# Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data

arXiv: https://arxiv.org/abs/1808.04327 · HTML: https://mlsanity.com/paper/1808.04327/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Aug 2018.

## Built on by (most cited first, 30 of 36)

- [Efficient training of physics-informed neural networks via importance sampling](https://mlsanity.com/paper/2104.12325/) (2021)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [Error estimates for physics informed neural networks approximating the Navier-Stokes equations](https://mlsanity.com/paper/2203.09346/) (2022)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [HyperPINN: Learning parameterized differential equations with physics-informed hypernetworks](https://mlsanity.com/paper/2111.01008/) (2021)
- [Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning](https://mlsanity.com/paper/2402.10926/) (2024)
- [Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNs](https://mlsanity.com/paper/2109.07143/) (2021)
- [wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws](https://mlsanity.com/paper/2207.08483/) (2022)
- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver](https://mlsanity.com/paper/2105.11045/) (2021)
- [IDRLnet: A Physics-Informed Neural Network Library](https://mlsanity.com/paper/2107.04320/) (2021)
- [On the Generalization of PINNs outside the training domain and the Hyperparameters influencing it](https://mlsanity.com/paper/2302.07557/) (2023)
- [WaveY-Net: Physics-augmented deep learning for high-speed electromagnetic simulation and optimization](https://mlsanity.com/paper/2203.01248/) (2022)
- [Physics-informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media](https://mlsanity.com/paper/2212.08965/) (2022)
- [Size Lowerbounds for Deep Operator Networks](https://mlsanity.com/paper/2308.06338/) (2023)
- [A Comparison of Mesh-Free Differentiable Programming and Data-Driven Strategies for Optimal Control under PDE Constraints](https://mlsanity.com/paper/2310.02286/) (2023)
- [Physics-informed neural networks modeling for systems with moving immersed boundaries: application to an unsteady flow past a plunging foil](https://mlsanity.com/paper/2306.13395/) (2023)
- [Learning solutions of parametric Navier-Stokes with physics-informed neural networks](https://mlsanity.com/paper/2402.03153/) (2024)
- [Learning Free-Surface Flow with Physics-Informed Neural Networks](https://mlsanity.com/paper/2111.09705/) (2021)
- [Leveraging Stochastic Predictions of Bayesian Neural Networks for Fluid Simulations](https://mlsanity.com/paper/2205.01222/) (2022)
- [Domain-informed graph neural networks: a quantum chemistry case study](https://mlsanity.com/paper/2208.11934/) (2022)
- [TSONN: Time-stepping-oriented neural network for solving partial differential equations](https://mlsanity.com/paper/2310.16491/) (2023)
- [Data-driven Modeling of Combined Sewer Systems for Urban Sustainability: An Empirical Evaluation](https://mlsanity.com/paper/2408.11619/) (2024)
- [Physics-constrained Unsupervised Learning of Partial Differential Equations using Meshes](https://mlsanity.com/paper/2203.16628/) (2022)
- [A comparison of PINN approaches for drift-diffusion equations on metric graphs](https://mlsanity.com/paper/2205.07195/) (2022)
- [Towards Long-Term predictions of Turbulence using Neural Operators](https://mlsanity.com/paper/2307.13517/) (2023)
- [Physics-informed Deep Learning to Solve Three-dimensional Terzaghi Consolidation Equation: Forward and Inverse Problems](https://mlsanity.com/paper/2401.05439/) (2024)
- [Transferable Model for Shape Optimization subject to Physical Constraints](https://mlsanity.com/paper/2103.10805/) (2021)
- [Physics-Guided Problem Decomposition for Scaling Deep Learning of High-dimensional Eigen-Solvers: The Case of Schr"odinger's Equation](https://mlsanity.com/paper/2202.05994/) (2022)

## Abstract (arXiv)

We present hidden fluid mechanics (HFM), a physics informed deep learning framework capable of encoding an important class of physical laws governing fluid motions, namely the Navier-Stokes equations. In particular, we seek to leverage the underlying conservation laws (i.e., for mass, momentum, and energy) to infer hidden quantities of interest such as velocity and pressure fields merely from spatio-temporal visualizations of a passive scaler (e.g., dye or smoke), transported in arbitrarily complex domains (e.g., in human arteries or brain aneurysms). Our approach towards solving the aforementioned data assimilation problem is unique as we design an algorithm that is agnostic to the geometry or the initial and boundary conditions. This makes HFM highly flexible in choosing the spatio-temporal domain of interest for data acquisition as well as subsequent training and predictions. Consequently, the predictions made by HFM are among those cases where a pure machine learning strategy or a mere scientific computing approach simply cannot reproduce. The proposed algorithm achieves accurate predictions of the pressure and velocity fields in both two and three dimensional flows for several benchmark problems motivated by real-world applications. Our results demonstrate that this relatively simple methodology can be used in physical and biomedical problems to extract valuable quantitative information (e.g., lift and drag forces or wall shear stresses in arteries) for which direct measurements may not be possible.