Despite great progress in simulating multiphysics problems using the numerical discretization of partial differential equations (PDEs), one still cannot seamlessly incorporate noisy data into existing algorithms, mesh generation remains complex, and high-dimensional problems governed by parameterized PDEs cannot be tackled. Moreover, solving inverse problems with hidden physics is often prohibitively expensive and requires different formulations and elaborate computer codes. Machine learning has emerged as a promising alternative, but training deep neural networks requires big data, not always available for scientific problems. Instead, such networks can be trained from additional information obtained by enforcing the physical laws (for example, at random points in the continuous space-time domain). Such physics-informed learning integrates (noisy) data and mathematical models, and implements them through neural networks or other kernel-based regression networks. Moreover, it may be possible to design specialized network architectures that automatically satisfy some of the physical invariants for better accuracy, faster training and improved generalization. Here, we review some of the prevailing trends in embedding physics into machine learning, present some of the current capabilities and limitations and discuss diverse applications of physics-informed learning both for forward and inverse problems, including discovering hidden physics and tackling high-dimensional problems. The rapidly developing field of physics-informed learning integrates data and mathematical models seamlessly, enabling accurate inference of realistic and high-dimensional multiphysics problems. This Review discusses the methodology and provides diverse examples and an outlook for further developments.

Physics-informed machine learning

\n Physics-informed neural networks (PINNs) have gained popularity across different engineering fields due to their effectiveness in solving realistic problems with noisy data and often partially missing physics. In PINNs, automatic differentiation is leveraged to evaluate differential operators without discretization errors, and a multitask learning problem is defined in order to simultaneously fit observed data while respecting the underlying governing laws of physics. Here, we present applications of PINNs to various prototype heat transfer problems, targeting in particular realistic conditions not readily tackled with traditional computational methods. To this end, we first consider forced and mixed convection with unknown thermal boundary conditions on the heated surfaces and aim to obtain the temperature and velocity fields everywhere in the domain, including the boundaries, given some sparse temperature measurements. We also consider the prototype Stefan problem for two-phase flow, aiming to infer the moving interface, the velocity and temperature fields everywhere as well as the different conductivities of a solid and a liquid phase, given a few temperature measurements inside the domain. Finally, we present some realistic industrial applications related to power electronics to highlight the practicality of PINNs as well as the effective use of neural networks in solving general heat transfer problems of industrial complexity. Taken together, the results presented herein demonstrate that PINNs not only can solve ill-posed problems, which are beyond the reach of traditional computational methods, but they can also bridge the gap between computational and experimental heat transfer.

Physics-Informed Neural Networks for Heat Transfer Problems

Parallel physics-informed neural networks via domain decomposition

The velocities measured by particle image velocimetry (PIV) and particle tracking velocimetry (PTV) commonly provide sparse information on flow motions. A dense velocity field with high resolution is indispensable for data visualization and analysis. In the present work, a physics-informed neural network (PINN) is proposed to reconstruct the dense velocity field from sparse experimental data. A PINN is a network-based data assimilation method. Within the PINN, both the velocity and pressure are approximated by minimizing a loss function consisting of the residuals of the data and the Navier–Stokes equations. Therefore, the PINN can not only improve the velocity resolution but also predict the pressure field. The performance of the PINN is investigated using two-dimensional (2D) Taylor's decaying vortices and turbulent channel flow with and without measurement noise. For the case of 2D Taylor's decaying vortices, the activation functions, optimization algorithms, and some parameters of the proposed method are assessed. For the case of turbulent channel flow, the ability of the PINN to reconstruct wall-bounded turbulence is explored. Finally, the PINN is applied to reconstruct dense velocity fields from the experimental tomographic PIV (Tomo-PIV) velocity in the three-dimensional wake flow of a hemisphere. The results indicate that the proposed PINN has great potential for extending the capabilities of PIV/PTV. 

Dense velocity reconstruction from particle image velocimetry/particle tracking velocimetry using a physics-informed neural network

Physics-informed neural networks (PINNs) are successful machine-learning methods for the solution and identification of partial differential equations. We employ PINNs for solving the Reynolds-averaged Navier–Stokes equations for incompressible turbulent flows without any specific model or assumption for turbulence and by taking only the data on the domain boundaries. We first show the applicability of PINNs for solving the Navier–Stokes equations for laminar flows by solving the Falkner–Skan boundary layer. We then apply PINNs for the simulation of four turbulent-flow cases, i.e., zero-pressure-gradient boundary layer, adverse-pressure-gradient boundary layer, and turbulent flows over a NACA4412 airfoil and the periodic hill. Our results show the excellent applicability of PINNs for laminar flows with strong pressure gradients, where predictions with less than 1% error can be obtained. For turbulent flows, we also obtain very good accuracy on simulation results even for the Reynolds-stress components. 

