# Deep learning observables in computational fluid dynamics

arXiv: https://arxiv.org/abs/1903.03040 · HTML: https://mlsanity.com/paper/1903.03040/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Mar 2019.

## Built on by (most cited first, 30 of 38)

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- [wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws](https://mlsanity.com/paper/2207.08483/) (2022)
- [An operator learning perspective on parameter-to-observable maps](https://mlsanity.com/paper/2402.06031/) (2024)
- [Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domains](https://mlsanity.com/paper/2305.19663/) (2023)
- [Data-driven Control of Agent-based Models: an Equation/Variable-free Machine Learning Approach](https://mlsanity.com/paper/2207.05779/) (2022)
- [Real-time simulation of parameter-dependent fluid flows through deep learning-based reduced order models](https://mlsanity.com/paper/2106.05722/) (2021)
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- [Novel DNNs for Stiff ODEs with Applications to Chemically Reacting Flows](https://mlsanity.com/paper/2104.01914/) (2021)
- [Neural network architectures using min-plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs](https://mlsanity.com/paper/2105.03336/) (2021)
- [Quantifying uncertainty for deep learning based forecasting and flow-reconstruction using neural architecture search ensembles](https://mlsanity.com/paper/2302.09748/) (2023)
- [Deep neural network surrogates for non-smooth quantities of interest in shape uncertainty quantification](https://mlsanity.com/paper/2101.07023/) (2021)
- [Theoretical Exploration of Solutions of Feedforward ReLU Networks](https://mlsanity.com/paper/2202.01919/) (2022)
- [Error analysis for deep neural network approximations of parametric hyperbolic conservation laws](https://mlsanity.com/paper/2207.07362/) (2022)
- [Looking at the posterior: accuracy and uncertainty of neural-network predictions](https://mlsanity.com/paper/2211.14605/) (2022)
- [PINNs error estimates for nonlinear equations in ℝ-smooth Banach spaces](https://mlsanity.com/paper/2305.11915/) (2023)
- [Full Domain Analysis in Fluid Dynamics](https://mlsanity.com/paper/2505.22275/) (2025)
- [Lattice-based Deep Neural Networks: Regularity and Tailored Regularization](https://mlsanity.com/paper/2603.02809/) (2026)
- [Cell-average based neural network method for hyperbolic and parabolic partial differential equations](https://mlsanity.com/paper/2107.00813/) (2021)
- [A comparison of PINN approaches for drift-diffusion equations on metric graphs](https://mlsanity.com/paper/2205.07195/) (2022)
- [Learning WENO for entropy stable schemes to solve conservation laws](https://mlsanity.com/paper/2403.14848/) (2024)
- [Multi-Stage Graph Neural Networks for Data-Driven Prediction of Natural Convection in Enclosed Cavities](https://mlsanity.com/paper/2509.06041/) (2025)
- [DepthWeave-KV: Token-Adaptive Cross-Layer Residual Factorization for Long-Context KV Cache Compression](https://mlsanity.com/paper/2607.06523/) (2026)
- [Efficient Quality Diversity Optimization of 3D Buildings through 2D Pre-optimization](https://mlsanity.com/paper/2303.15896/) (2023)

## Abstract (arXiv)

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.