The following list comprises articles related to computational fluid dynamics (CFD) and machine learning (ML). Some of the articles consider only heat and mass transfer or the solution of partial differential equations which are not strictly speaking computational fluid dynamics. There are also works where initial/boundary value problems are solved directly using machine learning without classical numerical techniques. Such articles are also listed here because they may contain techniques which help to improve data-driven CFD solutions. The articles are listed alphabetic order. There are many sensible ways to categorize articles, e.g. regarding the learning type (supervised, un-supervised, reinforced) or the physics of interest (turbulence, mass transfer, etc.). Therefore, each list item below contains a small description and several useful keywords to relate similar articles.
If you think there is some articles missing, if you would like to add/modify list items, or if you have general comments, please use the repository's issue tracker.
The following graphic illustrates the correlation between published articles related to ML+CFD together with several ML Google trends. The figure suggests that there is a strong connection between the progress in research and the availability of easy-to-use software packages. This connections seems to be particularly strong for open source frameworks like Tensorflow and PyTorch.
The number of articles per year is based on the literature list below. The trend curves were downloaded on the 26th of August 2019 (worldwide).
Abbreveation | Meaning |
---|---|
ADT | Adaboost Decision Tree |
BFGS | Broyden Fletcher Goldfarb Shanno method |
CNN | Convoloutional Neural Network |
DRL | Deep Reinforcement Learning |
DSC | Downsampled Skip Connection |
GAN | Generative Adversarial Network |
GP | Gaussian Process |
KDE | Kernel Density Estimation |
KNN | K-Nearest Neighbors |
LASSO | Least Absolute Shrinkage and Selection Operator |
LES | Large Eddy Simulation |
LSTM | Long Short Term Memory |
MLP | Multilayer Perceptron |
MSN | Multi Scale Network |
PINN | Physics Informed Neural Networks |
PCA | Principal Component Analysis |
PPO | Proximal Policy Optimization |
RANS | Reynolds Averaged Navier Stokes |
RBNN | Radial Basis Function Neural Network |
RF | Random Forest |
RNN | Recurrent Neural Network |
ROC | Receiver Operating Characteristic |
SBS | Sequential Backward Selection |
SOM | Self Organizing Map |
TBNN | Tensor Basis Neural Network |
M. Alsalman et al.: Training bioinspired sensors to classify flows (2018)
- Aim: classification of flow patterns based on local vorticity measurements; influence of sensor types on accuracy of flow classification
- Learning type: supervised learning (classification)
- ML algorithms: MLP, SGD
- ML frameworks: Mathematica 11.1.1
- CFD framework: inhouse
- Combination of CFD + ML: post
S. Bhalla et al.: Compact Representation of a Multi-dimensional Combustion Manifold Using Deep Neural Networks (2019)
- Aim: approximate combustion manifold using a neural network to predict temperature and composition of reaction; comparison to tabulation methods; publication of combustion data set
- Learning type: supervised learning (classification)
- ML algorithms: MLP, batch normalization, ensemble
- ML frameworks: not specified
- CFD framework: OpenFOAM
- Combination of CFD + ML: post, interactively
S. Bhatnagar et al.: Prediction of aerodynamic flow fields using convolutional neural networks (2019)
- Aim: predict velocity and pressure in unseen flow conditions given the pixelated shape of the object; predict impact of airfoil shape on aerodynamic forces in near real time
- Learning type: supervised learning (regression)
- ML algorithms: CNN, parameter sharing, gradient sharpening
- ML frameworks: Caffe
- CFD framework: overturns cfd
- Combination of CFD + ML: post
M. Bode et al.: Towards Prediction of Turbulent Flows at High Reynolds Numbers Using High Performance Computing Data and Deep Learning (2019)
- Aim: discuss suitability of various GAN architectures for understanding and modeling turbulence; use DNS data to train GANs and asses statistics of predicted turbulent fields
- Learning type: supervised learning (regression)
- ML algorithms: Wasserstein GAN
- ML frameworks: Keras, Tensorflow, Horovod
- CFD framework: database
- Combination of CFD + ML: post
S. L. Brunton et al.: Discovering governing equations from data by sparse identification of nonlinear dynamical systems (2016)
- Aim: extract governing equations from noisy measurement data; use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data; demonstration on linear and non-linear oscillators, chaotic Lorenz system, vortex shedding behind an obstacle
- Learning type: supervised learning (regression)
- ML algorithms: LASSO
- ML frameworks: Matlab
- CFD framework: inhouse
- Combination of CFD + ML: post
S. L. Brunton et al.: Machine Learning for Fluid Mechanics (2019)