Physics-informed neural networks for solving Reynolds-averaged Navier–Stokes equations

Tomographic background oriented schlieren (Tomo-BOS) imaging measures density or temperature fields in 3D using multiple camera BOS projections, and is particularly useful for instantaneous flow visualizations of complex fluid dynamics problems. We propose a new method based on physics-informed neural networks (PINNs) to infer the full continuous 3D velocity and pressure fields from snapshots of 3D temperature fields obtained by Tomo-BOS imaging. PINNs seamlessly integrate the underlying physics of the observed fluid flow and the visualization data, hence enabling the inference of latent quantities using limited experimental data. In this hidden fluid mechanics paradigm, we train the neural network by minimizing a loss function composed of a data mismatch term and residual terms associated with the coupled Navier-Stokes and heat transfer equations. We first quantify the accuracy of the proposed method based on a 2D synthetic data set for buoyancy-driven flow, and subsequently apply it to the Tomo-BOS data set, where we are able to infer the instantaneous velocity and pressure fields of the flow over an espresso cup based only on the temperature field provided by the Tomo-BOS imaging. Moreover, we conduct an independent PIV experiment to validate the PINN inference for the unsteady velocity field at a center plane. To explain the observed flow physics, we also perform systematic PINN simulations at different Reynolds and Richardson numbers and quantify the variations in velocity and pressure fields. The results in this paper indicate that the proposed deep learning technique can become a promising direction in experimental fluid mechanics.

Flow over an espresso cup: Inferring 3D velocity and pressure fields from tomographic background oriented schlieren videos via physics-informed neural networks

Ice scallops are a small-scale (5–20 cm) quasi-periodic ripple pattern that occurs at the ice–water interface. Previous work has suggested that scallops form due to a self-reinforcing interaction between an evolving ice-surface geometry, an adjacent turbulent flow field and the resulting differential melt rates that occur along the interface. In this study, we perform a series of laboratory experiments in a refrigerated flume to quantitatively investigate the mechanisms of scallop formation and evolution in high resolution. Using particle image velocimetry, we probe an evolving ice–water boundary layer at sub-millimetre scales and 15 Hz frequency. Our data reveal three distinct regimes of ice–water interface evolution: a transition from flat to scalloped ice; an equilibrium scallop geometry; and an adjusting scallop interface. We find that scalloped-ice geometry produces a clear modification to the ice–water boundary layer, characterized by a time-mean recirculating eddy feature that forms in the scallop trough. Our primary finding is that scallops form due to a self-reinforcing feedback between the ice-interface geometry and shear production of turbulent kinetic energy in the flow interior. The length of this shear production zone is therefore hypothesized to set the scallop wavelength.

/pdf/ice-scallops-a-laboratory-investigation-of-the-ice-water-4mzgvl474t.pdf

Ice Scallops: A Laboratory Investigation of the Ice-Water Interface.

Tomographic background oriented Schlieren (Tomo-BOS) imaging measures density or temperature fields in three dimensions using multiple camera BOS projections, and is particularly useful for instantaneous flow visualizations of complex fluid dynamics problems. We propose a new method based on physics-informed neural networks (PINNs) to infer the full continuous three-dimensional (3-D) velocity and pressure fields from snapshots of 3-D temperature fields obtained by Tomo-BOS imaging. The PINNs seamlessly integrate the underlying physics of the observed fluid flow and the visualization data, hence enabling the inference of latent quantities using limited experimental data. In this hidden fluid mechanics paradigm, we train the neural network by minimizing a loss function composed of a data mismatch term and residual terms associated with the coupled Navier–Stokes and heat transfer equations. We first quantify the accuracy of the proposed method based on a two-dimensional synthetic data set for buoyancy-driven flow, and subsequently apply it to the Tomo-BOS data set, where we are able to infer the instantaneous velocity and pressure fields of the flow over an espresso cup based only on the temperature field provided by the Tomo-BOS imaging. Moreover, we conduct an independent PIV experiment to validate the PINN inference for the unsteady velocity field at a centre plane. To explain the observed flow physics, we also perform systematic PINN simulations at different Reynolds and Richardson numbers and quantify the variations in velocity and pressure fields. The results in this paper indicate that the proposed deep learning technique can become a promising direction in experimental fluid mechanics.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/160E4A836637FE3996610389666DA030/S002211202100135Xa.pdf/div-class-title-flow-over-an-espresso-cup-inferring-3-d-velocity-and-pressure-fields-from-tomographic-background-oriented-schlieren-via-physics-informed-neural-networks-div.pdf

Flow over an espresso cup: inferring 3-D velocity and pressure fields from tomographic background oriented Schlieren via physics-informed neural networks

Callum Gray

Papers

Flow over an espresso cup: Inferring 3D velocity and pressure fields from tomographic background oriented schlieren videos via physics-informed neural networks

Ice Scallops: A Laboratory Investigation of the Ice-Water Interface.

Flow over an espresso cup: inferring 3-D velocity and pressure fields from tomographic background oriented Schlieren via physics-informed neural networks