[Paper Review] Physics-informed neural networks (PINNs) for fluid mechanics: A review

Shengze Cai, Zhiping Mao+3 more|arXiv (Cornell University)|May 20, 2021

Model Reduction and Neural NetworksPhysics and Astronomy55 references72 citations

TL;DR

A review of physics-informed neural networks (PINNs) for fluid mechanics, detailing how PINNs integrate data with Navier–Stokes models to enable forward and inverse problems and multi-physics applications.

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.

Motivation & Objective

• Motivate the use of PINNs to seamlessly fuse data with governing equations in fluid mechanics.
• Summarize basic PINN formulations and loss-function design for multi-physics problems.
• Discuss extensions such as domain decomposition, hp-Variational PINNs, and uncertainty quantification.
• Illustrate applications to 3D incompressible wakes, 2D compressible flows, and biomedical flow problems.

Proposed method

• Use a fully-connected neural network to approximate PDE solutions with space-time inputs and compute derivatives via automatic differentiation.
• Form a composite loss L = w1 L_PDE + w2 L_data + w3 L_IC + w4 L_BC to enforce physics and data.
• Train by minimizing L with stochastic optimization (e.g., ADAM) over collocation and data points.
• Infer PDE parameters and unknown fields by jointly optimizing network weights and physical parameters.
• Apply domain-decomposition, adaptive activations, and multi-fidelity strategies to improve convergence and accuracy.

Experimental results

Research questions

• RQ1 Can PINNs accurately reconstruct 3D flow fields from limited 2D observations in incompressible wakes?
• RQ2 Can PINNs recover pressure, velocity, and material parameters in compressible and biomedical flow settings from partial data?
• RQ3 What are the benefits and limitations of PINNs for forward/inverse problems in fluid mechanics compared to traditional CFD?
• RQ4 How do advanced PINN extensions (domain decomposition, hp-VPINN, uncertainty quantification) impact performance across flow regimes?

Key findings

• PINNs can reconstruct full 3D flow fields from a few 2D2C observations with small relative velocity errors in Case 1 and Case 2, while Case 3 remains acceptable.
• In 2D compressible flows, PINNs reproduce density, pressure, and velocity fields consistent with CFD data using limited surface pressure and density-gradient information.
• PINNs infer elusive quantities such as pressure fields and internal velocity components without full boundary-condition specification, given appropriate data and global constraints.
• In biomedical flows, PINNs can infer permeability-phi relationships and velocity fields from phase-field data in thrombus-involved FSI scenarios.
• Extensions like multi-fidelity training, adaptive activation, and uncertainty quantification enhance PINN robustness and applicability.