Model-parallel Fourier neural operators as learned surrogates for large-scale parametric PDEs

J., Grady II, Thomas; Khan, Rishi; Louboutin, Mathias; Yin, Ziyi; Witte, Patrick; Chandra, Ranveer; Hewett, Russell J.; Herrmann, Felix J.

doi:10.1016/j.cageo.2023.105402

Cited by 15 publications

(2 citation statements)

References 18 publications

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“…Particularly, the Fourier neural operators (FNOs) [14,16] learn mappings between infinite-dimensional function spaces. A brief description of FNOs is provided in Appendix A. FNOs have become a popular alternative to the conventional neural networks for a wide range of physical applications like climate modeling [22], multiphase flows in porous media [12,26], and wave propagation [9]. This popularity is due to their low computational cost, relatively small errors, and the support of features like zero-shot super-resolution for turbulent flows, which are limited in other machine learning methods.…”

Section: Methodsmentioning

confidence: 99%

Towards accelerating particle‐resolved direct numerical simulation with neural operators

Atif,

López‐Marrero,

Zhang

et al. 2024

Statistical Analysis

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We present our ongoing work aimed at accelerating a particle‐resolved direct numerical simulation model designed to study aerosol–cloud–turbulence interactions. The dynamical model consists of two main components—a set of fluid dynamics equations for air velocity, temperature, and humidity, coupled with a set of equations for particle (i.e., cloud droplet) tracing. Rather than attempting to replace the original numerical solution method in its entirety with a machine learning (ML) method, we consider developing a hybrid approach. We exploit the potential of neural operator learning to yield fast and accurate surrogate models and, in this study, develop such surrogates for the velocity and vorticity fields. We discuss results from numerical experiments designed to assess the performance of ML architectures under consideration as well as their suitability for capturing the behavior of relevant dynamical systems.

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Section: Methodsmentioning

confidence: 99%

Towards accelerating particle‐resolved direct numerical simulation with neural operators

Atif,

López‐Marrero,

Zhang

et al. 2024

Statistical Analysis

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show abstract

“…Due to the high computational costs of solving 3D Partial Differential Equations (PDEs), only very few 3D datasets are available. CO2 underground storage has been explored with SciML based on 3D numerical simulations (Grady et al, 2023;Wen et al, 2023;Witte et al, 2023). To support the study of Witte et al (2023), Annon (2022) provided 4,000 simulation results for 3D CO2 flow through geological models based on the Sleipner dataset complemented by random fields (Equinor, 2020).…”

Section: D 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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Comparison of neural FEM and neural operator methods for applications in solid mechanics

Hildebrand,

Klinge

2024

Neural Comput & Applic

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Machine learning methods are progressively investigated for a large amount of applications. Recently, the solution of partial differential equations (PDE) describing problems in elastostatics came into focus. The current work investigates two neural network-based classes of methods for their solution, namely the neural finite element method (FEM) and neural operator methods. The analysis of these approaches is carried out by means of numerical experiments with linear and nonlinear material behavior where the conventional FEM serves as a benchmark. The formulation of neural FEM allows for elegant integration of finite deformation hyperelasticity at medium training effort. Here, training data are replaced by the evaluation of the equilibrium PDE at sample points. In contrast, most neural operator methods require expensive training with large data sets, but then allow for solving multiple boundary value problems with the same machine learning model. For the comparative analysis, the maximal relative error values over the whole domain and over all components of the strain tensor are evaluated as accuracy measure. The current state of research shows that none of the methods investigated reaches the accuracy and computational performance of the conventional FEM. In many standard applications, the FEM achieves an accuracy of $$10^{-6}$$ 10 - 6 , whereas the numerical tests in the present work report a relative error of order of magnitude of $$10^{-4}$$ 10 - 4 for the neural FEM and $$10^{-2}$$ 10 - 2 to $$10^{-3}$$ 10 - 3 for neural operator methods.

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Model-parallel Fourier neural operators as learned surrogates for large-scale parametric PDEs

Cited by 15 publications

References 18 publications

Towards accelerating particle‐resolved direct numerical simulation with neural operators

Towards accelerating particle‐resolved direct numerical simulation with neural operators

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

Comparison of neural FEM and neural operator methods for applications in solid mechanics

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