# Deep learning observables in computational fluid dynamics

## TL;DR

This paper introduces a deep learning approach to efficiently approximate the input parameters to observable map in computational fluid dynamics, significantly reducing computational costs for uncertainty quantification and related tasks.

## Contribution

It develops a neural network-based method that accurately predicts the input-to-observable map with few training samples and combines it with Monte Carlo methods for faster uncertainty propagation.

## Key findings

- Neural networks achieve low prediction errors with minimal training samples.
- The combined deep learning and Monte Carlo methods outperform traditional approaches by one to two orders of magnitude.
- Numerical experiments validate the efficiency and accuracy of the proposed approach.

## 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 \emph{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.