├── figs ├── features.png ├── locations.png ├── PIML-algorithm.pdf └── PIML-algorithm.png ├── README.asciidoc ├── database └── pehill │ ├── XiEta-RANS │ └── Re10595 │ │ ├── line2_XiEta.xy │ │ ├── line10_XiEta.xy │ │ ├── line3_XiEta.xy │ │ ├── line1_XiEta.xy │ │ ├── line4_XiEta.xy │ │ ├── line5_XiEta.xy │ │ ├── line6_XiEta.xy │ │ ├── line7_XiEta.xy │ │ ├── line8_XiEta.xy │ │ └── line9_XiEta.xy │ └── deltaFields │ ├── Re5600 │ └── deltaField │ └── Re10595 │ └── deltaField └── regressionSolver.py /figs/features.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/xiaoh/turbulence-modeling-PIML/HEAD/figs/features.png -------------------------------------------------------------------------------- /figs/locations.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/xiaoh/turbulence-modeling-PIML/HEAD/figs/locations.png -------------------------------------------------------------------------------- /figs/PIML-algorithm.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/xiaoh/turbulence-modeling-PIML/HEAD/figs/PIML-algorithm.pdf -------------------------------------------------------------------------------- /figs/PIML-algorithm.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/xiaoh/turbulence-modeling-PIML/HEAD/figs/PIML-algorithm.png -------------------------------------------------------------------------------- /README.asciidoc: -------------------------------------------------------------------------------- 1 | # Data-driven Reynolds stress modeling with physics-informed machine learning 2 | 3 | Jinlong Wu, Carlos Michelén-Ströfer, Jianxun Wang, Heng Xiao 4 | 5 | 6 | A complex physical system characterized by a wide range of temporal and spatial scales, turbulence is among the last unsolved problems in classical physics that affects natural and engineered systems from sub-meter to planetary scales. Current simulations of turbulent flows rely on Reynolds Averaged Navier Stokes equations with closure models. In light of the decades-long stagnation in traditional turbulence modeling, we proposed data-driven, physics-informed machine learning method for turbulence modeling. 7 | 8 | This repository contains tutorial Python scripts used to generated results published in the <> and <>. Also see: <>. Review relevant review articles: <> and <> 9 | 10 | You can download the Jupyter Notebook to run and play with the code, or simply download the HTML version of the tutorial](regressionSolver.html) generated by the Jupyter Notebook. 11 | 12 | Please send bug reports and comments to: Heng Xiao (hengxiao@vt.edu) + 13 | 14 | 15 | [bibliography] 16 | References 17 | ---------- 18 | 19 | [bibliography] 20 | 21 | - [[[Wang2017Physics]]] J.-X. Wang, J.-L. Wu, and H. Xiao. Physics informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data. *Physical Review Fluids*, 2(3), 034603, 1-22, 2017. https://doi.org/10.1103/PhysRevFluids.2.034603[DOI:10.1103/PhysRevFluids.2.034603] 22 | 23 | - [[[Wu2018Physics]]] J.-L. Wu, H. Xiao and E. G. Paterson. Physics-informed machine learning approach for augmenting turbulence models: A comprehensive framework. *Physical Review Fluids*, 3, 074602, 1-28, 2018. https://doi.org/10.1103/PhysRevFluids.3.074602[DOI: 10.1103/PhysRevFluids.3.074602] 24 | 25 | - [[[Wu2018Representation]]] J.