Back

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

Shengze CaiZhiping MaoZhicheng WangMinglang YinGeorge Em Karniadakis

article2021Acta Mechanica Sinica2,175 citations

Explains how physics-informed neural networks integrate Navier-Stokes equations with observational data to overcome mesh-generation bottlenecks and solve complex inverse problems in supersonic, biomedical, and three-dimensional wake flows.

Brown UniversityCenter for Biomedical EngineeringDalian University of TechnologyDivision of Applied MathematicsLaboratory of Ocean Energy Utilization of Ministry of EducationSchool of EngineeringXiamen Universityinverse problemsneural networksnumerical solutionPhysics-informed neural networksfluid mechanicscomputational fluid dynamics
arXiv: 2105.09506
Cover for Physics-informed neural networks (PINNs) for fluid mechanics: a review

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.

Topics

inverse problemsneural networksnumerical solution

Keywords

Physics-informed neural networksfluid mechanicscomputational fluid dynamicsNavier-Stokes equationsflow controlsurrogate modelingadaptive activation functionsturbulent flowCFD solvers

Table of Contents

  • 1 Introduction
  • 2 PINNs: Physics-Informed Neural Networks
  • 2.1 PINNs: Basic Concepts
  • 2.2 Recent Advances of PINNs
  • 3 Case Study for 3D Incompressible Flows
  • 3.1 Problem setup
  • 3.2 Implementation of PINNs
  • 3.3 Inference results
  • 4 Case Study for Compressible Flows
  • 4.1 Problem setup
  • 4.2 Inference results
  • 5 Case Study for Biomedical Flows
  • 5.1 PINNs
  • 5.2 Problem setup
  • 5.3 Inference Results
  • 6 Summary
  • References

Knowls

  1. Knowl 1 — Physics-Informed Neural Networks Formulation for Fluid Mechanics

    model/method

    Physics-Informed Neural Networks (PINNs) solve forward and inverse partial differential equation (PDE) problems in fluid mechanics by approximating the unknown field variables using a fully-connected deep neural network and embedding governing physical laws directly into the optimization loss function.

    For a general parametrized PDE system defined on spatial domain Ω⊂Rd\Omega \subset \mathbb{R}^d and time domain [0,T][0, T]: f(x,t,u^,∂xu^,∂tu^,… ;λ)=0,x∈Ω,  t∈[0,T]f(\mathbf{x}, t, \hat{\mathbf{u}}, \partial_{\mathbf{x}}\hat{\mathbf{u}}, \partial_t \hat{\mathbf{u}}, \dots; \boldsymbol{\lambda}) = 0, \quad \mathbf{x} \in \Omega, \; t \in [0, T] subject to initial conditions u^(x,t0)=g0(x)\hat{\mathbf{u}}(\mathbf{x}, t_0) = \mathbf{g}_0(\mathbf{x}) and boundary conditions u^(x,t)=gΓ(t)\hat{\mathbf{u}}(\mathbf{x}, t) = \mathbf{g}_{\Gamma}(t) on ∂Ω\partial \Omega, where λ=[λ1,λ2,… ]\boldsymbol{\lambda} = [\lambda_1, \lambda_2, \dots] are PDE parameters and ff represents the PDE residual.

    A feed-forward neural network maps space-time coordinates (x,t)(\mathbf{x}, t) to predicted state variables u^(x,t)\hat{\mathbf{u}}(\mathbf{x}, t) through LL layers: z0=(x,t)\mathbf{z}^0 = (\mathbf{x}, t) zk=σ(Wkzk−1+bk),1≤k≤L−1\mathbf{z}^k = \sigma\left(\mathbf{W}^k \mathbf{z}^{k-1} + \mathbf{b}^k\right), \quad 1 \le k \le L-1 u^≈zL=WLzL−1+bL\hat{\mathbf{u}} \approx \mathbf{z}^L = \mathbf{W}^L \mathbf{z}^{L-1} + \mathbf{b}^L where Wk\mathbf{W}^k and bk\mathbf{b}^k denote the weight matrices and bias vectors collectively parameterized by θ\boldsymbol{\theta}, and σ(⋅)\sigma(\cdot) is a nonlinear activation function. Partial derivatives of u^\hat{\mathbf{u}} with respect to input coordinates (x,t)(\mathbf{x}, t) are computed via exact automatic differentiation (AD) using the chain rule on the computational graph, eliminating spatial discretization grids and truncation errors.

