American Institute of Mathematical Sciences

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April  2022, 15(4): 913-930. doi: 10.3934/dcdss.2021138

An out-of-distribution-aware autoencoder model for reduced chemical kinetics

Pei Zhang 1, , Siyan Liu 1, , Dan Lu 1, , Ramanan Sankaran 1, and  Guannan Zhang 2,,

1. 

Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA
2. 

Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA

* Corresponding author: Guannan Zhang

Notice: This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan)

Received  August 2021 Revised  September 2021 Published  April 2022 Early access  November 2021

While detailed chemical kinetic models have been successful in representing rates of chemical reactions in continuum scale computational fluid dynamics (CFD) simulations, applying the models in simulations for engineering device conditions is computationally prohibitive. To reduce the cost, data-driven methods, e.g., autoencoders, have been used to construct reduced chemical kinetic models for CFD simulations. Despite their success, data-driven methods rely heavily on training data sets and can be unreliable when used in out-of-distribution (OOD) regions (i.e., when extrapolating outside of the training set). In this paper, we present an enhanced autoencoder model for combustion chemical kinetics with uncertainty quantification to enable the detection of model usage in OOD regions, and thereby creating an OOD-aware autoencoder model that contributes to more robust CFD simulations of reacting flows. We first demonstrate the effectiveness of the method in OOD detection in two well-known datasets, MNIST and Fashion-MNIST, in comparison with the deep ensemble method, and then present the OOD-aware autoencoder for reduced chemistry model in syngas combustion.

Keywords: Chemical kinetics, dimension reduction, neural network, uncertainty quantification, out of distribution.
Mathematics Subject Classification: 37M05, 37N99, 35K57, 68T07.
Citation: Pei Zhang, Siyan Liu, Dan Lu, Ramanan Sankaran, Guannan Zhang. An out-of-distribution-aware autoencoder model for reduced chemical kinetics. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 913-930. doi: 10.3934/dcdss.2021138
References:
[1]

A. Amini, W. Schwarting, A. Soleimany and D. Rus, Deep evidential regression, Advances in Neural Information Processing Systems, 33 (2020), 14927-14937. 

[2]

R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, John Wiley and Sons, New York, 1960.

[3]

J. H. Chen, A. Choudhary, B. de Supinski, M. DeVries, E. R. Hawkes, S. Klasky, W. K. Liao, K. L. Ma, J. Mellor-Crummey, N. Podhorszki, R. Sankaran, S. Shende and C. S. Yoo, Terascale direct numerical simulations of turbulent combustion using S3D, Computational Science & Discovery, 2 (2009), 015001.  doi: 10.1088/1749-4699/2/1/015001.

[4]

A. Coussement, O. Gicquel and A. Parente, MG-local-PCA method for reduced order combustion modeling, Proceedings of the Combustion Institute, 34 (2013), 1117-1123.  doi: 10.1016/j.proci.2012.05.073.

[5]

G. Esposito and H. Chelliah, Skeletal reaction models based on principal component analysis: Application to ethylene-air ignition, propagation, and extinction phenomena, Combustion and Flame, 158 (2011), 477-489.  doi: 10.1016/j.combustflame.2010.09.010.

[6]

Y. Gal and Z. Ghahramani, Dropout as a Bayesian approximation: Representing model uncertainty in deep learning, Proceedings of The 33rd International Conference on Machine Learning, 48 (2016), 1050-1059. 

[7]

E. R. Hawkes, R. Sankaran, J. C. Sutherland and J. H. Chen, Scalar mixing in direct numerical simulations of temporally evolving plane jet flames with skeletal {CO/H$_2$} kinetics, Proceedings of the Combustion Institute, 31 (2007), 1633-1640. 

[8]

M. D. Hoffman, D. M. Blei, C. Wang and J. Paisley, Stochastic variational inference, J. Mach. Learn. Res., 14 (2013), 1303–1347, URL http://jmlr.org/papers/v14/hoffman13a.html.

