Transformer-based Dimensionality Reductionmentioning

confidence: 99%

Deep Learning for Real-Time Inverse Problems and Data Assimilation with Uncertainty Quantification for Digital Twins

Mücke

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We develop novel approaches to enhance the functionality and efficiency of digital twins through deep learning techniques. Digital twins, sophisticated virtual models of physical systems, serve as dynamic counterparts, mirroring real-world entities' behavior and performance. Their primary role includes real-time monitoring, flaw detection, automated control, and proactive maintenance. To perform such tasks it is necessary to use various methods from scientific computing to combine data with physics models. Furthermore, to assess the reliability of a digital twin, it is important to not only make predictions, but also quantify the uncertainty of estimations and predictions. In this regard, Bayesian methods serve as a natural framework to pose the problems. However, the complexity and real-time operation requirements pose significant computational challenges. The thesis explores the integration of deep learning in scientific computing for enabling digital twins. The focus is on overcoming the computational hurdles in real-time data assimilation and inverse problem-solving through surrogate modeling. Deep learning, with its capability to handle high-dimensional functions in both deterministic and stochastic settings is shown to be a viable solution to these challenges. This work investigates several key areas, beginning with a brief introduction to digital twins, data assimilation, and inverse problems. It proceeds to elaborate on the role of Bayesian inversion problems in static and dynamic contexts, highlighting the importance of uncertainty quantification. The focus is on the challenges of real-time computation and the potential of Bayesian methods, despite their computational intensity. The thesis investigates deep learning architectures like dense and residual neural networks, convolutional neural networks, recurrent neural networks and transformers. The core contributions are presented in the four main chapters of the thesis: 1) Reduced Order Modeling for Parameterized Time-Dependent Partial Differential Equations: Introducing a deep learning framework for solving parameterized PDEs using spatially and memory-aware neural networks, demonstrated on linear advection and incompressible Navier-Stokes equations. 2) Markov Chain Generative Adversarial Neural Networks: Enhancing Bayesian inversion for static problems with sparse observations using a unique algorithm combining MCMC and GANs, demonstrated on a Darcy flow problem and pipe flow leakage detection. 3) Probabilistic Digital Twin for Leak Localization: Developing a probabilistic framework using supervised Wasserstein Autoencoders for leak localization in water distribution networks, tested on several network models. 4) The Deep Latent Space Particle Filter: Proposing a real-time nonlinear data assimilation method for dynamic PDEs with uncertainty quantification, using transformer-based dimensionality reduction and time-stepping in particle filters, demonstrated on various dynamic systems. This thesis contributes significantly to digital twins, addressing computational limitations and opening new avenues for practical applications in various industries.

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Beyond the Courant-Friedrichs-Lewy condition: Numerical methods for the wave problem using deep learning

Cited by 12 publications

References 10 publications

Synthetic ground motions in heterogeneous geologies: the HEMEW-3D dataset for scientific machine learning

Synthetic ground motions in heterogeneous geologies: the HEMEW-3D dataset for scientific machine learning

Numerical Methods for Pdes Over Manifolds Using Spectral Physics Informed Neural Networks

Deep Learning for Real-Time Inverse Problems and Data Assimilation with Uncertainty Quantification for Digital Twins

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