CDAnet: A Physics‐Informed Deep Neural Network for Downscaling Fluid Flows

Abstract

Generating high-resolution flow fields is of paramount importance for various applications in engineering and climate sciences. This is typically achieved by solving the governing dynamical equations on high-resolution meshes, suitably nudged towards available coarse-scale data. To alleviate the computational cost of such downscaling process, we develop a physics-informed deep neural network (PI-DNN) that mimics the mapping of coarse-scale information into their fine-scale counterparts of continuous data assimilation (CDA). Specifically, the PI-DNN is trained within the theoretical framework described by Foias et al. (2014) to generate a surrogate of the theorized determining form map from the coarse-resolution data to the fine-resolution solution. We demonstrate the PI-DNN methodology through application to 2D Rayleigh-Bénard convection, and assess its performance by contrasting its predictions against those obtained by dynamical downscaling using CDA. The analysis suggests that the surrogate is constrained by similar conditions, in terms of spatio-temporal resolution of the input, as the ones required by the theoretical determining form map. The numerical results also suggest that the surrogate's downscaled fields are of comparable accuracy to those obtained by dynamically downscaling using CDA. Consistent with the analysis of Farhat, Jolly, and Titi (2015), temperature observations are not needed for the PI-DNN to predict the fine-scale velocity, pressure and temperature fields.

Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data

“…We find that the assimilation of the tracers improves the level of convergence of the kinetic energy of the system, however we do not obtain convergence to the level of machine precision, except in certain cases with simplified physics. We note that the level of convergence we obtain is comparable to those of recent studies of this algorithm for ocean models including [12] where the simulations were done in the context of an idealized mesoscale eddy test case and [29], where the performance This work is organized as follows. In Section 2 we discuss the formulation of the AOT algorithm equations and the ocean model equations.…”

Spatio‐Temporal Super‐Resolution Data Assimilation (SRDA) Utilizing Deep Neural Networks With Domain Generalization

“…First, despite the reduced accuracy due to domain shift, the NN inference can remain more accurate than that of baselines, making domain‐generalization methods less necessary. Second, loss functions that include physics‐based terms, such as the residuals of the governing equations, can act as regularizers that improve generalizability (e.g., Bao et al., 2022; Hammoud et al., 2022; Jiang et al., 2020; C. Wang et al., 2020). Third, incorporating geometric symmetries as prior knowledge can enhance the generalization performance (e.g., Chattopadhyay et al., 2022; Ling et al., 2016; R. Wang et al., 2021; Yasuda & Onishi, 2023a).…”

“…Figure 3 outlines the network architecture, which is based on U‐Net (Ronneberger et al., 2015). U‐Net architectures have been employed in the SR for fluid systems (e.g., Hammoud et al., 2022; Jiang et al., 2020; L. Wang et al., 2021). The input and output of the NN are vorticity fields.…”

“…The error bars show the maxima and minima of the MAE ratios from five experiments where the random seed for the initializer of neural-network weights was varied. Bao et al, 2022;Hammoud et al, 2022;Jiang et al, 2020;C. Wang et al, 2020).…”