[9]

B. J. Isaac, A. Coussement, O. Gicquel, P. J. Smith and A. Parente, Reduced-order PCA models for chemical reacting flows, Combustion and Flame, 161 (2014), 2785-2800.  doi: 10.1016/j.combustflame.2014.05.011.

[10]

B. J. Isaac, J. N. Thornock, J. Sutherland, P. J. Smith and A. Parente, Advanced regression methods for combustion modelling using principal components, Combustion and Flame, 162 (2015), 2592-2601.  doi: 10.1016/j.combustflame.2015.03.008.

[11]

B. Lakshminarayanan, A. Pritzel and C. Blundell, Simple and scalable predictive uncertainty estimation using deep ensembles, Proceedings of the 31st International Conference on Neural Information Processing Systems, 17 (2017), 6405-6416. 

[12]

Y. Li, C.-W. Zhou, K. P. Somers, K. Zhang and H. J. Curran, The oxidation of 2-butene: A high pressure ignition delay, kinetic modeling study and reactivity comparison with isobutene and 1-butene, Proceedings of the Combustion Institute, 36 (2017), 403-411.  doi: 10.1016/j.proci.2016.05.052.

[13]

T. Lu and C. K. Law, Toward accommodating realistic fuel chemistry in large-scale computations, Progress in Energy and Combustion Science, 35 (2009), 192-215.  doi: 10.1016/j.pecs.2008.10.002.

[14]

D. J. C. MacKay, A practical Bayesian framework for backpropagation networks, Neural Comput., 4 (1992), 448-472.  doi: 10.1162/neco.1992.4.3.448.

[15]

M. R. Malik, B. J. Isaac, A. Coussement, P. J. Smith and A. Parente, Principal component analysis coupled with nonlinear regression for chemistry reduction, Combustion and Flame, 187 (2018), 30-41.  doi: 10.1016/j.combustflame.2017.08.012.

[16]

H. Mirgolbabaei and T. Echekki, A novel principal component analysis-based acceleration scheme for LES–ODT: An a priori study, Combustion and Flame, 160 (2013), 898-908.  doi: 10.2514/6.2013-168.

[17]

H. Mirgolbabaei and T. Echekki, Nonlinear reduction of combustion composition space with kernel principal component analysis, Combustion and Flame, 161 (2014), 118-126.  doi: 10.1016/j.combustflame.2013.08.016.

[18]

H. Mirgolbabaei and T. Echekki, The reconstruction of thermo-chemical scalars in combustion from a reduced set of their principal components, Combustion and Flame, 162 (2015), 1650-1652.  doi: 10.1016/j.combustflame.2014.11.027.

[19]

H. Mirgolbabaei, T. Echekki and N. Smaoui, A nonlinear principal component analysis approach for turbulent combustion composition space, International Journal of Hydrogen Energy, 39 (2014), 4622-4633.  doi: 10.1016/j.ijhydene.2013.12.195.

[20]

D. A. Nix and A. S. Weigend, Learning local error bars for nonlinear regression, In Advances in neural information processing systems, (1995), 489–496.

[21]

A. Parente, J. C. Sutherland, L. Tognotti and P. J. Smith, Identification of low-dimensional manifolds in turbulent flames, Proceedings of the Combustion Institute, 32 (2009), 1579-1586.  doi: 10.1016/j.proci.2008.06.177.

[22]

T. Pearce, A. Brintrup, M. Zaki and A. Neely, High-quality prediction intervals for deep learning: A distribution-free, ensembled approach, Proceedings of the 35th International Conference on Machine Learning, 80 (2018), 4075-4084. 

[23]

T. Pearce, F. Leibfried and A. Brintrup, Uncertainty in neural networks: Approximately Bayesian ensembling, In Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, Proceedings of Machine Learning Research, 108 (2020), 234–244.

[24]

T. S. Salem, H. Langseth and H. Ramampiaro, Prediction intervals: Split normal mixture from quality-driven deep ensembles, In Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI), Proceedings of Machine Learning Research, 124 (2020), 1179–1187.

[25]

R. Shan and T. Lu, Ignition and extinction in perfectly stirred reactors with detailed chemistry, Combustion and Flame, 159 (2012), 2069-2076.  doi: 10.1016/j.combustflame.2012.01.023.

