Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data (1808.04327v1)

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.

• The paper introduces a Hidden Fluid Mechanics framework that integrates Navier-Stokes equations into a deep learning model to reconstruct unobserved flow fields without explicit boundary conditions.
• The methodology harnesses physics-informed neural networks and passive scalar data to accurately infer velocity, pressure fields, and aerodynamic parameters.
• Robust numerical experiments on canonical problems demonstrate its reliability for applications in aerodynamic design and non-invasive biomedical diagnostics.

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

The paper "Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data" introduces an innovative approach to fluid mechanics by employing deep learning frameworks to assimilate data and infer hidden flow quantities. The proposed framework, termed Hidden Fluid Mechanics (HFM), leverages the Navier-Stokes equations to predict unobserved velocity and pressure fields from spatio-temporal data of passive scalars, such as dye or smoke, within fluid systems. By doing so, this method bypasses the need for explicit boundary and initial conditions, which traditional computational fluid dynamics (CFD) methods rely heavily upon.

Key Methodologies

• Physics Informed Neural Networks (PINNs): The paper builds upon the concept of PINNs, which integrate physical laws into deep learning models. By encoding the Navier-Stokes equations into the neural network architectures, the authors ensure that the predictions adhere to the fundamental laws of fluid dynamics.
• Data Assimilation via Passive Scalars: The methodology exploits the transport equation of passive scalars to reconstruct the velocity and pressure fields. This approach uniquely enables predictions in complex domains, such as biological flows in arteries, where direct measurement could be invasive or impractical.
• Automatic Differentiation: The authors utilized automatic differentiation via TensorFlow to efficiently compute derivatives necessary for enforcing the Navier-Stokes informed neural networks framework. This method maintains computational efficiency and precision.

Numerical Results and Implications

The paper presents robust numerical experiments on several benchmark problems, illustrating the capability of HFM to accurately predict velocity and pressure fields in both two and three-dimensional flows. Importantly, results show that this method can estimate quantities of interest such as lift and drag forces, and wall shear stress, which are significant in applications like aerodynamic design and biomedical diagnostics.

• Benchmark Validation: The framework is validated against canonical problems like flow past a cylinder and a stenotic vessel model. The results are compared with high-fidelity spectral element solutions showing excellent agreement, with relative errors remaining minimal.
• Parameter Estimation: Besides the flow field reconstruction, the paper demonstrates the framework's capability to estimate physical parameters such as Reynolds and Peclet numbers as latent variables, underscoring the model's potential in settings where these parameters may not be directly measurable.

Practical and Theoretical Implications

The paper's findings propose significant advancements in fluid dynamics modeling and biomedical applications. The methodology could transform CFD by reducing dependencies on precise initial and boundary conditions and geometrical configurations. More practically, this approach offers novel pathways in applications where traditional measurements are challenging, such as cardiovascular diagnostics through non-invasive imaging modalities.

Future Directions

The paper lays out future research paths, including tackling turbulent and chaotic flow regimes where the optimization landscape might become complex due to the inherent nature of turbulence. Another intriguing prospective endeavor involves applying this methodology to non-Newtonian fluid flows, which are prevalent in biological and industrial contexts.

In essence, "Hidden Fluid Mechanics" bridges data-driven and physics-based modeling objectives, advancing the simulation and comprehension of complex fluid flow phenomena beyond conventional methods. As the development of PINNs progresses, this framework could potentially reshape numerous scientific and engineering disciplines reliant on fluid dynamics.

Overall, the marriage of traditional fluid mechanics principles with contemporary machine learning tools as explored in this paper presents a promising avenue for both theoretical exploration and practical application in a wide array of domains where fluid flow plays a crucial role.