    The composite loss function L(θ,λ)\mathcal{L}(\boldsymbol{\theta}, \boldsymbol{\lambda}) is minimized over the parameters θ\boldsymbol{\theta} (and unknown physics parameters λ\boldsymbol{\lambda} in inverse problems): L=ω1LPDE+ω2Ldata+ω3LIC+ω4LBC\mathcal{L} = \omega_1 \mathcal{L}_{\text{PDE}} + \omega_2 \mathcal{L}_{\text{data}} + \omega_3 \mathcal{L}_{\text{IC}} + \omega_4 \mathcal{L}_{\text{BC}} where ω1−4\omega_{1-4} are user-defined or adaptive weighting coefficients, and each loss term is typically evaluated as a mean squared error (L2L_2-norm) over sampling collocation points: LPDE=1Nf∑i=1Nf∥f(xif,tif;θ,λ)∥2\mathcal{L}_{\text{PDE}} = \frac{1}{N_f} \sum_{i=1}^{N_f} \|f(\mathbf{x}_i^f, t_i^f; \boldsymbol{\theta}, \boldsymbol{\lambda})\|^2 Ldata=1Ndata∑i=1Ndata∥u^(xid,tid)−uid∥2\mathcal{L}_{\text{data}} = \frac{1}{N_{\text{data}}} \sum_{i=1}^{N_{\text{data}}} \|\hat{\mathbf{u}}(\mathbf{x}_i^d, t_i^d) - \mathbf{u}_i^d\|^2 LIC=1NIC∑i=1NIC∥u^(xiIC,t0)−g0(xiIC)∥2\mathcal{L}_{\text{IC}} = \frac{1}{N_{\text{IC}}} \sum_{i=1}^{N_{\text{IC}}} \|\hat{\mathbf{u}}(\mathbf{x}_i^{\text{IC}}, t_0) - \mathbf{g}_0(\mathbf{x}_i^{\text{IC}})\|^2 LBC=1NBC∑i=1NBC∥u^(xiBC,tiBC)−gΓ(tiBC)∥2\mathcal{L}_{\text{BC}} = \frac{1}{N_{\text{BC}}} \sum_{i=1}^{N_{\text{BC}}} \|\hat{\mathbf{u}}(\mathbf{x}_i^{\text{BC}}, t_i^{\text{BC}}) - \mathbf{g}_{\Gamma}(t_i^{\text{BC}})\|^2

  2. Knowl 2 — 3D Flow Reconstruction Formulation from Planar 2D2C Velocity Data

    model/method

    In 3D incompressible flow fields where experimental diagnostics (such as planar Particle Image Velocimetry, PIV) only provide two-dimensional, two-component (2D2C) velocity measurements on isolated intersecting planes, PINNs can infer the full 3D velocity field u=(u,v,w)\mathbf{u} = (u, v, w) and the pressure field pp throughout the continuous space-time volume without requiring initial or boundary condition specifications.

    The neural network takes (x,y,z,t)(x, y, z, t) as inputs and outputs (u,v,w,p)(u, v, w, p). The total loss function is: L=Ldata+LPDE\mathcal{L} = \mathcal{L}_{\text{data}} + \mathcal{L}_{\text{PDE}}

    The data loss accounts only for the specific velocity components measured on each cross-sectional plane: Ldata=1Nu∑i=1Nu∥u(xdatai,tdatai)−udatai∥2+1Nv∑i=1Nv∥v(xdatai,tdatai)−vdatai∥2+1Nw∑i=1Nw∥w(xdatai,tdatai)−wdatai∥2\mathcal{L}_{\text{data}} = \frac{1}{N_u} \sum_{i=1}^{N_u} \|u(\mathbf{x}_{\text{data}}^i, t_{\text{data}}^i) - u_{\text{data}}^i\|^2 + \frac{1}{N_v} \sum_{i=1}^{N_v} \|v(\mathbf{x}_{\text{data}}^i, t_{\text{data}}^i) - v_{\text{data}}^i\|^2 + \frac{1}{N_w} \sum_{i=1}^{N_w} \|w(\mathbf{x}_{\text{data}}^i, t_{\text{data}}^i) - w_{\text{data}}^i\|^2 where on an xx-plane only (v,w)(v, w) are known, on a yy-plane only (u,w)(u, w) are known, and on a zz-plane only (u,v)(u, v) are known (orthogonal components and pressure remain completely unmeasured).

    The physical residual loss penalizes deviations from the 3D dimensionless incompressible Navier-Stokes equations at NfN_f sampled space-time collocation points: LPDE=1Nf∑i=1Nf∑j=14∥fj(xfi,tfi)∥2\mathcal{L}_{\text{PDE}} = \frac{1}{N_f} \sum_{i=1}^{N_f} \sum_{j=1}^4 \|f_j(\mathbf{x}_f^i, t_f^i)\|^2 f1,2,3=∂u∂t+(u⋅∇)u+∇p−1Re∇2uf_{1, 2, 3} = \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} + \nabla p - \frac{1}{Re} \nabla^2 \mathbf{u} f4=∇⋅uf_4 = \nabla \cdot \mathbf{u} where ReRe is the Reynolds number.

  3. Knowl 3 — Setup for 3D Wake Flow Inference Behind a Cylinder from 2D Cross-Sections

    experimental setup

    The 3D wake flow behind a circular cylinder of unit diameter at Reynolds number Re=200Re = 200 is simulated numerically on a full domain Ω:[−7.5,28.5]×[−20,20]×[0,12.5]\Omega: [-7.5, 28.5] \times [-20, 20] \times [0, 12.5] with uniform inflow u=1u = 1, periodic lateral boundaries at y=±20y = \pm 20, zero outlet pressure at x=28.5x = 28.5, and no-slip cylinder wall conditions.