- Aim: review article ML + fluid dynamics;
- Learning type: supervised learning, unsupervised learning, reinforcement learning
- ML algorithms: MLP, RNN, LSTM, PCA, SOM, k-means, GAN
- ML frameworks: -
- CFD framework: -
- Combination of CFD + ML: -
S. Chanda et al.: Estimation of principal thermal conductivities of layered honeycomb composites using ANN-GA based inverse technique (2017)
- Aim: estimation of principal thermal conductivities of layered honeycomb composite material; minimization of the difference between measured and computed temperatures
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: Matlab
- CFD framework: Matlab
- Combination of CFD + ML: interactively
C. W. Chang et al.: A Study of Physics-Informed Deep Learning for System Fluid Dynamics Closures (2016)
- Aim: study stability of DL-based closure models for fluid dynamics; test influence of activation function and model complexity
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: Tensorflow
- CFD framework: inhouse, Modelica
- Combination of CFD + ML: post
C. W. Chang et al.: Classification of machine learning frameworks for data-driven thermal fluid models (2019)
- Aim: classification system for usage of ML in data-driven modeling of thermal fluids; demonstrate ML-model types on thermal conductivity problem
- Learning type: supervised learning (regression)
- ML algorithms: MLP, CNN
- ML frameworks: Tensorflow
- CFD framework: inhouse, various
- Combination of CFD + ML: post, interactively
S. H. Cheung et al.: Bayesian uncertainty analysis with applications to turbulence modeling (2011)
- Aim: apply Bayesian uncertainty quantification to the process of calibrating mathematical models and the to prediction of quantities to interest; comparison of Spalart-Allmaras model clases in terms of their posterior probabilities to fit experimental observations; selection of appropriate model classes for given task
- Learning type: -
- ML algorithms: Bayesian statistics, markov chains
- ML frameworks: inhouse
- CFD framework: OpenFOAM
- Combination of CFD + ML: post
B. Colvert et al.: Classifying vortex wakes using neural networks (2018)
- Aim: flows contain information about objects creating them; classification of flow patterns based on local vorticity measurements
- Learning type: supervised learning (classification)
- ML algorithms: MLP, SGD
- ML frameworks: Mathematica 11.1.1
- CFD framework: inhouse
- Combination of CFD + ML: post
S. J. Daniels et al.: A Suite of Computationally Expensive Shape Optimisation Problems Using Computational Fluid Dynamics (2018)
- Aim: create a set of computationally expensive benchmark test problems for design optimisation using CFD; development of a Python software package for parameterized geometry creation and object function computation
- Learning type: -
- ML algorithms: generic
- ML frameworks: generic
- CFD framework: OpenFOAM
- Combination of CFD + ML:
K. Duraisamy et al.: Turbulence Modeling in the Age of Data (2019)
- Aim: review article for developments in bounding uncertainties in RANS models via physical constrains, adopting statistical inference to characterize model coefficients and discrepancies, and in using machine learning to improve turbulence models;
- Learning type: -
- ML algorithms: -
- ML frameworks: -
- CFD framework: -
- Combination of CFD + ML: -
W. Edeling et al.: Bayesian Predictions of Reynolds-Averaged Navier–Stokes Uncertainties Using Maximum a Posteriori Estimates (2018)
- Aim: Bayesian model-scenario averaging to combine the predictions of several competing models validated on various sets of data; stochastic estimate of a quantity of interest in an unmeasured prediction scenario
- Learning type: supervised learning (regression)
- ML algorithms: Bayesian averaging
- ML frameworks: -
- CFD framework: Fluent
- Combination of CFD + ML: post
N. B. Erichson et al.: Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Prediction (2019)
- Aim: use physics-informed prior knowledge (Lyapunov stability) for improving model quality (generalization performance, sensitivity to parameter tuning, robustness to noise)
- Learning type: supervised learning (regression)
- ML algorithms: MLP, Autoencoder
- ML frameworks: PyTorch
- CFD framework: -
- Combination of CFD + ML: post
K. Fukami et al.: Super-resolution reconstruction of turbulent flows with machine learning (2019)
- Aim: reconstruct laminar flow around a cylinder and 2D homogeneous turbulence based on low-resolution simulation data
- Learning type: supervised learning (regression)
- ML algorithms: CNN, DSC, MSN, autoencoder
- ML frameworks: Tensorflow, Keras, Scikit Learn
- CFD framework: database
- Combination of CFD + ML: post
K. Fukami et al.: Synthetic turbulent inflow generator using machine learning (2019)
- Aim: generate time-dependent turbulent inflow data to replace conventional driver simulations or synthetic inflow generators; investigation of spurious periodicity; lower computational cost of DNS
