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Physics-informed neural networks (PINNs) for fluid mechanics: A review
Shengze Cai, Zhiping Mao, Zhicheng Wang, Minglang Yin, George Em Karniadakis
TL;DR
Existing flow solvers struggle to integrate multifidelity data, avoid costly mesh generation, and solve expensive inverse problems. This paper reviews PINNs, which combine governing equations with data, and demonstrates inference across wake, supersonic, and biomedical flows.
Problem
Existing algorithms cannot seamlessly incorporate multifidelity data, mesh generation remains time-consuming, and inverse flow problems require expensive formulations and new codes.
Method
The paper reviews PINNs, which use automatic differentiation and loss functions to combine Navier–Stokes equations, constraints, measurements, and initial or boundary conditions.
Results
PINNs accurately infer three-dimensional wake flows, reconstruct high-speed flow fields from alternative measurements, and recover biomedical material parameters with good agreement to reference data.
Takeaways & Limitations
PINNs provide a mesh-free framework for integrating mathematical models with available data in forward and inverse flow problems across several application settings.
Takeaways & Limitations
For inverse problems with unknown parameters or initial or boundary conditions, measurements must be included to make the optimization problem solvable.
Abstract
from arXiv · showhide
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.
1 Introduction
Traditional CFD has advanced substantially, but real-world flow applications still face difficulties integrating multifidelity data, generating meshes, solving inverse problems, and maintaining complex codes. Physics-informed learning addresses these challenges by combining experimental data with Navier–Stokes formulations in PINNs, which the paper applies across several flow settings.
- 1 Introduction: Industrial-complexity flow problems still require time-consuming mesh generation, while inverse problems can demand expensive formulations and new computer codes.The paper also notes that large CFD programs can exceed 100,000 lines of code, complicating long-term maintenance.
- 1 Introduction: PINNs integrate multifidelity and multimodality experimental data with incompressible, compressible, and biomedical Navier–Stokes formulations.They incorporate equations and constraints into a neural-network loss function alongside measurements.
- 1 Introduction: Automatic differentiation removes the explicit need for mesh generation when PINNs represent differential operators.Governing-equation residuals and kinematic or thermodynamic constraints are penalized in the loss function.
- 1 Introduction: Forward and inverse PINN formulations are identical, avoiding separate expensive data-assimilation schemes for flow optimization and design.This unified formulation applies when data are available for the flow problem.
- 1 Introduction: The paper reviews PINN principles and extensions, then presents results for 3D cylinder wakes, 2D supersonic flow, and biomedical thrombus deformation.The applications include inference of material parameters in the biomedical-flow case.
2 PINNs: Physics-Informed Neural Networks
PINNs approximate PDE solutions with neural networks and train them by minimizing losses that combine governing-equation residuals with data and initial or boundary conditions. The review covers automatic differentiation, domain-decomposition and training extensions, and applications to fluid-mechanics problems.
- 2.1 PINNs: Basic Concepts: A fully connected feed-forward network takes space and time coordinates as inputs and approximates the PDE solution at its final layer.The trainable weights and biases are collected in θ.
- 2.1 PINNs: Basic Concepts: PINNs convert PDE-system solution into optimization by updating neural-network parameters to minimize a weighted loss function.The loss terms can represent PDE residuals, measurements, initial conditions, and boundary conditions.
- 2.1 PINNs: Basic Concepts: The loss function is problem-dependent: data loss may be omitted for known forward problems but is needed for unknown parameters or initial and boundary conditions.Measurements help make inverse optimization problems solvable.
- 2.1 PINNs: Basic Concepts: Automatic differentiation computes derivatives directly in the computational graph, supplying the differential operators needed for PDE-loss construction.This avoids introducing truncation errors associated with explicit numerical differentiation.
- 2.2 Recent Advances of PINNs: Domain decomposition, variational formulations, uncertainty quantification, adaptive training, hard constraints, and alternative architectures extend PINNs across PDE and flow problems.The review also cites theoretical work providing convergence guarantees and insights.
- 2.2 Recent Advances of PINNs: PINNs provide a unified framework for forward and inverse fluid problems and integrate flow observations with governing equations.Reported applications include flow visualization from concentration fields and images.
3 Case Study for 3D Incompressible Flows
This case study reconstructs unsteady three-dimensional cylinder-wake velocity and pressure fields from limited 2D2C observations using PINNs. Across three observation setups, reconstruction is highly accurate with four or five planes and remains acceptable with only three cross-planes, although errors increase.
- Problem setup: The study reconstructs a 3D, unsteady cylinder wake at Re = 200 from limited 2D2C velocity observations using PINNs.The reference solution is generated numerically, and the investigated wake sub-domain is evaluated over time.