    The PINN reconstruction focuses on an unsteady wake sub-domain Ωs:[1.5,7.5]×[−3,3]×[4,9]\Omega_s: [1.5, 7.5] \times [-3, 3] \times [4, 9] over 50 time snapshots with time step Δt=0.2\Delta t = 0.2 (t∈[0,10]t \in [0, 10], covering approximately two vortex shedding periods). Three observation configurations are evaluated:

    • Case 1 (5 cross-planes): two xx-planes (x=1.5,7.5x = 1.5, 7.5), one yy-plane (y=0y = 0), two zz-planes (z=4.0,9.0z = 4.0, 9.0).
    • Case 2 (4 cross-planes): two xx-planes (x=1.5,7.5x = 1.5, 7.5), one yy-plane (y=0y = 0), one zz-plane (z=6.4z = 6.4).
    • Case 3 (3 cross-planes): one xx-plane (x=1.5x = 1.5), one yy-plane (y=0y = 0), one zz-plane (z=6.4z = 6.4).

    The 2D2C measurement grid resolutions on each cross-plane are 61×2661 \times 26 points on xx-planes measuring (v,w)(v, w), 61×2661 \times 26 points on yy-planes measuring (u,w)(u, w), and 61×6161 \times 61 points on zz-planes measuring (u,v)(u, v).

    The neural network consists of 8 hidden layers with 200 neurons per layer using σ(x)=sin⁡(x)\sigma(x) = \sin(x) activation. Training is executed using the Adam optimizer with a batch size of 10410^4 for both data points and residual collocation points (Nf=3×106N_f = 3 \times 10^6 total residual points sampled across Ωs×[0,10]\Omega_s \times [0, 10]). The learning rate starts at 10−310^{-3} for epochs 1–50, decays to 5×10−45 \times 10^{-4} for epochs 51–100, and reaches 10−410^{-4} for epochs 101–150.

  4. Knowl 4 — Empirical Accuracy of 3D Wake Flow and Pressure Field Reconstruction

    empirical result

    Reconstruction accuracy is quantified over the continuous space-time sub-domain using the relative L2L_2-norm error: εV=∥VCFD−V^∥2∥VCFD∥2×100%,V∈{u,v,w,p}\varepsilon_V = \frac{\|V_{\text{CFD}} - \hat{V}\|_2}{\|V_{\text{CFD}}\|_2} \times 100\%, \quad V \in \{u, v, w, p\} where VCFDV_{\text{CFD}} is the high-order spectral/hp element reference simulation and V^\hat{V} is the PINN prediction.

    • In Case 1 (5 planes) and Case 2 (4 planes), the PINN achieves close agreement with reference CFD data for all velocity components and pressure across the domain.
    • In Case 3 (3 planes), errors increase but remain low; the relative error in streamwise velocity uu stays below 2%2\% throughout most of the domain.
    • Across all cases, the relative error for the spanwise velocity ww is larger than for uu and vv due to the smaller absolute magnitude of ww.
    • Temporal error profiles display localized increases near the initial (t=0t = 0) and final (t=10t = 10) temporal boundaries because no initial condition is enforced, leaving temporal derivatives constrained only by interior data points.

  5. Knowl 5 — Compressible Euler PINN with Density Gradient and Global Conservation Constraints

    model/method

    For steady 2D compressible high-speed flows, PINNs reconstruct complete thermodynamic and kinematic fields from Schlieren-inspired density gradients ∇ρ\nabla \rho and sparse surface pressure measurements without requiring standard outflow or shock boundary conditions.

    The 2D steady compressible Euler equations are: ∇⋅f(U)=0,x∈Ω⊂R2\nabla \cdot \mathbf{f}(\mathbf{U}) = 0, \quad \mathbf{x} \in \Omega \subset \mathbb{R}^2 U=[ρ,ρu,ρv,ρE]T,f=(G1,G2)\mathbf{U} = [\rho, \rho u, \rho v, \rho E]^T, \quad \mathbf{f} = (\mathbf{G}_1, \mathbf{G}_2) G1(U)=[ρu, p+ρu2, ρuv, (p+ρE)u]T\mathbf{G}_1(\mathbf{U}) = [\rho u, \, p + \rho u^2, \, \rho u v, \, (p + \rho E)u]^T G2(U)=[ρv, ρuv, p+ρv2, (p+ρE)v]T\mathbf{G}_2(\mathbf{U}) = [\rho v, \, \rho u v, \, p + \rho v^2, \, (p + \rho E)v]^T closed by the polytropic gas equation of state: p=(γ−1)(ρE−12ρ∥u∥2)p = (\gamma - 1)\left(\rho E - \frac{1}{2}\rho \|\mathbf{u}\|^2\right) where ρ\rho is density, pp is static pressure, u=(u,v)\mathbf{u} = (u, v) is the velocity vector, EE is total specific energy, and γ\gamma is the adiabatic index.