- Learning type: supervised learning (regression)
- ML algorithms: autoencoder, CNN, MLP
- ML frameworks: Tensorflow, Keras, Scikit Learn
- CFD framework: inhouse
- Combination of CFD + ML:
P. Garnier et al.: A review on Deep Reinforcement Learning for Fluid Mechanics (2019)
- Aim: review article of DRL application to fluid dynamics problems; discussion of advantages and disadvantages of different coupling methods (deep Q learning, policy gradients)
- Learning type: (deep) reinforcement learning
- ML algorithms: deep Q-networks, deep policy gradient, advantage actor-critic, proximal policy optimization
- ML frameworks: Tensorflow
- CFD framework: Fenics
- Combination of CFD + ML:
F. Gueniat et al.: A statistical learning strategy for closed-loop control of fluid flows (2016)
- Aim: discuss closed-loop control for complex systems; derive Markov process model based on sensor measurements
- Learning type: reinforcement learning
- ML algorithms: Markov process, Q-learning
- ML frameworks: inhouse
- CFD framework: inhouse
- Combination of CFD + ML:
X. Guo et al.: Convolutional Neural Networks for Steady Flow Approximation (2016)
- Aim: real-time prediction of non-uniform steady laminar flow in a 2D or 3D domain based on CNNs; explore alternatives for geometry representation
- Learning type: supervised learning
- ML algorithms: CNN
- ML frameworks: Caffe
- CFD framework: OpenLB
- Combination of CFD + ML: post
B. N. Hanna et al.: Coarse-Grid Computational Fluid Dynamics Error Prediction using Machine Learning (2017)
- Aim: ML-based surrogate model for the prediction of errors in coarse grid CFD simulations; application to 3D lid-driven cavity
- Learning type: supervised learning (regression)
- ML algorithms: MLP, RF regression
- ML frameworks: inhouse
- CFD framework: OpenFOAM
- Combination of CFD + ML: post
J. R. Holland et al.: Towards Integrated Field Inversion and Machine Learning With Embedded Neural Networks for RANS Modeling (2019)
- Aim: reduce model form errors in RANS simulations; integrate learning step into field inversion process; reduce training time by "layered approach"
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: inhouse
- CFD framework: SU2
- Combination of CFD + ML: post
M. Ihme et al.: Optimal artificial neural networks and tabulation methods for chemistry representation in LES of a bluff-body swirl-stabilized flame (2009)
- Aim: LES of bluff-body swirl-stabilized methane-hydrogen flame with structured tabulation and neural networks; comparison of tabulation and network based methods regarding accuracy, data retrieval time and storage requirements
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: not specified
- CFD framework: not specified
- Combination of CFD + ML: post, interactively
K. Jambunathan et al.: Evaluating convective heat transfer coefficients using neural networks (1996)
- Aim: predict heat transfer coefficients based on temperature measurements over time; generate explicit relationship for heat transfer coefficients based on neural network
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: NeuralWorks Professional Plus
- CFD framework: -
- Combination of CFD + ML: post
M. L. A. Kaandorp et al.: Stochastic Random Forests with Invariance for RANS Turbulence Modelling (2018)
- Aim: predict Reynolds-stress anisotropy tensor while guaranteeing Galilean invariance by making use of the tensor basis; usage of random forest to detect outliers in training data, to prevent overfitting, and to provide uncertainty estimates;
- Learning type: supervised learning (regression, classification)
- ML algorithms: RF, MLP
- ML frameworks: not specified
- CFD framework: OpenFOAM
- Combination of CFD + ML: post, interactively
B. Kim et al.: Deep Fluids: A Generative Network for Parameterized Fluid Simulations (2019)
- Aim: synthesize fluid simulations from a set of reduced parameters; propose loss function that guarantees divergence-free velocity fields; handle complex parameterizations in reduced spaces and advance simulations in time by integrating the latent space with a second network
- Learning type: supervised learning (regression), unsupervised learning
- ML algorithms: CNN, autoencoder, MLP, GAN
- ML frameworks: Tensorflow
- CFD framework: Mantaflow
- Combination of CFD + ML:
J. N. Kutz et al.: Deep learning in fluid dynamics (2017)
- Aim: short overview article for deep learning in turbulence modeling; pose open questions regarding data-driven models
- Learning type: -
- ML algorithms: -
- ML frameworks: -
- CFD framework: -
- Combination of CFD + ML: -
L. Ladicky et al.: Data-driven Fluid Simulations using Regression Forests (2015)
- Aim: formulate machine learning and physics based fluid simulation as regression problem for time-critical simulations
- Learning type: supervised learning (regression)
- ML algorithms: RF
- ML frameworks: not specified
- CFD framework: PCISPH
- Combination of CFD + ML: post