-L. Wu, R. Sun, S. Laizet and H. Xiao. Representation of Reynolds stress perturbations with application in machine-learning-assisted turbulence modeling. *Computer Methods in Applied Mechanics and Engineering*，346, 707-726, 2019. https://doi.org/10.1016/j.cma.2018.09.010[DOI: 10.1016/j.cma.2018.09.010] 26 | 27 | - [[[Duraisamy2019Turbulence]]] K. Duraisamy, G. Iaccarino, and H. Xiao. Turbulence modeling in the age of data. *Annual Review of Fluid Mechanics*, 51, 357-377, 2019. https://doi.org/10.1146/annurev-fluid-010518-040547[DOI: 10.1146/annurev-fluid-010518-040547] 28 | 29 | - [[[Xiao2019RANS]]] H. Xiao and P. Cinnella. Quantification of model uncertainty in RANS simulations: A review. *Progress in Aerospace Sciences*. In Press, 2018. Invited review article. Also available at: https://arxiv.org/abs/1806.10434[arXiv:1806.10434] 30 | -------------------------------------------------------------------------------- /database/pehill/XiEta-RANS/Re10595/line2_XiEta.xy: -------------------------------------------------------------------------------- 1 | 4.496069921697303240e-01 6.041331252195787371e-01 2 | 4.448249362737060064e-01 5.789282853326185307e-01 3 | 4.425839887215842761e-01 5.672049543098548474e-01 4 | 4.418712526788732831e-01 5.636241131324162268e-01 5 | 4.419120371171237349e-01 5.640502986175401912e-01 6 | 4.421164183277975779e-01 5.653545732701323878e-01 7 | 4.421078710683494783e-01 5.655686706154372523e-01 8 | 4.417017387070436563e-01 5.636620008805688364e-01 9 | 4.408016037107002627e-01 5.591993914976594038e-01 10 | 4.394023784209574179e-01 5.521527685639017191e-01 11 | 4.375450112067337249e-01 5.427400208173226437e-01 12 | 4.353005123304059532e-01 5.313279201310753264e-01 13 | 4.327533843604198238e-01 5.183536509668178471e-01 14 | 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6.083599119145702838e-01 77 | 4.470209678234502437e-01 5.907382773876066473e-01 78 | 4.440666082548396787e-01 5.753888890881093676e-01 79 | 4.415703192669920840e-01 5.624158766856443004e-01 80 | 4.395353512998517731e-01 5.518438322621700420e-01 81 | 4.379551417132167002e-01 5.436328391959008854e-01 82 | 4.368099164228549380e-01 5.376840770032982331e-01 83 | 4.360764032893007514e-01 5.338706518943533430e-01 84 | 4.357271503230154552e-01 5.320558883098210901e-01 85 | 4.357394788871651881e-01 5.321199557104416833e-01 86 | 4.361001966890032389e-01 5.339943058167183620e-01 87 | 4.368083339125368725e-01 5.376779520006080837e-01 88 | 4.378861027922537064e-01 5.432761945758064481e-01 89 | 4.393724999540656873e-01 5.509977213919131644e-01 90 | 4.413249966586811901e-01 5.611452962730792970e-01 91 | 4.438142847173578431e-01 5.740779934574127275e-01 92 | 4.469362843363707816e-01 5.903004844093144854e-01 93 | 4.509217576121944959e-01 6.110098227636443502e-01 94 | 4.564960310763532259e-01 6.399750498707449387e-01 95 | 4.655944177765708458e-01 6.872368562815293069e-01 96 | 4.783249730193667171e-01 7.533986598267895518e-01 97 | 4.882355213466053057e-01 8.048953795277210554e-01 98 | 4.933879440796026805e-01 8.316719652729291923e-01 99 | -------------------------------------------------------------------------------- /regressionSolver.py: -------------------------------------------------------------------------------- 1 | 2 | # coding: utf-8 3 | 4 | # In[1]: 5 | 6 | 7 | from IPython.display import Image 8 | 9 | 10 | # # Overview 11 | # This code implements the PIML regresson procedure described in Wang et al. 2017, with the case of flow over periodic hills as example. 12 | # 13 | # * J.-X. Wang, J.-L. Wu, and H. Xiao. Physics informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data. Physical Review Fluids. 2(3), 034603, 1-22, 2017. https://doi.org/10.1103/PhysRevFluids.2.034603[DOI:10.1103/PhysRevFluids.2.034603] 14 | 15 | # # Algorithm of PIML-Based Turbulence Modeling 16 | # The overall procedure can be summarized as follows: 17 | # 18 | # 1. Perform baseline RANS simulations on both the training flows and the test flow. 19 | # 1. Compute the input feature field $\mathbf{q}(\mathbf{x})$ based on the local 20 | # RANS flow variables. 