    The PINN loss function combines sub-domain density gradients, surface pressure, inflow conditions, surface tangency, and domain-wide integral conservation laws: L=ω1LF+ω2L∇ρ∣D+ω3Linflow+ω4Lp∗+ω5(LMass+LMomentum+LEnergy)+ω6Ln⋅u\mathcal{L} = \omega_1 \mathcal{L}_F + \omega_2 \mathcal{L}_{\nabla \rho|_D} + \omega_3 \mathcal{L}_{\text{inflow}} + \omega_4 \mathcal{L}_{p^*} + \omega_5 (\mathcal{L}_{\text{Mass}} + \mathcal{L}_{\text{Momentum}} + \mathcal{L}_{\text{Energy}}) + \omega_6 \mathcal{L}_{\mathbf{n} \cdot \mathbf{u}} where L∇ρ∣D\mathcal{L}_{\nabla \rho|_D} penalizes the difference from density gradient data in an experimental observation sub-domain D⊂ΩD \subset \Omega, Lp∗\mathcal{L}_{p^*} enforces measured wall surface pressure, Ln⋅u\mathcal{L}_{\mathbf{n} \cdot \mathbf{u}} enforces zero normal velocity on the obstacle surface, and LMass,LMomentum,LEnergy\mathcal{L}_{\text{Mass}}, \mathcal{L}_{\text{Momentum}}, \mathcal{L}_{\text{Energy}} penalize residual imbalances of the macroscopic conservation laws.

  6. Knowl 6 — Supersonic Bow Shock Flow Reconstruction Results

    empirical result

    A 2D supersonic bow shock wave past a blunt body is reconstructed using PINNs with free-stream inflow conditions: M∞=4,p∞=101253.6 Pa,ρ∞=1.225 kg/m3,u∞=1360.7 m/s,v∞=0,T∞=288 KM_\infty = 4, \quad p_\infty = 101253.6\,\text{Pa}, \quad \rho_\infty = 1.225\,\text{kg/m}^3, \quad u_\infty = 1360.7\,\text{m/s}, \quad v_\infty = 0, \quad T_\infty = 288\,\text{K}

    The network comprises 6 hidden layers with 60 neurons per layer, trained with layer-wise adaptive tanh⁡\tanh activation functions and dynamic loss weighting over 3×1053 \times 10^5 epochs using the Adam optimizer (learning rate 6×10−46 \times 10^{-4}).

    Given density gradients ∇ρ\nabla \rho in a localized sub-region D⊂ΩD \subset \Omega and surface pressure on the body, the trained PINN accurately recovers the full 2D detached bow shock location, continuous post-shock pressure distribution, and velocity field uu, matching reference CFD solutions without requiring numerical artificial viscosity or explicit shock-fitting boundaries.

  7. Knowl 7 — Coupled Navier-Stokes and Cahn-Hilliard System for Thrombus FSI

    equation

    The mechanical interaction between a deformable thrombus and surrounding blood flow is governed by a coupled fluid-structure interaction (FSI) system combining the incompressible Navier-Stokes equations with Darcy flow resistance and the Cahn-Hilliard phase-field model:

    ρ(∂u∂t+u⋅∇u)+∇p=∇⋅(σvis+σcoh)−μ(1−ϕ)u2κ(ϕ)\rho\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right) + \nabla p = \nabla \cdot (\boldsymbol{\sigma}_{\text{vis}} + \boldsymbol{\sigma}_{\text{coh}}) - \mu \frac{(1 - \phi)\mathbf{u}}{2\kappa(\phi)} ∇⋅u=0\nabla \cdot \mathbf{u} = 0 ∂ψ∂t+u⋅∇ψ=0\frac{\partial \boldsymbol{\psi}}{\partial t} + \mathbf{u} \cdot \nabla \boldsymbol{\psi} = \mathbf{0} ∂ϕ∂t+u⋅∇ϕ=τΔω\frac{\partial \phi}{\partial t} + \mathbf{u} \cdot \nabla \phi = \tau \Delta \omega ω=Δϕ+γg(ϕ)\omega = \Delta \phi + \gamma g(\phi)

    where u(x,t)\mathbf{u}(\mathbf{x}, t) is blood velocity, p(x,t)p(\mathbf{x}, t) is pressure, ϕ(x,t)\phi(\mathbf{x}, t) is the phase field (with ϕ=1\phi = 1 in pure fluid, ϕ=0\phi = 0 in the permeable shell, and ϕ=−1\phi = -1 in the clot core), and ψ=(ψ1,ψ2)\boldsymbol{\psi} = (\psi_1, \psi_2) is an auxiliary vector representing the deformation gradient tensor: F=[−∂ψ1∂y−∂ψ2∂y∂ψ1∂x∂ψ2∂x]\mathbf{F} = \begin{bmatrix} -\frac{\partial \psi_1}{\partial y} & -\frac{\partial \psi_2}{\partial y} \\[4pt] \frac{\partial \psi_1}{\partial x} & \frac{\partial \psi_2}{\partial x} \end{bmatrix}