I. E. Lagaris et al.: Artificial Neural Networks for Solving Ordinary and Partial Differential Equations (1998)
- Aim: formulate IBVs as optimization problem; approximate ODE (system) PDS solution based on neural networks; compare to finite elements
- Learning type: supervised learning (regression)
- ML algorithms: MLP, BFGS
- ML frameworks: Merlin
- CFD framework: inhouse
- Combination of CFD + ML: post
I. E. Lagaris et al.: Neural-Network Methods for Boundary ValueProblems with Irregular Boundaries (2000)
- Aim: approximate solution of IBV problems on irregular domains; test on 2D and 3D domains
- Learning type: supervised learning (regression)
- ML algorithms: MLP, RBNN
- ML frameworks: Merlin
- CFD framework: -
- Combination of CFD + ML: -
J. Ling et al.: Evaluation of machine learning algorithms for prediction of regions of high Reynolds averaged Navier Stokes uncertainty (2015)
- Aim: train ML algorithms to identify regions of high RANS uncertainty; point-by-point classification of high or low uncertainty; show that classifiers can generalize to flows substantially different from training data
- Learning type: supervised learning (classification)
- ML algorithms: SVM, ADT, RF, ROC
- ML frameworks: Scikit Learn
- CFD framework: Fluent 12.0, Sierra Fuego, Sigma CFD
- Combination of CFD + ML: post
J. Ling et al.: Reynolds averaged turbulence modelling using deep neural networks with embedded invariance (2016)
- Aim: learn a model for the Reynolds stress anisotropy from high-fidelity simulation data; propose neural network architecture with invariant tensor basis to embed Galilean invariance of predicted anisotropy tensor
- Learning type: supervised learning (regression)
- ML algorithms: MLP, TBNN
- ML frameworks: FFNet
- CFD framework: Fluent, database
- Combination of CFD + ML: post, interactively
J. Ling et al.: Machine learning strategies for systems with invariance properties (2016)
- Aim: address machine learning for systems with symmetry or invariance properties; explore two ways of incorporating invariance in models: based on invariant features and based on specifically transformed data
- Learning type: supervised learning (regression)
- ML algorithms: RF, MLP
- ML frameworks: scikit-learn, FFNet
- CFD framework: Fluent 12.0
- Combination of CFD + ML: post
J. C. Loiseau et al.: Constrained sparse Galerkin regression (2018)
- Aim: combine approach for creation of nonlinear low-order models with dimensionality reduction and enforce physical constraints in the regression; create method without need for high-fidelity data
- Learning type: supervised learning (regression)
- ML algorithms: PCA, LASSO
- ML frameworks: scikit-learn
- CFD framework: database
- Combination of CFD + ML: post
L. Lu et al.: DeepXDE: A deep learning library for solving differential equations (2019)
- Aim: propose a residual-based adaptive refinement to improve training efficiency of PINNs; present a Python library for PINNs
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: DeepXDE, Tensorflow
- CFD framework: -
- Combination of CFD + ML: -
M. Ma et al.: Using statistical learning to close two-fluid multiphase flow equations for bubbly flows in vertical channels (2015)
- Aim: use DNS of bubbly flows to find closure terms for a simple model of the average flow; train neural network on one simulation and use it to predict evolution of different initial conditions
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: Matlab
- CFD framework: inhouse
- Combination of CFD + ML: post
M. Ma et al.: Using statistical learning to close two-fluid multiphase flow equations for bubbly flows in vertical channels (2016)
- Aim: use DNS of bubbly up-flow in a periodic vertical channel to generate closure relationships for a simplified two-fluid model for the average flow; generate database by averaging DNS results over planes parallel to the walls; test closure realation for different initial conditions
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: Matlab
- CFD framework: inhouse
- Combination of CFD + ML: post
P. Ma et al.: Fluid Directed Rigid Body Control using Deep Reinforcement Learning (2018)
- Aim: control a 2D coupled system of fluids and rigid bodies by applying control forces at the domain boundaries; create fluid controller to accomplish challenging 2D tasks as keeping a rigid body balanced, playing ping pong and driving a rigid body to hit specific points on a wall
- Learning type: reinforcement learning
- ML algorithms: MLP, autoencoder, CNN
- ML frameworks: not specified
- CFD framework: inhouse
- Combination of CFD + ML: interactively
R. Maulik et al.: Subgrid modelling for two-dimensional turbulence using neural networks (2019)
- Aim: use high-fidelity data to train neural networks to predict a turbulent source term through localized grid resolved information; present hyper-parameter optimization analysis and learning quantification through probability-density-function-based validation of subgrid predictions; validate with 2D decaying turbulence