[26]

E. Simhayev, G. Katz and L. Rokach, PIVEN: A deep neural network for prediction intervals with specific value prediction, 2021.

[27]

J. C. Sutherland and A. Parente, Combustion modeling using principal component analysis, Proceedings of the Combustion Institute, 32 (2009), 1563-1570. 

[28]

S. Vajda, P. Valko and T. Turányi, Principal component analysis of kinetic models, International Journal of Chemical Kinetics, 17 (1985), 55-81.  doi: 10.1002/kin.550170107.

[29]

P. Zhang, S. Liu, D. Lu, G. Zhang and R. Sankaran, A prediction interval method for uncertainty quantification of regression models, In ICLR 2021 SimDL Workshop, Virtual, 2021.

[30]

P. Zhang, R. Sankaran and E. R. Hawkes, A priori examination of reduced chemistry models derived from canonical stirred reactors using three-dimensional direct numerical simulation datasets, In AIAA Scitech, (2021), 1784 doi: 10.2514/6.2021-1784.

[31]

P. Zhang, R. Sankaran, M. Stoyanov, D. Lebrun-Grandie and C. E. Finney, Reduced models for chemical kinetics derived from parallel ensemble simulations of stirred reactors, In AIAA Scitech Forum, (2020), 0177. doi: 10.2514/6.2020-0177.

show all references

References:
[1]

A. Amini, W. Schwarting, A. Soleimany and D. Rus, Deep evidential regression, Advances in Neural Information Processing Systems, 33 (2020), 14927-14937. 

[2]

R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, John Wiley and Sons, New York, 1960.

[3]

J. H. Chen, A. Choudhary, B. de Supinski, M. DeVries, E. R. Hawkes, S. Klasky, W. K. Liao, K. L. Ma, J. Mellor-Crummey, N. Podhorszki, R. Sankaran, S. Shende and C. S. Yoo, Terascale direct numerical simulations of turbulent combustion using S3D, Computational Science & Discovery, 2 (2009), 015001.  doi: 10.1088/1749-4699/2/1/015001.

[4]

A. Coussement, O. Gicquel and A. Parente, MG-local-PCA method for reduced order combustion modeling, Proceedings of the Combustion Institute, 34 (2013), 1117-1123.  doi: 10.1016/j.proci.2012.05.073.

[5]

G. Esposito and H. Chelliah, Skeletal reaction models based on principal component analysis: Application to ethylene-air ignition, propagation, and extinction phenomena, Combustion and Flame, 158 (2011), 477-489.  doi: 10.1016/j.combustflame.2010.09.010.

[6]

Y. Gal and Z. Ghahramani, Dropout as a Bayesian approximation: Representing model uncertainty in deep learning, Proceedings of The 33rd International Conference on Machine Learning, 48 (2016), 1050-1059. 

[7]

E. R. Hawkes, R. Sankaran, J. C. Sutherland and J. H. Chen, Scalar mixing in direct numerical simulations of temporally evolving plane jet flames with skeletal {CO/H$_2$} kinetics, Proceedings of the Combustion Institute, 31 (2007), 1633-1640. 

[8]

M. D. Hoffman, D. M. Blei, C. Wang and J. Paisley, Stochastic variational inference, J. Mach. Learn. Res., 14 (2013), 1303–1347, URL http://jmlr.org/papers/v14/hoffman13a.html.

[9]

B. J. Isaac, A. Coussement, O. Gicquel, P. J. Smith and A. Parente, Reduced-order PCA models for chemical reacting flows, Combustion and Flame, 161 (2014), 2785-2800.  doi: 10.1016/j.combustflame.2014.05.011.

[10]

B. J. Isaac, J. N. Thornock, J. Sutherland, P. J. Smith and A. Parente, Advanced regression methods for combustion modelling using principal components, Combustion and Flame, 162 (2015), 2592-2601.  doi: 10.1016/j.combustflame.2015.03.008.