21 | # 1. Compute the discrepancies field $\Delta \boldsymbol{\tau}(\mathbf{x})$ in the RANS-modeled 22 | # Reynolds stresses for the training flows based on the high-fidelity data. 23 | # 1. **Construct regression functions $ f: \mathbf{q} \mapsto \Delta \boldsymbol{\tau}$ for the 24 | # discrepancies based on the training data prepared in Step 3, using machine learning algorithms.** 25 | # 1. Compute the Reynolds stress discrepancies for the test flow by querying the regression 26 | # functions. The Reynolds stresses can subsequently be obtained by correcting the baseline RANS 27 | # predictions with the evaluated discrepancies. 28 | # 1. Propagate the corrected Reynolds stresses to the mean velocity field by solving the RANS 29 | # equations with the corrected Reynolds stress field. 30 | # 31 | # 32 | # **This code only performs Step 4.** (see the green-shaded box below in the flow chart). The training data prepared in Steps 1-3 are saved in _database_ folder. 33 | # 34 | 35 | # In[2]: 36 | 37 | 38 | Image(filename='figs/PIML-algorithm.png') 39 | 40 | 41 | # # Machine learning algorithms 42 | # 43 | # The procedure implented here consists of three parts: 44 | # 45 | # 1. load training and test data 46 | # 2. construct regression function $\Delta \boldsymbol{\tau} (\mathbf{q})$ (detailed below) 47 | # 3. plot the anisotropy parameters $\xi$ and $\eta$ (componebnts of $\Delta \boldsymbol{\tau}$) and compare with ground truth (DNS) 48 | # 49 | # We used two algorithms to build the regression function: 50 | # 51 | # * Random Forests (based on scikit-learn). This is what was used in Wang et al. 52 | # * Neural networks (based on Tensorflow) 53 | # 54 | # Both algorithms yielded similar results, but the former is cheaper computationally. 55 | 56 | # The input features consist of 12 variables (see Table 1 below and also Wang et al.) 57 | 58 | # In[3]: 59 | 60 | 61 | Image(filename='figs/features.png') 62 | 63 | 64 | # In[4]: 65 | 66 | 67 | get_ipython().run_line_magic('matplotlib', 'inline') 68 | ## Import system modules 69 | # sci computing 70 | import numpy as np 71 | # sklearn importing 72 | from sklearn.ensemble.forest import RandomForestRegressor 73 | # plotting 74 | import matplotlib.pyplot as plt # for plotting 75 | #import matplotlib as mp 76 | 77 | # keras importing 78 | from keras.models import Sequential 79 | from keras.layers import Dense 80 | 81 | import time 82 | 83 | 84 | # In[5]: 85 | 86 | 87 | def loadTrainingData(caseName, ReNum): 88 | trainFeaturesFile = './database/' + caseName + '/markers/' + ReNum + '/markerFile' 89 | trainResponsesFile = './database/' + caseName + '/deltaFields/' + ReNum + '/deltaField' 90 | trainFeatures = np.loadtxt(trainFeaturesFile) 91 | trainResponses = np.loadtxt(trainResponsesFile) 92 | return trainFeatures, trainResponses 93 | 94 | 95 | # In[6]: 96 | 97 | 98 | def loadTestData(caseName, ReNum): 99 | testFeaturesFile = './database/' + caseName + '/markers/' + ReNum + '/markerFile' 100 | testResponsesFile = './database/' + caseName + '/deltaFields/' + ReNum + '/deltaField' 101 | testFeatures = np.loadtxt(testFeaturesFile) 102 | testResponses = np.loadtxt(testResponsesFile) 103 | return testFeatures, testResponses 104 | 105 | 106 | # In[7]: 107 | 108 | 109 | def randomForest(trainFeatures, trainResponses, testFeatures, maxFeatures = 'log2', nTree=100): 110 | ## Settings of random forests regressor 111 | regModel = RandomForestRegressor(n_estimators=nTree, max_features=maxFeatures) 112 | ## Train the random forests regressor 113 | regModel.fit(trainFeatures, trainResponses) 114 | ## Prediction 115 | testResponsesPred = regModel.predict(testFeatures) 116 | return testResponsesPred 117 | 118 | 119 | # In[8]: 120 | 121 | 122 | def keras_nn(trainFeatures, trainResponses, testFeatures, Nepochs = 100): 123 | 124 | ''' 125 | This function is to construct neural network based on the training data and predict the 126 | response given test data. 127 | Two hidded layers are used, the number of neurals are 64 and 32, respectively. 128 | ''' 129 | model = Sequential() 130 | # The first hidder layer of NN 131 | model.add(Dense(64, input_dim=trainFeatures.shape[1], activation='relu')) 132 | # The second hidder layer of NN 133 | model.add(Dense(32, activation='relu')) 134 | model.add(Dense(2, activation='tanh')) 135 | model.compile(loss='mean_squared_error', optimizer='adam') 136 | # Training 137 | model.fit(trainFeatures, trainResponses, epochs=Nepochs, batch_size=200, verbose=0) 138 | # Prediction 139 | testResponsesPred = model.predict(testFeatures) 140 | return testResponsesPred 141 | 142 | 143 | # In[61]: 144 | 145 | 146 | def plotXiEta(XiEta_RANS, testResponses, testResponsesPred, name, symbol='r^'): 147 | # Reconstruct Barycentric coordinates 148 | XiEta_DNS = XiEta_RANS + testResponses 149 | XiEta_ML = XiEta_RANS + testResponsesPred 150 | # Plot Reynolds stress anisotropy in Barycentric triangle 151 | interval = 2 152 | pointsNum = int(XiEta_RANS.shape[0]) 153 | plt.figure() 154 | plt.plot([0,1,0.5,0.5,0],[0,0,3**0.5/2.0,3**0.5/2.0,0],'g-') 155 | p1, = plt.plot(XiEta_RANS[:pointsNum:interval,0],XiEta_RANS[:pointsNum:interval,1], 156 | 'bo', markerfacecolor='none', markeredgecolor='b', 157 | markeredgewidth=2, markersize=10) 158 | p2, = plt.plot(XiEta_DNS[:pointsNum:interval,0],XiEta_DNS[:pointsNum:interval,1], 159 | 'ks', markerfacecolor='none', markeredgecolor='k', 160 | markeredgewidth=2, markersize=10) 161 | p3, = plt.plot(XiEta_ML[:pointsNum:interval,0],XiEta_ML[:pointsNum:interval,1], 162 | symbol, markerfacecolor='none', #markeredgecolor='r', 163 | markeredgewidth=2, markersize=10) 164 | lg = plt.legend([p1,p2,p3], ['RANS', 'DNS', name], loc = 0) 165 | lg.draw_frame(False) 166 | plt.ylim([0,3**0.5/2.0]) 167 | plt.show() 168 | 169 | 170 | # In[62]: 171 | 172 | 173 | def comparePlotRFNN(XiEta_RANS, testResponses, testResponsesPred_RF, testResponsesPred_NN): 174 | 175 | XiEta_DNS = XiEta_RANS + testResponses 176 | XiEta_RF = XiEta_RANS + testResponsesPred_RF 177 | XiEta_NN = XiEta_RANS + testResponsesPred_NN 178 | # Plot Reynolds stress anisotropy in Barycentric triangle 179 | interval = 2 180 | pointsNum = int(XiEta_RANS.shape[0]) 181 | plt.figure() 182 | plt.plot([0,1,0.5,0.5,0],[0,0,3**0.5/2.0,3**0.5/2.0,0],'g-') 183 | p1, = plt.plot(XiEta_RANS[:pointsNum:interval,0],XiEta_RANS[:pointsNum:interval,1], 184 | 'bo', markerfacecolor='none', markeredgecolor='b', 185 | markeredgewidth=1.5, markersize=8) 186 | p2, = plt.plot(XiEta_DNS[:pointsNum:interval,0],XiEta_DNS[:pointsNum:interval,1], 187 | 'ks', markerfacecolor='none', markeredgecolor='k', 188 | markeredgewidth=1.5, markersize=8) 189 | p3, = plt.plot(XiEta_RF[:pointsNum:interval,0],XiEta_RF[:pointsNum:interval,1], 190 | 'r^', markerfacecolor='none', markeredgecolor='r', 191 | markeredgewidth=1.5, markersize=8) 192 | p4, = plt.plot(XiEta_NN[:pointsNum:interval,0],XiEta_NN[:pointsNum:interval,1], 193 | 'r+', markerfacecolor='none', markeredgecolor='g', 194 | markeredgewidth=1.5, markersize=8) 