[Paper Review] Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data

Maziar Raissi, Alireza Yazdani+1 more|arXiv (Cornell University)|Aug 13, 2018

Model Reduction and Neural Networks22 references84 citations

TL;DR

HFM uses physics-informed neural networks constrained by Navier-Stokes to infer velocity, pressure, and related quantities from passive scalar visualizations in complex geometries, without predefined boundaries.

ABSTRACT

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.

Motivation & Objective

• Leverage Navier-Stokes laws to infer hidden velocity and pressure fields from passive scalar data (e.g., dye, smoke) in arbitrary geometries.
• Develop a geometry-agnostic data assimilation approach that requires minimal boundary information.
• Demonstrate accurate 2D and 3D flow reconstructions in external and internal flows relevant to biomedical and engineering applications.

Proposed method

• Represent c, d, u, v, w, p as outputs of a deep neural network and train with a physics-informed loss.
• Enforce six residuals e1–e6 derived from non-dimensional transport and Navier–Stokes equations using automatic differentiation.
• Introduce auxiliary variable d = 1 - c to enhance boundary-geometry inference and training stability.
• Use a training domain boundary selection that relies on concentration gradients to infer flow boundaries without explicit velocity/pressure BCs.
• Apply the Adam optimizer with a staged learning-rate schedule and 10-layer networks with sin activations.
• Demonstrate NS-informed networks with data-driven regression of c and its complement d, while enforcing PDE residuals at sampled points.

Experimental results

Research questions

• RQ1 Can velocity and pressure fields be uniquely inferred from passive scalar concentration data within arbitrarily chosen training domains?
• RQ2 How well can a NS-informed neural network recover flow fields in external and internal flows using only passive scalar data?
• RQ3 Does including the auxiliary variable d = 1 - c improve accuracy and geometry detection?
• RQ4 Can the framework estimate derived quantities such as lift, drag, and wall stresses from learned fields?
• RQ5 What is the impact of training domain design and data availability on inference quality?

Key findings

• The framework accurately reconstructs velocity and pressure in 2D and 3D flows from passive scalar data alone.
• Including the auxiliary variable d improves prediction accuracy and boundary geometry detection with no extra cost.
• Predicted lift and drag on a cylinder closely match exact solutions, with errors around 1% for tested cases.
• The method can infer Reynolds and Peclet numbers when treated as learnable parameters, showing good agreement with exact values.
• Training domains do not need to coincide with physical boundaries if passive scalar gradients are sufficiently informative.