[11]

B. Lakshminarayanan, A. Pritzel and C. Blundell, Simple and scalable predictive uncertainty estimation using deep ensembles, Proceedings of the 31st International Conference on Neural Information Processing Systems, 17 (2017), 6405-6416. 

[12]

Y. Li, C.-W. Zhou, K. P. Somers, K. Zhang and H. J. Curran, The oxidation of 2-butene: A high pressure ignition delay, kinetic modeling study and reactivity comparison with isobutene and 1-butene, Proceedings of the Combustion Institute, 36 (2017), 403-411.  doi: 10.1016/j.proci.2016.05.052.

[13]

T. Lu and C. K. Law, Toward accommodating realistic fuel chemistry in large-scale computations, Progress in Energy and Combustion Science, 35 (2009), 192-215.  doi: 10.1016/j.pecs.2008.10.002.

[14]

D. J. C. MacKay, A practical Bayesian framework for backpropagation networks, Neural Comput., 4 (1992), 448-472.  doi: 10.1162/neco.1992.4.3.448.

[15]

M. R. Malik, B. J. Isaac, A. Coussement, P. J. Smith and A. Parente, Principal component analysis coupled with nonlinear regression for chemistry reduction, Combustion and Flame, 187 (2018), 30-41.  doi: 10.1016/j.combustflame.2017.08.012.

[16]

H. Mirgolbabaei and T. Echekki, A novel principal component analysis-based acceleration scheme for LES–ODT: An a priori study, Combustion and Flame, 160 (2013), 898-908.  doi: 10.2514/6.2013-168.

[17]

H. Mirgolbabaei and T. Echekki, Nonlinear reduction of combustion composition space with kernel principal component analysis, Combustion and Flame, 161 (2014), 118-126.  doi: 10.1016/j.combustflame.2013.08.016.

[18]

H. Mirgolbabaei and T. Echekki, The reconstruction of thermo-chemical scalars in combustion from a reduced set of their principal components, Combustion and Flame, 162 (2015), 1650-1652.  doi: 10.1016/j.combustflame.2014.11.027.

[19]

H. Mirgolbabaei, T. Echekki and N. Smaoui, A nonlinear principal component analysis approach for turbulent combustion composition space, International Journal of Hydrogen Energy, 39 (2014), 4622-4633.  doi: 10.1016/j.ijhydene.2013.12.195.

[20]

D. A. Nix and A. S. Weigend, Learning local error bars for nonlinear regression, In Advances in neural information processing systems, (1995), 489–496.

[21]

A. Parente, J. C. Sutherland, L. Tognotti and P. J. Smith, Identification of low-dimensional manifolds in turbulent flames, Proceedings of the Combustion Institute, 32 (2009), 1579-1586.  doi: 10.1016/j.proci.2008.06.177.

[22]

T. Pearce, A. Brintrup, M. Zaki and A. Neely, High-quality prediction intervals for deep learning: A distribution-free, ensembled approach, Proceedings of the 35th International Conference on Machine Learning, 80 (2018), 4075-4084. 

[23]

T. Pearce, F. Leibfried and A. Brintrup, Uncertainty in neural networks: Approximately Bayesian ensembling, In Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, Proceedings of Machine Learning Research, 108 (2020), 234–244.

[24]

T. S. Salem, H. Langseth and H. Ramampiaro, Prediction intervals: Split normal mixture from quality-driven deep ensembles, In Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI), Proceedings of Machine Learning Research, 124 (2020), 1179–1187.

[25]

R. Shan and T. Lu, Ignition and extinction in perfectly stirred reactors with detailed chemistry, Combustion and Flame, 159 (2012), 2069-2076.  doi: 10.1016/j.combustflame.2012.01.023.

[26]

E. Simhayev, G. Katz and L. Rokach, PIVEN: A deep neural network for prediction intervals with specific value prediction, 2021.

[27]

J. C. Sutherland and A. Parente, Combustion modeling using principal component analysis, Proceedings of the Combustion Institute, 32 (2009), 1563-1570. 