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: Tensorflow
- CFD framework: database
- Combination of CFD + ML: post
M. Milano et al.: Neural Network Modeling for Near Wall Turbulent Flow (2002)
- Aim: reconstruct the near wall field in a turbulent flow by exploiting flow fields provided by direct numerical simulations; comparison to results obtained from PCA; present outlook towards usages in wall functions for RANS or LES
- Learning type: supervised learning, autoencoder
- ML algorithms: MLP
- ML frameworks: inhouse
- CFD framework: inhouse
- Combination of CFD + ML: post
K. Mills et al.: Deep learning and the Schrödinger equation (2017)
- Aim: train neual network to predict the ground-state energy of an electron in four classes of confining two-dimensional electrostatic potentials; investigate the performance of the model in predicting other quantities such as the kinetic energy and the first excited-state energy; compare different machine learning algorithms for the task
- Learning type: supervised learning (regression)
- ML algorithms: CNN
- ML frameworks: Tensorflow
- CFD framework: inhouse
- Combination of CFD + ML: post
M. Moioli et al.: Parametric data-based turbulence modelling for vortex dominated flows (2019)
- Aim: enhance Spalart–Allmaras model with additional source terms, which are exclusively active in the vortex field; optimize model based on experimental reference data
- Learning type: parameter calibration
- ML algorithms: gradient decent
- ML frameworks: inhouse
- CFD framework: TAU
- Combination of CFD + ML: post
T. Murata et al.: Nonlinear mode decomposition with machine learning for fluid dynamics (2019)
- Aim: decompose flow field using a CNN; create nonlinear decomposition to be used for feature extraction of flow fields in lower dimension than PCA
- Learning type: unsupervised learning
- ML algorithms: CNN, autoencoder
- ML frameworks: not specified
- CFD framework: not specified
- Combination of CFD + ML: post
G. Novati et al.: Synchronisation through learning for two self-propelled swimmers (2017)
- Aim: conduct simulations of two, self-propelled, fish-like bodies that employ a learning algorithm to synchronise their swimming patterns; demonstrate that two rigid bodies executing pre-specified motions, with an alternating leader and follower, can result in substantial drag-reduction and intermittent thrust generation
- Learning type: reinforcement learning
- ML algorithms: not specified
- ML frameworks: not specified
- CFD framework: not specified
- Combination of CFD + ML: interactively
E. J. Parish et al.: A paradigm for data-driven predictive modeling using field inversion and machine learning (2016)
- Aim: propose a modeling paradigm that seeks to comprehensively harness data from sources such as high-fidelity simulations and experiments to aid the creation of improved closure models for computational physics applications; use inverse modeling to obtain corrective, spatially distributed functional terms
- Learning type: supervised learning (regression)
- ML algorithms: GP
- ML frameworks: not specified
- CFD framework: not specified
- Combination of CFD + ML: post
H. V. Patel et al.: Computing interface curvature from volume fractions: A machine learning approach (2019)
- Aim: develop a model which predicts the local interface curvature from neighbouring volume fractions; introduce data generation methodology is devised which generates well-balanced randomized data sets comprising of spherical interface patches of different configurations/orientations
- ML algorithms: MLP
- ML frameworks: Matlab
- CFD framework: inhouse
- Combination of CFD + ML: interactively
S. Pierret et al.: Turbomachinery Blade Design Using a Navier–Stokes Solver and Artificial Neural Network (1999)
- Aim: describe a knowledge-based method for the automated design of more efficient turbine blades; optimize shape parameters based on simulated annealing and neural networks; use neural networks to learn Mach number distribution on wing surface
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: not specified
- CFD framework: TRAF2D
- Combination of CFD + ML: post
Y. Qi et al.: Computing curvature for volume of fluid methods using machine learning (2019)
- Aim: create data set with volume fractions and corresponding curvature for well-defined shapes, and use machine learning to fit the data; test model on shapes not used for training and in an interface capturing solver
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: Matlab
- CFD framework: inhouse
- Combination of CFD + ML: interactively
J. Rabault et al.: Artificial neural networks trained through deep reinforcement learning discover control strategies for active flow control (2019)
- Aim: application of an artificial neural network trained through a deep reinforcement learning agent to perform active flow control for the flow around a cylinder
- Learning type: reinforcement learning
- ML algorithms: MLP
