Deep learning observables in computational fluid dynamics

Abstract

Many large scale problems in computational fluid dynamics such as uncertainty quantification, Bayesian inversion, data assimilation and PDE constrained optimization are considered very challenging computationally as they require a large number of expensive (forward) numerical solutions of the corresponding PDEs. We propose a machine learning algorithm, based on deep artificial neural networks, that predicts the underlying input parameters to observable map from a few training samples (computed realizations of this map). By a judicious combination of theoretical arguments and empirical observations, we find suitable network architectures and training hyperparameters that result in robust and efficient neural network approximations of the parameters to observable map. Numerical experiments are presented to demonstrate low prediction errors for the trained network networks, even when the network has been trained with a few samples, at a computational cost which is several orders of magnitude lower than the underlying PDE solver.Moreover, we combine the proposed deep learning algorithm with Monte Carlo (MC) and Quasi-Monte Carlo (QMC) methods to efficiently compute uncertainty propagation for nonlinear PDEs. Under the assumption that the underlying neural networks generalize well, we prove that the deep learning MC and QMC algorithms are guaranteed to be faster than the baseline (quasi-) Monte Carlo methods. Numerical experiments demonstrating one to two orders of magnitude speed up over baseline QMC and MC algorithms, for the intricate problem of computing probability distributions of the observable, are also presented.

Statistical solutions of the incompressible Euler equations

“…This necessitates a very high computational cost, particularly in three space dimensions. We plan to consider efficient variants such as multi-level Monte Carlo, 15,28,40 Quasi-Monte Carlo and deep learning algorithms, 37 for computing statistical solutions of the incompressible Euler equations in three space dimensions, in forthcoming papers.…”

Three ways to solve partial differential equations with neural networks — A review

“…In addition, PINNs have been applied successfully in a wide range of applications, including fluid dynamics [113,115,117,160,177], continuum mechanics and elastodynamics [66,132,162], inverse problems [91,121], fractional advection–diffusion equations [135], stochastic advection–diffusion–reaction equations [34], stochastic differential equations [179] and power systems [127]. Finally, we mention that Gaussian processes as an alternative to neural networks for approximating complex multivariate functions have also been studied extensively for solving PDEs and inverse problems [136,155,158,164].…”

Heating Homes with Servers: Workload Scheduling for Heat Reuse in Distributed Data Centers

“…Tradeoffs should be done among the accuracy of the simulation, execution time, and resource overheads. Mathematical models and machine-leaning-based approaches are used to address such tradeoffs [ 4 , 29 , 34 ] with varying levels of success. Mathematical models of the temperature evolution in a server room are presented in [ 35 , 36 ] addressing the thermal behavior concerning heat generation, circulation, and air-cooling system using Navier–Stokes equations expressing thermal laws or by using fast approximate solvers [ 37 , 38 ].…”