Physics-informed neural networks (PINNs) for fluid mechanics: A review

TL;DR

This review surveys physics-informed neural networks (PINNs) as a framework to fuse data with $NSE$-based flow models, addressing data assimilation, mesh-generation challenges, and inverse problems in fluid mechanics. It covers foundational PINN concepts, key methodological advances (domain-decomposition, multi-fidelity, and uncertainty quantification), and demonstrates three case studies: 3D incompressible wake reconstruction, 2D compressible flow inference, and thrombus-permeability estimation in biomedical flows. The results show PINNs can recover full 3D fields from sparse measurements, infer unknown parameters, and solve forward/inverse problems within a unified, mesh-free paradigm, albeit with current limitations in forward accuracy compared to high-order CFD. The authors discuss practical challenges (training non-convexity, data requirements) and outline future directions, including active flow control, transfer learning for high-Re flows, closure modeling, and scalable, GPU-accelerated PINN implementations for industrial-scale problems.

Abstract

Despite the significant progress over the last 50 years in simulating flow problems using numerical discretization of the Navier-Stokes equations (NSE), we still cannot incorporate seamlessly noisy data into existing algorithms, mesh-generation is complex, and we cannot tackle high-dimensional problems governed by parametrized NSE. Moreover, solving inverse flow problems is often prohibitively expensive and requires complex and expensive formulations and new computer codes. Here, we review flow physics-informed learning, integrating seamlessly data and mathematical models, and implementing them using physics-informed neural networks (PINNs). We demonstrate the effectiveness of PINNs for inverse problems related to three-dimensional wake flows, supersonic flows, and biomedical flows.

Figures (9)

• Figure 1: Schematic of a physics-informed neural network (PINN). A fully-connected neural network, with time and space coordinates ($t,\mathbf{x}$) as inputs, is used to approximate the multi-physics solutions $\hat{u}=[u,v,p,\phi]$. The derivatives of $\hat{u}$ with respect to the inputs are calculated using automatic differentiation (AD) and then used to formulate the residuals of the governing equations in the loss function, that is generally composed of multiple terms weighted by different coefficients. The parameters of the neural network $\theta$ and the unknown PDE parameters $\lambda$ can be learned simultaneously by minimizing the loss function.
• Figure 2: Case study of PINNs for incompressible flows: illustration of simulating the 3D wake flow over a circular cylinder. (a) Iso-surface of the vorticity (x-component) in the whole domain color-coded by the streamwise velocity. The cube with blue edges represents the computational domain in this case. (b) Velocity and pressure fields in the domain. The simulation was performed by the CFD solver Nektar, which is based on the spectral/ hp element method GK_CFDbook.
• Figure 3: Case study of PINNs for incompressible flows: problem setup for 3D flow reconstruction from 2D2C observations. (a) Case 1: two x-planes ($x=1.5, 7.5$), one y-plane ($y=0$) and two z-planes ($z=4.0,9.0$) are observed. (b) Case 2: two x-planes ($x=1.5, 7.5$), one y-plane ($y=0$) and one z-plane ($z=6.4$) are observed. (c) Case 3: one x-plane ($x=1.5$), one y-plane ($y=0$) and one z-plane ($z=6.4$) are observed. Note that for the cross-planes, only the projected vectors are measured. The goal is to infer the 3D flow in the investigated domain using PINNs from these 2D2C observations.
• Figure 4: Case study of PINNs for incompressible flows: relative $L_2$-norm errors of velocities and pressure for different flow reconstruction setups. These three cases correspond to those shown in Fig. \ref{['fig:incomp_setup']}. The errors are computed over the entire investigated domain.
• Figure 5: Case study of PINNs for incompressible flows: inference result of PINNs for Case 2. (a) Iso-surfaces of vorticity magnitude (top) and pressure (bottom) at $t=8.0$ from CFD data. (b) Iso-surfaces of vorticity magnitude (top) and pressure (bottom) at $t=8.0$ inferred by PINNs. (c) Point measurement $(x=3,y=0,z=6.4)$ of velocity $(u,v)$ against time. In this case, the 3D flow is inferred by PINNs from four cross-planes.