- ML frameworks: Tensorflow, Tensorforce
- CFD framework: Fenics
- Combination of CFD + ML: interactively
- Code: Github
J. Rabault et al.: Accelerating Deep Reinforcement Leaning strategies of Flow Control through a multi-environment approach (2019)
- Aim: introduce a simple strategy is to use several independent simulations running in parallel to collect the required experiences for reinforcement learning faster
- Learning type: reinforcement learning
- ML algorithms: MLP
- ML frameworks: Tensorflow, Tensorforce
- CFD framework: Fenics
- Combination of CFD + ML: interactively
M. Raissi et al.: Hidden physics models: Machine learning of nonlinear partial differential equations (2018)
- Aim: present a new paradigm of learning partial differential equations from small data; extract patterns from high-dimensional data generated from experiments
- Learning type: supervised learning (regression)
- ML algorithms: GP
- ML frameworks: Mathematica, Matlab
- CFD framework: -
- Combination of CFD + ML: -
- Code: Github
M. Raissi et al.: Numerical Gaussian Processes for Time-Dependent and Nonlinear Partial Differential Equations (2018)
- Aim: introduce the concept of numerical Gaussian processes, which are defined as Gaussian processes with covariance functions resulting from temporal discretization of time-dependent partial differential equations
- Learning type: supervised learning (regression)
- ML algorithms: GP
- ML frameworks: Matlab, Mathematica
- CFD framework: inhouse
- Combination of CFD + ML: -
- Code: Github
M. Raissi et al.: Deep Learning of Turbulent Scalar Mixing (2018)
- Aim: describe a framework for discovering turbulence models from scattered and potentially noisy spatio-temporal measurements of the probability density function
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: Tensorflow
- CFD framework: not specified
- Combination of CFD + ML: post
M. Raissi et al.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations (2019)
- Aim: present developments in the context of solving two main classes of problems: data-driven solution and data-driven discovery of partial differential equations
- Learning type: supervised learning
- ML algorithms: MLP
- ML frameworks: Tensorflow
- CFD framework: inhouse
- Combination of CFD + ML: interactively
- Code: Github
J. Ray et al.: Bayesian calibration of a k-e turbulence model for predictive jet-in-crossflow simulations (2016)
- Aim: calibrate parameters of a RANS model to improve its predictive skill in jet-in-crossflow simulations; estimate parameters from Reynolds stress measurements obtained from an incompressible flow-over-a-square-cylinder experiment
- Learning type: supervised learning (regression)
- ML algorithms: Bayesian calibration
- ML frameworks: not specified
- CFD framework: SIGMA CFD
- Combination of CFD + ML: post
K. Rudd et al.: A Constrained Backpropagation Approach for the Adaptive Solution of Partial Differential Equations (2014)
- Aim: solve nonlinear elliptic and parabolic partial differential equations (PDEs) adaptively; solve the constrained optimization problem associated with training a neural network to approximate the PDE solution by means of direct elimination
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: Matlab
- CFD framework: Matlab
- Combination of CFD + ML: -
K. Rudd et al.: A constrained integration (CINT) approach to solving partial differential equations using artificial neural networks (2015)
- Aim: combine classical Galerkin methods with a constrained backpropogation training approach to obtain an artificial neural network representation of the PDE solution that approximately satisfies the boundary conditions at every integration step
- Learning type: supervised learning
- ML algorithms: MLP
- ML frameworks: Matlab
- CFD framework: Matlab
- Combination of CFD + ML: interactively
S. H. Rudy et al.: Data-driven discovery of partial differential equations (2017)
- Aim: propose a sparse regression method capable of discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain; select the nonlinear and partial derivative terms of the governing equations that most accurately represent the data, bypassing a combinatorially large search through all possible candidate models
- Learning type: supervised learning (regression)
- ML algorithms: RIDGE regression
- ML frameworks: inhouse
- CFD framework: inhouse
- Combination of CFD + ML: post
- Code: Github
H. Schaefer: Learning partial differential equations via data discovery and sparse optimization (2017)
- Aim: develop a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data
- Learning type: supervised learning (regression)
- ML algorithms: LASSO
- ML frameworks: not specified
- CFD framework: not specified
- Combination of CFD + ML:
M. Schmelzer et al.: Machine Learning of Algebraic Stress Models using Deterministic Symbolic Regression (2019)
- Aim: use deterministic symbolic regression to infer algebraic stress models for the closure of RANS equations directly from high-fidelity LES or DNS data;