    The constitutive stresses and potentials are: σvis=μ∇u\boldsymbol{\sigma}_{\text{vis}} = \mu \nabla \mathbf{u} σcoh=λ∇⋅(∇ϕ⊗∇ϕ)\boldsymbol{\sigma}_{\text{coh}} = \lambda \nabla \cdot (\nabla \phi \otimes \nabla \phi) g(ϕ)=(ϕ2−1)ϕh2g(\phi) = \frac{(\phi^2 - 1)\phi}{h^2} where μ\mu is fluid dynamic viscosity, ρ\rho is fluid density, λ\lambda is mixing energy density, τ\tau is relaxation mobility, γ\gamma is interfacial parameter, hh is interfacial thickness, and κ(ϕ)\kappa(\phi) is the phase-dependent permeability parameterized by: κ(ϕ)=eaϕ+b\kappa(\phi) = e^{a\phi} + b

  8. Knowl 8 — Dual-Network PINN Architecture for Biomedical Thrombus Parameter Inference

    model/method

    To infer the unknown nonhomogeneous permeability κ(ϕ)=eaϕ+b\kappa(\phi) = e^{a\phi} + b (parameters aa and bb) and the hidden velocity and pressure fields from observable phase-field data ϕ\phi, a dual neural network architecture is employed:

    • Net U: A fully-connected neural network taking (x,t)(\mathbf{x}, t) as input and outputting surrogate state variables (u^,v^,p^,ϕ^)(\hat{u}, \hat{v}, \hat{p}, \hat{\phi}).
    • Net W: A fully-connected neural network taking (x,t)(\mathbf{x}, t) as input and outputting auxiliary variables (ω^,ψ^1,ψ^2)(\hat{\omega}, \hat{\psi}_1, \hat{\psi}_2).

    Both networks comprise 9 hidden layers with 20 neurons per layer. The parameters of both networks θ\boldsymbol{\theta} and the PDE parameters λ=[a,b]\boldsymbol{\lambda} = [a, b] are optimized simultaneously by minimizing the total loss: L(θ,λ)=ω1LPDE+ω2LIC+ω3LBC+ω4Ldata\mathcal{L}(\boldsymbol{\theta}, \boldsymbol{\lambda}) = \omega_1 \mathcal{L}_{\text{PDE}} + \omega_2 \mathcal{L}_{\text{IC}} + \omega_3 \mathcal{L}_{\text{BC}} + \omega_4 \mathcal{L}_{\text{data}} LPDE(θ,λ)=1∣XPDE∣∑x∈XPDE∥f(x,∂tu^,∂xu^,… ;λ)∥22\mathcal{L}_{\text{PDE}}(\boldsymbol{\theta}, \boldsymbol{\lambda}) = \frac{1}{|X_{\text{PDE}}|} \sum_{\mathbf{x} \in X_{\text{PDE}}} \|f(\mathbf{x}, \partial_t \hat{\mathbf{u}}, \partial_{\mathbf{x}} \hat{\mathbf{u}}, \dots; \boldsymbol{\lambda})\|_2^2 LBC(θ,λ)=1∣XBC∣∑x∈XBC∥B(u^,x)∥22\mathcal{L}_{\text{BC}}(\boldsymbol{\theta}, \boldsymbol{\lambda}) = \frac{1}{|X_{\text{BC}}|} \sum_{\mathbf{x} \in X_{\text{BC}}} \|\mathcal{B}(\hat{\mathbf{u}}, \mathbf{x})\|_2^2 LIC(θ,λ)=1∣XIC∣∑x∈XIC∥u^−ut0∥22\mathcal{L}_{\text{IC}}(\boldsymbol{\theta}, \boldsymbol{\lambda}) = \frac{1}{|X_{\text{IC}}|} \sum_{\mathbf{x} \in X_{\text{IC}}} \|\hat{\mathbf{u}} - \mathbf{u}_{t_0}\|_2^2 Ldata(θ,λ)=1∣Xdata∣∑x∈Xdata∥ϕ^(x)−ϕdata(x)∥22\mathcal{L}_{\text{data}}(\boldsymbol{\theta}, \boldsymbol{\lambda}) = \frac{1}{|X_{\text{data}}|} \sum_{\mathbf{x} \in X_{\text{data}}} \|\hat{\phi}(\mathbf{x}) - \phi_{\text{data}}(\mathbf{x})\|_2^2 where B\mathcal{B} enforces Dirichlet velocity at the inlet, no-slip on channel walls, and homogeneous Neumann conditions ∂nϕ=∂nω=0\partial_n \phi = \partial_n \omega = 0 on all boundaries.

  9. Knowl 9 — Empirical Inference of Thrombus Permeability and Velocity Fields

    empirical result

    In a 2D channel flow past a semi-circular thrombus with an impermeable core (ϕ=−1\phi = -1, true permeability κ=0.001\kappa = 0.001) and a permeable outer shell (ϕ=0\phi = 0, true permeability κ=1.0\kappa = 1.0), the ground-truth permeability model parameters are a=6.90a = 6.90 and b=0.0b = 0.0.