[28]

S. Vajda, P. Valko and T. Turányi, Principal component analysis of kinetic models, International Journal of Chemical Kinetics, 17 (1985), 55-81.  doi: 10.1002/kin.550170107.

[29]

P. Zhang, S. Liu, D. Lu, G. Zhang and R. Sankaran, A prediction interval method for uncertainty quantification of regression models, In ICLR 2021 SimDL Workshop, Virtual, 2021.

[30]

P. Zhang, R. Sankaran and E. R. Hawkes, A priori examination of reduced chemistry models derived from canonical stirred reactors using three-dimensional direct numerical simulation datasets, In AIAA Scitech, (2021), 1784 doi: 10.2514/6.2021-1784.

[31]

P. Zhang, R. Sankaran, M. Stoyanov, D. Lebrun-Grandie and C. E. Finney, Reduced models for chemical kinetics derived from parallel ensemble simulations of stirred reactors, In AIAA Scitech Forum, (2020), 0177. doi: 10.2514/6.2020-0177.

Figure 1.  Elemental mass conservation errors. An elemental error of 20% is expected to cause large errors in temperature prediction thus false extinction/ignition events in engine simulations. With OOD-aware model, non-physical solutions with conservation laws violated can be detected before leading to poor engine designs
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Figure 2.  Estimation of 95% PI prediction in toy regression task $ y = x^3 + \varepsilon $ with asymmetric noise $ \varepsilon $. The 95% PI produced by PI3NN captures about 95% of training data with tight bounds within training region, while DE produces an unnecessarily wide lower bound. Both methods produce reasonably wide PIs in the OOD region
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Figure 3.  Architecture diagrams of AE with DE and PI3NN. Neurons are represented by the circles with different edge colors for different types of layers, i.e., red for input and output layers, green for bottleneck layer, and blue for other hidden layers. The red circles filled with gray are outputs for variance/standard deviation variables. The diagrams shown here are simplified for illustration only. The actual numbers of layers and neurons in each layer vary in different experiments
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Figure 4.  Comparison of OOD detection accuracy in MNIST (ID) and Fashion-MNIST (OOD) test sets between DE and PI3NN methods. Both methods capture the difference between ID and OOD with larger uncertainties for OOD samples. The confusion matrices show that the majority of test samples are correctly detected as OOD (true positive) or ID (true negative)
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Figure 5.  Receiver operating characteristic curves of DE and PI3NN in OOD detection. All methods can capture the OOD samples well in the image experiment. Though DE with ensemble size of 2 shows better accuracy than PI3NN, it requires two runs while PI3NN only needs a single run. DE with a single run shows lower OOD detection accuracy than PI3NN
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Figure 6.  Normalized joint histogram of predicted values by AE (with $ n_z = 2 $) and true values for 12 thermo-chemical state variables, i.e., temperature and mass fractions of 11 species, in PSR test set. The dark red area along the diagonal line shows that AE can reduce the state dimension of syngas CO/H$ _2 $ combustion from $ n_x = 12 $ to $ n_z = 2 $ without much loss of accuracy
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Figure 7.  Predictive uncertainty (the width of PI) vs. predictive error of the ID (red) and OOD (blue) test sets from the PI3NN and DE methods. Scatter points are samples randomly selected from the two test sets. The filled contours show the normalized joint histogram for samples in ID/OOD test sets, with light red/blue colors representing fewer samples and dark red/blue representing more samples. DE with a single run $ \left((a)-(c)\right) $ fails to capture the difference in uncertainty for ID and OOD samples. DE with 10 runs $ \left((d)-(f)\right) $ shows improved but still limited separation of OOD samples from ID samples and it fails to produce the uncertainty-error correlation. PI3NN $ \left((g)-(i)\right) $ shows a strong correlation between the uncertainty and the error and clearly demonstrates that OOD and ID have different uncertainty magnitudes
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Figure 8.  Receiver operating characteristic curves of PI3NN and DE ($ M = 1, 10, 20 $) methods in OOD detection. PI3NN shows the best OOD detection accuracy. DE shows improved detection accuracy with increasing ensemble size ($ M $)
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