- Learning type: supervised learning (regression)
- ML algorithms: elastic net
- ML frameworks: not specified
- CFD framework: OpenFOAM
- Combination of CFD + ML: post
N. Shah et al.: Prediction of Autoignition and Flame Properties for Multicomponent Fuels Using Machine Learning Techniques (2019)
- Aim: use RFs and MLPs to predict ignition delay times, flame speeds, octane ratings and CA50 values in homogeneous charge compression ignition engine for multicomponent gasoline surrogates
- Learning type: supervised learning (regression)
- ML algorithms: RF, MLP
- ML frameworks: not specified
- CFD framework: TPRF surrogate model
- Combination of CFD + ML: post
Y. B. Sinai et al.: Learning data-driven discretizations for partial differential equations (2019)
- Aim: introduce data-driven discretization, a method for learning optimized approximations to PDEs based on actual solutions to the known underlying equations; use neural networks to estimate spatial derivatives, which are optimized end to end to best satisfy the equations on a low-resolution grid
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: Tensorflow
- CFD framework: inhouse
- Combination of CFD + ML: interactively
- Code: Github
A. P. Singh et al.: Machine-Learning-Augmented Predictive Modeling of Turbulent Separated Flows over Airfoils (2017)
- Aim: apply inverse modeling to infer the spatial distribution of model discrepancies, and, machine learning to reconstruct discrepancy information from a large number of inverse problems into corrective model forms; reconstruct model forms using neural networks and embedded model within a standard solver
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: FANN
- CFD framework: ADTURNS
- Combination of CFD + ML: post, interactively
A. P. Singh et al.: Characterizing and Improving Predictive Accuracy in Shock-Turbulent Boundary Layer Interactions Using Data-driven Models (2017)
- Aim: apply data-driven solution to enhance RANS predictions of flows involving shock-boundary layer interactions; solve inverse problems to infer spatial discrepancies in the Spalart Allmaras model and project these discrepancies to locally non-dimensional flow features
- Learning type: supervised learning (regression)
- ML algorithms: Adaboost
- ML frameworks: scikit learn
- CFD framework: database, not specified
- Combination of CFD + ML: post, interactively
C. M. Stroefer et al.: Data-Driven, Physics-Based Feature Extraction from Fluid Flow Fields (2018)
- Aim: present a physics-based, data-driven method capable of identifying any flow feature it is trained to; identify any type of feature, even distinguish between similar ones, without the need to explicitly define the physics
- Learning type: supervised learning (classification)
- ML algorithms: CNN
- ML frameworks: Lasagne, Theano
- CFD framework: database
- Combination of CFD + ML:
L. Sun et al.: On developing data-driven turbulence model for DG solution of RANS (2019)
- Aim: develop a neural network model with low complexity acting as an algebraic turbulence model to estimate the turbulence eddy viscosity for RANS
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: not specified
- CFD framework: not specified
- Combination of CFD + ML: post, interactively
L. Sun et al.: Surrogate Modeling for Fluid Flows Based on Physics-Constrained Deep Learning Without Simulation Data (2019)
- Aim: provide a physics-constrained DL approach for surrogate modeling of fluid flows without relying on any simulation data; a structured neural network is devised to enforce the initial and boundary conditions, and the governing partial differential equations are incorporated into the loss of the network to drive the training
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: Tensorflow
- CFD framework: -
- Combination of CFD + ML: -
J. Tompson et al.: Accelerating Eulerian Fluid Simulation With Convolutional Networks (2017)
- Aim: propose a data-driven approach that leverages the approximation power of deep learning with the precision of standard solvers to obtain fast and highly realistic simulations
- Learning type: supervised learning (regression)
- ML algorithms: CNN
- ML frameworks: Torch
- CFD framework: Mantaflow
- Combination of CFD + ML: post, interactively
B. D. Tracey et al.: Application of Supervised Learning to Quantify Uncertainties in Turbulence and Combustion Modeling (2013)
- Aim: introduce a methodology aimed at improving low-fidelity models of turbulence and combustion and obtaining error bounds; develop a new machine learning algorithm to construct a stochastic model of the error of low-fidelity models using information from higher-fidelity data sets; obtain better approximations of uncertain model outputs and generate confidence intervals on the prediction of simulation outputs
- Learning type: supervised learning (regression)
- ML algorithms: kernel regression
- ML frameworks: not specified