    Using 10,000 phase-field measurement points sampled across 30 snapshots (t∈[0.03,0.93]t \in [0.03, 0.93]), 10,000 PDE interior points, 1,000 boundary points, and 1,000 initial points, the dual-network PINN trained over 300,000 epochs achieved:

    • Inferred model parameters: a=7.10a = 7.10 and b=0.0003b = 0.0003.
    • Inferred core permeability: κ(ϕ=−1)=0.0011\kappa(\phi = -1) = 0.0011 (reference: 0.00100.0010).
    • Inferred shell permeability: κ(ϕ=0)=1.0003\kappa(\phi = 0) = 1.0003 (reference: 1.00001.0000).

    The full velocity field u=(u,v)\mathbf{u} = (u, v) and pressure field pp throughout the thrombus and channel were accurately reconstructed using exclusively phase-field measurements ϕ\phi, with velocity errors strictly localized inside the permeable shell layer.

  10. Knowl 10 — Strengths and Limitations of PINNs vs. Traditional Computational Fluid Dynamics Solvers

    limitation

    Physics-informed neural networks exhibit distinct trade-offs when compared with classical CFD numerical discretization schemes (e.g., finite volume, finite element, and spectral/hp methods):

    • Forward Problem Limitations: For standard well-posed forward simulations with fully specified initial and boundary conditions, current PINNs are neither as accurate nor as computationally efficient as high-order CFD codes. This limitation arises from the non-convex, high-dimensional optimization landscape of deep neural network loss functions, which often encounters gradient pathologies and slow convergence.
    • Inverse and Data Assimilation Strengths: For inverse problems (identifying unknown physical parameters, material properties, or unmeasured boundary conditions), ill-posed systems, and sparse multi-fidelity/multimodal data assimilation, PINNs execute forward and inverse estimation within an identical, unified optimization framework. Unlike conventional CFD solvers, PINNs require no grid/mesh generation, avoid truncation errors in spatial derivatives via automatic differentiation, and seamlessly infer unobserved flow variables (such as 3D pressure and velocity fields) from sparse local observations.

Coverage note — All primary contributed case studies and formulations—including the general PINN framework for fluid mechanics, 3D incompressible flow reconstruction from 2D2C cross-sections, 2D compressible Euler bow shock flow inversion, biomedical thrombus FSI parameter inference, and solver capability comparisons—are fully captured. External background literature and survey references on domain decomposition (XPINNs, cPINNs) and general operator theoretical convergence bounds were omitted as they are citations to earlier work.