- Problem setup: Three setups vary the observed planes from five planes in Case 1 to four in Case 2 and three in Case 3.The setups use different combinations of x-, y-, and z-planes within the investigated domain.
- Problem setup: Only projected two-component velocity vectors are known on cross-planes, while the orthogonal velocity component remains unknown.This setup mimics planar particle image velocimetry observations.
- Implementation of PINNs: The PINN takes (x,y,z,t) as input and predicts (u,v,w,p), combining observation losses with PDE residual losses over 3×10^6 sampled space-time points.Boundary and initial conditions are not required in this study, and no pressure information is provided.
- Inference results: PINNs infer the 3D flow very accurately for Cases 1 and 2, with five planes performing slightly better than four planes.Case 3 has much larger errors, but its streamwise-velocity error is mostly less than 2% and the main flow features remain accurately inferred.
- Inference results: The Case 2 inference is very consistent with CFD and captures the unsteadiness of vortex shedding accurately from a few 2D2C observations.The comparison includes vorticity and pressure iso-surfaces at t = 8.0 and time histories of u and v at one point.
4 Case Study for Compressible Flows
This case study applies PINNs to an inverse 2D compressible Euler problem, using limited pressure data, inflow conditions, density-gradient information, and physical constraints. The resulting pressure and velocity fields agree well with CFD solutions.
- Problem formulation: The compressible-flow model uses the Euler equations together with an equation of state for a polytropic gas.The state variables include density, pressure, velocity components, and total energy.
- Problem formulation: PINNs infer density, pressure, and velocity fields for a 2D steady compressible Euler problem from density gradients, limited surface-pressure data, inflow conditions, and global physical constraints.The approach addresses an inverse problem rather than relying on all boundary conditions required by traditional CFD.
- Training: A 6×60 neural network is trained with a layer-wise adaptive tanh activation, Adam optimization, dynamic weights, and 3 × 10^5 epochs.The reported learning rate is 6 × 10−4.
- Inference results: The PINN pressure and velocity solutions are in good agreement with CFD data for the bow-shock problem.The comparison is shown for pressure p and velocity component u.
- Inference results: The study indicates that high-speed flow fields can be reconstructed using available knowledge other than the boundary conditions required by traditional CFD simulation.The stated reconstruction uses the supplied flow information and physical constraints.
5 Case Study for Biomedical Flows
This case study uses PINNs to infer thrombus material properties and hidden flow fields in a coupled Navier–Stokes and Cahn–Hilliard model. Using phase-field and point measurements, the method recovers permeability values close to reference values and accurately infers velocity.
- Model and objective: The biomedical-flow model describes thrombus–blood interaction using coupled Navier–Stokes and Cahn–Hilliard equations.The formulation represents velocity, pressure, stress, and phase-field variables in a fluid–structure interaction setting.
- PINN formulation: Two fully connected networks approximate the PDE solutions, while the total loss combines PDE, initial-condition, boundary-condition, and data losses.The model parameters λ = [κ] are optimized jointly with the network parameters.
- Model and objective: The inverse task infers unknown core and shell permeability together with the velocity field from measurable phase-field data.The idealized thrombus has an impermeable core and permeable shell.
- Inference results: After 300,000 epochs, inferred parameters a = 7.1 and b = 0.0003 yield core permeability κ(φ = −1) = 0.0011 and shell permeability κ(φ = 0) = 1.0003.The reference values are core permeability 0.001 and shell permeability 1; other errors are lower than O(10−3).
- Inference results: The phase-field prediction agrees well with reference data, and the hidden velocity field is accurately inferred using only phase-field data.Phase-field errors are concentrated near the thrombus outlet, while velocity errors are mainly confined to the shell layer.
6 Summary
PINNs are presented as a complementary approach for realistic fluid flows involving multimodality data and unknown boundary or initial conditions. The review identifies future directions spanning active flow control, transfer learning, closure modeling, and scalable implementations.
- PINNs complement existing CFD methods for realistic flows with multimodality measurements and unknown boundary or initial conditions.The paper contrasts this setting with CFD solvers designed for idealized problems.
- Active flow control with PINNs could replace expensive experiments and time-consuming large-scale simulations.
- Transfer learning could predict flow rapidly at a new high Reynolds number when solutions at lower Reynolds numbers are available.
- Future work includes closure models for unresolved dynamics at very high Reynolds numbers using PINN-based automatic data assimilation.
- Efficient multi-GPU implementations are needed for flow problems at scale, while reported parallel speedup suggests future industrial-complexity applications.The proposed implementations include data-parallel and hybrid data-parallel/model-parallel paradigms.