- CFD framework: database
- Combination of CFD + ML: post, interactively
B. D. Tracey et al.: A Machine Learning Strategy to Assist Turbulence Model Development (2015)
- Aim: use supervised learning algorithms to build a representation of turbulence modeling closure terms; investigate the feasibility of such an approach by attempting to reproduce, through a machine learning methodology, the results obtained with the well-established Spalart-Allmaras model
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: not specified
- CFD framework: SU2
- Combination of CFD + ML: post, interactively
G. Tryggvason et al.: DNS–Assisted Modeling of Bubbly Flows in Vertical Channels (2017)
- Aim: discuss usage of DNS results to provide values for the unresolved closure terms in a simple average model for the flow found by statistical learning from the data using neural networks; explore using the results from simulations of large systems with bubbles of different sizes in turbulent flows for large eddy–like simulations
- Learning type: supervised learning (regression)
- ML algorithms: MLP
- ML frameworks: Matlab
- CFD framework: inhouse
- Combination of CFD + ML: post
S. Verma et al.: Efficient collective swimming by harnessing vortices through deep reinforcement learning (2018)
- Aim: show that fish can improve their sustained propulsive efficiency by placing themselves in appropriate locations in the wake of other swimmers and intercepting judiciously their shed vortices; swimming strategy leads to collective energy savings and is revealed through a combination of high-fidelity flow simulations with a deep reinforcement learning algorithm
- Learning type: reinforcement learning
- ML algorithms: LSTM
- ML frameworks: nt specified
- CFD framework: CUBISM
- Combination of CFD + ML: interactively
J. Viquerat et al.: Direct shape optimization through deep reinforcement learning (2019)
- Aim: present the first application of DRL to direct shape optimization; show that artificial neural network trained through DRL is able to generate optimal shapes on its own, without any prior knowledge and in a constrained time
- Learning type: reinforcement learning
- ML algorithms: MLP, PPO
- ML frameworks: Tensorflow, Tensorforce
- CFD framework: Fenics
- Combination of CFD + ML:
J. X. Wang et al.: Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data (2017)
- Aim: propose a data-driven, physics-informed machine learning approach for reconstructing discrepancies in RANS modeled Reynolds stresses; discrepancy functions are trained by existing direct numerical simulation databases
- Learning type: supervised learning (regression)
- ML algorithms: RF
- ML frameworks: not specified
- CFD framework: database, OpenFOAM
- Combination of CFD + ML: post, interactively
M. Wang et al.: Detecting exotic wakes with hydrodynamic sensors (2019)
- Aim: wake sensing for exotic wake types that arise in swimming; classification library to classify unknown wakes from hydrodynamic signal measurements; locomotion control
- Learning type: supervised learning (classification)
- ML algorithms: KNN
- ML frameworks: not specified
- CFD framework: inhouse
- Combination of CFD + ML: post
A. Weiner et al.: Data‐Driven Subgrid‐Scale Modeling for Convection‐Dominated Concentration Boundary Layers (2019)
- Aim:
- Learning type: supervised learning (regression)
- ML algorithms: MLP, SBS, KNN
- ML frameworks: PyTorch, Scikit Learn
- CFD framework: OpenFOAM
- Combination of CFD + ML: post, interactively
J. L. Wu et al.: A Priori Assessment of Prediction Confidence for Data-Driven Turbulence Modeling (2017)
- Aim: identify quantitative measure for a priori estimation of prediction confidence in data-driven turbulence modeling; use Mahalanobis distance and kernel density estimation to quantify distance in feature space; provide an approach to aid in the choice of data sources and to asses prediction performance
- Learning type: unsupervised learning
- ML algorithms: KDE, RF
- ML frameworks: not specified
- CFD framework: database
- Combination of CFD + ML: post
- Aim: develop data-driven, physics-informed Bayesian framework for quantifying model form uncertainties in RANS simulations; evaluate framework performance on periodic hill and square duct
- Learning type: supervised learning
- ML algorithms: GP, ensemble Kalman method, Bayesian inference
- ML frameworks: inhouse, UQTk
- CFD framework: OpenFOAM
- Combination of CFD + ML: post
Z. J. Zhang et al.: Machine Learning Methods for Data-Driven Turbulence Modeling (2015)
- Aim: investigate machine learning capability to reconstruct the functional forms of fields extracted from high-fidelity simulations and experimental data; application to turbulent channel flow and bypass transition
- Learning type: supervised learning (regression)
- ML algorithms: multiscale GP, GP, MLP
- ML frameworks: Fast Artificial Neural Network library
- CFD framework: database
- Combination of CFD + ML: post