195 | lg = plt.legend([p1,p2,p3, p4], ['RANS', 'DNS', 'RF', 'NN'], loc = 0) 196 | lg.draw_frame(False) 197 | plt.ylim([0,3**0.5/2.0]) 198 | plt.show() 199 | 200 | 201 | # In[63]: 202 | 203 | 204 | def iterateLines(dataFolderRANS, testResponses, testResponsesPred, name, symbol='r^'): 205 | # Start index of different sample lines 206 | indexList = [0, 98, 191, 287, 385, 483, 581, 679, 777, 875, 971] 207 | # Make plots at x=2 and x=4 208 | for iterN in [3,5]: 209 | XiEta = np.loadtxt(dataFolderRANS + 'line' + str(iterN) + '_XiEta.xy') 210 | startIndex = indexList[iterN-1] 211 | endIndex = indexList[iterN] 212 | plotXiEta(XiEta, testResponses[startIndex:endIndex,:], 213 | testResponsesPred[startIndex:endIndex,:], name, symbol) 214 | #plt.show() 215 | 216 | 217 | # In[64]: 218 | 219 | 220 | def compareResults(dataFolderRANS, testResponses, testResponsesPred_RF, testResponsesPred_NN): 221 | ## compare the results in one plot 222 | # Start index of different sample lines 223 | indexList = [0, 98, 191, 287, 385, 483, 581, 679, 777, 875, 971] 224 | # Make plots at x=2 and x=4 225 | for iterN in [3,5]: 226 | XiEta = np.loadtxt(dataFolderRANS + 'line' + str(iterN) + '_XiEta.xy') 227 | startIndex = indexList[iterN-1] 228 | endIndex = indexList[iterN] 229 | comparePlotRFNN(XiEta, testResponses[startIndex:endIndex,:], 230 | testResponsesPred_RF[startIndex:endIndex,:], 231 | testResponsesPred_NN[startIndex:endIndex,:]) 232 | 233 | 234 | # Now, plot the anisotropy at the two locations $x/H = 2$ and 4: 235 | 236 | # In[65]: 237 | 238 | 239 | Image(filename='figs/locations.png') 240 | 241 | 242 | # In[66]: 243 | 244 | 245 | # if __name__== "__main__": 246 | # Load data 247 | trainFeatures, trainResponses = loadTrainingData('pehill', 'Re5600') 248 | testFeatures, testResponses = loadTestData('pehill', 'Re10595') 249 | time_begin_RF = time.time() 250 | # Make prediction via the random forest regressor 251 | testResponsesPred_RF = randomForest(trainFeatures, trainResponses, testFeatures, 6, 100) 252 | time_end_RF = time.time() 253 | # Make plots of Reynolds stress anisotropy 254 | dataFolderRANS = './database/pehill/XiEta-RANS/Re10595/' 255 | iterateLines(dataFolderRANS, testResponses, testResponsesPred_RF, name='RF') 256 | plt.show() 257 | 258 | 259 | # In[67]: 260 | 261 | 262 | Nepochs = 1000 263 | time_begin_NN = time.time() 264 | # Make prediction via the neural network 265 | testResponsesPred_NN = keras_nn(trainFeatures, trainResponses, testFeatures, Nepochs) 266 | time_end_NN = time.time() 267 | 268 | 269 | # ### Make plots of Reynolds stress anisotropy (NN results) 270 | 271 | # In[72]: 272 | 273 | 274 | dataFolderRANS = './database/pehill/XiEta-RANS/Re10595/' 275 | symbol = 'g+' 276 | iterateLines(dataFolderRANS, testResponses, testResponsesPred_NN, name='NN', symbol='m+') 277 | plt.show() 278 | 279 | 280 | # ## Compare the results of random forest and neural network 281 | 282 | # In[73]: 283 | 284 | 285 | compareResults(dataFolderRANS, testResponses, testResponsesPred_RF, testResponsesPred_NN) 286 | plt.show() 287 | 288 | 289 | # ## Comparison of computational cost between RF and NN 290 | # 291 | # The cost depends on the number of epoches, which is written in the title of the plot. 292 | 293 | # In[74]: 294 | 295 | 296 | cost_time_RF = time_end_RF - time_begin_RF 297 | cost_time_NN = time_end_NN - time_begin_NN 298 | 299 | xlabel = np.arange(2) 300 | plt.bar(xlabel, [cost_time_RF, cost_time_NN], 0.4) 301 | plt.ylabel('CPU time (sec') 302 | plt.xticks(xlabel, ('RF', 'NN')) 303 | 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