References

  1. 1.Brooks, A.N., Hughes, T.J.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering 32(1-3), 199–259 (1982)
  2. 2.Karniadakis, G.E., Sherwin, S.: Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd edition. Oxford University Press, Oxford,UK (2005)
  3. 3.Katz, A.J.: Meshless methods for computational fluid dynamics. Stanford University Stanford, CA (2009)
  4. 4.Liu, M., Liu, G.: Smoothed particle hydrodynamics (SPH): an overview and recent developments. Archives of Computational Methods in Engineering 17(1), 25–76 (2010)
  5. 5.Beck, J.V., Blackwell, B., Clair Jr, C.R.S.: Inverse heat conduction: Ill-posed problems. James Beck (1985)
  6. 6.Jasak, H., Jemcov, A., Tukovic, Z., et al.: OpenFOAM: A C++ library for complex physics simulations. In: International Workshop on Coupled Methods in Numerical Dynamics, vol. 1000, pp. 1–20. IUC Dubrovnik Croatia (2007)
  7. 7.Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics informed learning machine (2021). US Patent 10,963,540
  8. 8.Raissi, M., Perdikaris, P., Karniadakis, G.E.: Machine learning of linear differential equations using Gaussian processes. Journal of Computational Physics 348, 683–693 (2017)
  9. 9.Raissi, M., Perdikaris, P., Karniadakis, G.E.: Numerical Gaussian processes for time-dependent and nonlinear partial differential equations. SIAM Journal on Scientific Computing 40(1), A172–A198 (2018)
  10. 10.Raissi, M., Wang, Z., Triantafyllou, M.S., Karniadakis, G.E.: Deep learning of vortex-induced vibrations. Journal of Fluid Mechanics 861, 119–137 (2019)
  11. 11.Jin, X., Cai, S., Li, H., Karniadakis, G.E.: NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics 426, 109951 (2021)
  12. 12.Raissi, M., Yazdani, A., Karniadakis, G.E.: Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. Science 367(6481), 1026–1030 (2020)
  13. 13.Mao, Z., Jagtap, A.D., Karniadakis, G.E.: Physics-informed neural networks for high-speed flows. Computer Methods in Applied Mechanics and Engineering 360, 112789 (2020)
  14. 14.Yin, M., Zheng, X., Humphrey, J.D., Karniadakis, G.E.: Non-invasive inference of thrombus material properties with physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering 375, 113603 (2021)
  15. 15.Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019)
  16. 16.Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014)
  17. 17.Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., Devin, M., Ghemawat, S., Irving, G., Isard, M., et al.: Tensorflow: A system for large-scale machine learning. In: 12th {USENIX} symposium on operating systems design and implementation ({OSDI} 16), pp. 265–283 (2016)
  18. 18.Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. arXiv preprint arXiv:1912.01703 (2019)
  19. 19.Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations. arXiv preprint arXiv:1711.10561 (2017)
  20. 20.Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics informed deep learning (part ii): Data-driven discovery of nonlinear partial differential equations. arxiv. arXiv preprint arXiv:1711.10561 (2017)
  21. 21.Pang, G., Lu, L., Karniadakis, G.E.: fPINNs: Fractional physics-informed neural networks. SIAM Journal on Scientific Computing 41(4), A2603–A2626 (2019)
  22. 22.Fang, Z., Zhan, J.: A physics-informed neural network framework for PDEs on 3D surfaces: Time independent problems. IEEE Access 8, 26328–26335 (2019)
  23. 23.Zhang, D., Guo, L., Karniadakis, G.E.: Learning in modal space: Solving time-dependent stochastic PDEs using physics-informed neural networks. SIAM Journal on Scientific Computing 42(2), A639–A665 (2020)
  24. 24.Kharazmi, E., Zhang, Z., Karniadakis, G.E.: hp-VPINNs: Variational physics-informed neural networks with domain decomposition. Computer Methods in Applied Mechanics and Engineering 374, 113547 (2021)
  25. 25.Jagtap, A.D., Kharazmi, E., Karniadakis, G.E.: Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems. Computer Methods in Applied Mechanics and Engineering 365, 113028 (2020)
  26. 26.Jagtap, A.D., Karniadakis, G.E.: Extended Physics-Informed Neural Networks (XPINNs): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations. Communications in Computational Physics 28(5), 2002–2041 (2020)
  27. 27.Shukla, K., Jagtap, A.D., Karniadakis, G.E.: Parallel physics-informed neural networks via domain decomposition. arXiv preprint arXiv:2104.10013 (2021)
  28. 28.Hennigh, O., Narasimhan, S., Nabian, M.A., Subramaniam, A., Tangsali, K., Rietmann, M., Ferrandis, J.d.A., Byeon, W., Fang, Z., Choudhry, S.: NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation framework. arXiv preprint arXiv:2012.07938 (2020)
  29. 29.Yang, Y., Perdikaris, P.: Adversarial uncertainty quantification in physics-informed neural networks. Journal of Computational Physics 394, 136–152 (2019)
  30. 30.Zhang, D., Lu, L., Guo, L., Karniadakis, G.E.: Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems. Journal of Computational Physics 397, 108850 (2019)
  31. 31.Zhu, Y., Zabaras, N., Koutsourelakis, P.S., Perdikaris, P.: Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. Journal of Computational Physics 394, 56–81 (2019)
  32. 32.Sun, L., Wang, J.X.: Physics-constrained Bayesian neural network for fluid flow reconstruction with sparse and noisy data. Theoretical and Applied Mechanics Letters 10(3), 161–169 (2020)
  33. 33.Yang, L., Meng, X., Karniadakis, G.E.: B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data. Journal of Computational Physics 425, 109913 (2021)
  34. 34.Meng, X., Karniadakis, G.E.: A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems. Journal of Computational Physics 401, 109020 (2020)
  35. 35.Jagtap, A.D., Kawaguchi, K., Karniadakis, G.E.: Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. Journal of Computational Physics 404, 109136 (2020)
  36. 36.Wang, S., Teng, Y., Perdikaris, P.: Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536 (2020)
  37. 37.Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. arXiv preprint arXiv:2102.04626 (2021)
  38. 38.Gao, H., Sun, L., Wang, J.X.: PhyGeoNet: physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain. Journal of Computational Physics 428, 110079 (2021)
  39. 39.Shin, Y., Darbon, J., Karniadakis, G.E.: On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs. Communications in Computational Physics 28, 2042–2074 (2020)
  40. 40.Mishra, S., Molinaro, R.: Estimates on the generalization error of physics informed neural networks (PINNs) for approximating PDEs. arXiv preprint arXiv:2006.16144 (2020)
  41. 41.Wang, S., Yu, X., Perdikaris, P.: When and why PINNs fail to train: A neural tangent kernel perspective. arXiv preprint arXiv:2007.14527 (2020)
  42. 42.Kissas, G., Yang, Y., Hwuang, E., Witschey, W.R., Detre, J.A., Perdikaris, P.: Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering 358, 112623 (2020)
  43. 43.Yang, X., Zafar, S., Wang, J.X., Xiao, H.: Predictive large-eddy-simulation wall modeling via physics-informed neural networks. Physical Review Fluids 4(3), 034602 (2019)
  44. 44.Lou, Q., Meng, X., Karniadakis, G.E.: Physics-informed neural networks for solving forward and inverse flow problems via the Boltzmann-BGK formulation. arXiv preprint arXiv:2010.09147 (2020)
  45. 45.Cai, S., Wang, Z., Wang, S., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks for heat transfer problems. Journal of Heat Transfer 143(6), 060801 (2021)
  46. 46.Cai, S., Wang, Z., Fuest, F., Jeon, Y.J., Gray, C., Karniadakis, G.E.: Flow over an espresso cup: inferring 3-D velocity and pressure fields from tomographic background oriented Schlieren via physics-informed neural networks. Journal of Fluid Mechanics 915 (2021)
  47. 47.Wang, S., Perdikaris, P.: Deep learning of free boundary and stefan problems. Journal of Computational Physics 428, 109914 (2021)
  48. 48.Lucor, D., Agrawal, A., Sergent, A.: Physics-aware deep neural networks for surrogate modeling of turbulent natural convection. arXiv preprint arXiv:2103.03565 (2021)
  49. 49.Mahmoudabadbozchelou, M., Caggioni, M., Shahsavari, S., Hartt, W.H., Karniadakis, G.E., Jamali, S.: Data-driven physics-informed constitutive metamodeling of complex fluids: A multifidelity neural network (mfnn) framework. Journal of Rheology 65(2), 179–198 (2021)
  50. 50.Arzani, A., Wang, J.X., D’Souza, R.M.: Uncovering near-wall blood flow from sparse data with physics-informed neural networks. arXiv preprint arXiv:2104.08249 (2021)
  51. 51.Cai, S., Li, H., Zheng, F., Kong, F., Dao, M., Karniadakis, G.E., Suresh, S.: Artificial intelligence velocimetry and microaneurysm-on-a-chip for three-dimensional analysis of blood flow in physiology and disease. Proceedings of the National Academy of Sciences 118(13) (2021)
  52. 52.Wang, R., Kashinath, K., Mustafa, M., Albert, A., Yu, R.: Towards physics-informed deep learning for turbulent flow prediction. In: Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 1457–1466 (2020)
  53. 53.Goswami, S., Anitescu, C., Chakraborty, S., Rabczuk, T.: Transfer learning enhanced physics informed neural network for phase-field modeling of fracture. Theoretical and Applied Fracture Mechanics 106, 102447 (2020)
  54. 54.Zhang, E., Yin, M., Karniadakis, G.E.: Physics-informed neural networks for nonhomogeneous material identification in elasticity imaging. arXiv preprint arXiv:2009.04525 (2020)
  55. 55.Wufsus, A., Macera, N., Neeves, K.: The hydraulic permeability of blood clots as a function of fibrin and platelet density. Biophysical journal 104(8), 1812–1823 (2013)
  56. 56.Zheng, X., Yazdani, A., Li, H., Humphrey, J.D., Karniadakis, G.E.: A three-dimensional phase-field model for multiscale modeling of thrombus biomechanics in blood vessels. PLOS Computational Biology 16(4), e1007709 (2020)
  57. 57.Xu, Z., Chen, N., Kamocka, M.M., Rosen, E.D., Alber, M.: A multiscale model of thrombus development. Journal of the Royal Society Interface 5(24), 705–722 (2008)
  58. 58.Yazdani, A., Li, H., Humphrey, J.D., Karniadakis, G.E.: A general shear-dependent model for thrombus formation. PLoS Computational Biology 13(1), e1005291 (2017)
  59. 59.Fan, D., Yang, L., Wang, Z., Triantafyllou, M.S., Karniadakis, G.E.: Reinforcement learning for bluff body active flow control in experiments and simulations. Proceedings of the National Academy of Sciences 117(42), 26091–26098 (2020)

