Source-linked AI summary
Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data
Maziar Raissi, Alireza Yazdani, George Em Karniadakis
TL;DR
The paper addresses inferring fluid velocity and pressure fields from spatio-temporal observations of a transported passive scalar, without directly measuring those hidden quantities. It uses Navier-Stokes-informed neural networks and demonstrates accurate predictions for complex two- and three-dimensional flows, while noting scope limitations for turbulent regimes and boundary-condition-free uniqueness.
Problem
Inferring spatial fluid velocity and pressure fields typically requires direct measurements or precisely specified geometry, initial conditions, and boundary conditions.
Method
HFM approximates velocity and pressure with neural networks constrained by the Navier-Stokes equations and uses passive-scalar evolution to infer these hidden fields.
Results
The algorithm made accurate predictions for complex 2D and 3D flows and produced differentiable velocity and pressure surrogates for estimating quantities such as shear stresses and vorticity.
Takeaways & Limitations
Passive-scalar visualizations can provide quantitative flow information, including shear stresses, in settings where direct measurements may be difficult or invasive.
Takeaways & Limitations
The methodology avoids chaotic and turbulent Navier-Stokes regimes because optimization may fail to converge when computing the required derivatives.
Abstract
from arXiv · showhide
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.
1. Introduction
The paper frames fluid-flow inference as an inverse data-assimilation problem: recovering velocity and pressure from passive-scalar visualizations when conventional measurements or fully specified models are limited. It proposes using Navier–Stokes-informed deep learning to infer these hidden quantities in complex flow settings.
- Experimental inference of fluid velocity and pressure is difficult, while forward simulations require precisely defined geometry and initial and boundary conditions.
- Passive scalars such as smoke and dye provide visual information about transport in applications including aerodynamics, biofluid mechanics, and vascular flow.
- The proposed approach uses Navier–Stokes equations within deep neural networks to infer latent velocity and pressure fields from passive-scalar data.
- The method targets passive-scalar transport by incompressible Newtonian flows in both unbounded external geometries and confined internal flows.
2. Problem Setup and Solution Methodology
HFM formulates passive-scalar assimilation as a coupled neural-network problem constrained by transport, Navier–Stokes, and continuity equations. It uses automatic differentiation and complementary scalar information to infer hidden flow fields without requiring the training boundaries to coincide with physical boundaries.
- The normalized passive-scalar concentration satisfies an advection–diffusion equation governed by velocity and the Péclet number.
- The velocity and pressure fields satisfy nondimensional Navier–Stokes momentum equations together with incompressibility.
- The method assumes noisy, scattered concentration observations and infers the hidden velocity components and pressure while remaining agnostic to geometry and initial or boundary conditions.
- An auxiliary variable d := 1 − c complements the passive scalar, helping the algorithm detect geometry and corresponding boundary conditions.
- A deep network represents c, d, u, v, w, and p, while automatic differentiation supplies residual derivatives for six physics-informed equations.
- Using d can make training boundaries differ from physical boundaries and improve prediction accuracy without additional cost.
- The squared-error loss combines concentration-data errors for c and d with residual penalties enforcing the governing equations at selected measurement points.
3. Results
Across external, internal, and biomedical flows, HFM reconstructs velocity and pressure fields from passive-scalar data, even in arbitrary or boundary-poor training domains. These reconstructions support accurate force and wall-shear estimates, while errors increase near sparsely observed initial and final times.
- External flows: Velocity and pressure fields are accurately reconstructed in a small domain containing the cylinder without imposing the cylinder-wall no-slip condition.Only the left-boundary velocity is prescribed; normal concentration gradients provide information needed to infer the wall condition.
- External flows: Lift and drag predictions agree excellently with exact values in both oscillation frequency and amplitude, with error within 1% of test data.Discrepancies occur mainly at initial and final times because concentration data are unavailable outside 0 ≤ t ≤ 16.
- External flows: The method accurately reconstructs velocity and pressure in 3D cylinder flow without velocity observations on the training-domain boundaries.3D velocity and pressure errors are slightly higher than in 2D, and insufficient data near the initial and final times produces larger errors.
- Internal flows: In internal-flow benchmarks, predictions agree excellently with numerical estimates, including smooth wall-shear-stress distributions and learned Reynolds and Péclet numbers.The framework also infers fields in physiologic intracranial aneurysm flow using only scalar data inside the cropped aneurysm sac, with accurate analysis supported over 10 ≤ t ≤ 25.
4. Discussion and Concluding Remarks
The algorithm infers fluid velocity and pressure while remaining agnostic to geometry and initial or boundary conditions. Demonstrations support accurate predictions for complex 2D and 3D flows, but turbulence and realistic measurement conditions remain important boundaries.
- The algorithm is agnostic to geometry, initial conditions, and boundary conditions, allowing flexible selection of acquisition and prediction domains.
- It constructs computationally efficient, fully differentiable velocity and pressure surrogates that can estimate shear stresses and vorticity.
- Conventional alternatives require expensive, high-dimensional optimization involving parameterized conditions, repeated solvers, or solver backpropagation when geometry is known.
- The study avoids chaotic and turbulent Navier–Stokes regimes because derivative-based loss optimization may fail to converge there.
- The method produced accurate predictions for complex 2D and 3D flows in prototypical and realistic examples.
- The evaluation uses relatively clean synthetic passive-scalar data, while real applications must address 2D projections, modality-dependent resolution, and sparse temporal snapshots.