Citation

MLA
Cai, S., et al. “Physics-informed Neural Networks (PINNs) for Fluid Mechanics: A Review”. arXiv, 2021, http://arxiv.org/abs/2105.09506v1.
APA
Cai, S., Mao, Z., Wang, Z., Yin, M., & Karniadakis, G. E. (2021). Physics-informed neural networks (PINNs) for fluid mechanics: A review. arXiv. http://arxiv.org/abs/2105.09506v1
Chicago
Cai, S., Z. Mao, Z. Wang, M. Yin, and G. E. Karniadakis. 2021. “Physics-informed Neural Networks (PINNs) for Fluid Mechanics: A Review”. arXiv. http://arxiv.org/abs/2105.09506v1.
Harvard
Cai, S. et al. (2021) “Physics-informed neural networks (PINNs) for fluid mechanics: A review”, arXiv [Preprint]. Available at: http://arxiv.org/abs/2105.09506v1.
Vancouver
1. Cai S, Mao Z, Wang Z, Yin M, Karniadakis GE (2021) Physics-informed neural networks (PINNs) for fluid mechanics: A review. arXiv

BibTeX

@article{cai2021physics,
  title = {Physics-informed neural networks (PINNs) for fluid mechanics: A review},
  author = {Cai, Shengze and Mao, Zhiping and Wang, Zhicheng and Yin, Minglang and Karniadakis, George Em},
  year = {2021},
  journal = {arXiv},
  url = {http://arxiv.org/abs/2105.09506v1},
  eprint = {2105.09506}
}
Metadata:arXiv

Access the Paper

This paper is available from its original source. Click below to access the PDF.

Open PDF