1 Introduction
Combustion is a ubiquitous process with applications that vary extensively, ranging from heating and transportation to mass production of metallic and ceramic nanoparticles. Combustion involves different processes that span a wide range of length and time scales, and are controlled by the delicate interplay among several chemical and physical phenomena, such as chemical kinetics, thermodynamics, fluid mechanics, and heat transfer. Over the past decades, chemical modeling has played a great role due to the continuing expansion in computational capabilities, development of diagnostic methods, quantum chemistry methods and kinetic theory. Within this context, the approach to developing detailed kinetic models (mechanisms) of fuel combustion involves compiling a set of elementary reactions whose rate parameters may be determined from individual rate measurements, reactionrate theory, or a combination of both. These detailed mechanisms consist of a large number of chemical species and reactions: for example, the mechanisms of large hydrocarbons, as the ones used in surrogates for real transportation fuels, typically describe the interactions of – 10^{3} species via 10^{3} – 10^{4} reactions Law (2007). The resulting networks of reactions show high nonlinearity and dimensionality. The relevance of the different species and reactions not only changes with time during combustion, but also depends on the specific system and conditions (e.g., temperature, pressure, equivalence ratio).
The analysis of chemical kinetic mechanisms, particularly understanding the relevance of different species and reactions in various conditions, has several important applications. While chemical mechanisms have been active research area for last several decades, refinements in implemented pathways and reaction rate parameters are still necessary. Thus, identification of most influential reactions in the network provides a critical information on which chemistry in the mechanism should be improved. The analysis of chemical reaction networks can be leveraged to generate reduced chemical mechanisms that have similar combustion characteristics to the parent mechanism but present a significantly smaller number of species and reactions. A direct application of this work is the use of reduced mechanisms into computational fluid dynamics simulations to describe realistic combustion systems like engines, overcoming the complexity and computational cost of large kinetic mechanisms that make full kinetic models unusable in most cases Tomlin et al. (1997); Turányi (1990); Lu and Law (2009). For example, computational simulations of internal combustion engines require the coupling of chemical models for the conversion of the fuel into combustion products with numerical treatments of the fluid dynamics of reacting flows Law (2007). Evaluating the influence of each species or reaction on the combustion process is a widely used strategy for developing reduced kinetic mechanisms Tomlin et al. (1997); Turányi (1990); Vajda et al. (1985); Maas and Pope (1992a, b); Lam and Goussis (1994); Lam (1993); Bhattacharjee et al. (2003); Lu and Law (2005, 2008); Pepiotdesjardins and Pitsch (2008); Niemeyer et al. (2010). Perturbation sensitivity analysis is to date the most commonly used method. Of particular interest is the influence of kinetics parameters on combustion characteristics, such as ignition delay or heat release trends. The brute force approach to such sensitivity analysis is to evaluate the impact of each reaction on the combustion feature of interest by perturbing the rates one by one Chang et al. (1987); Lifshitz and Frenklach (1980); Fridlyand et al. (2017). This brute force scheme is straightforward to implement but becomes overly computationally expensive for large mechanisms as a full simulation run is required for every change in the kinetic parameters. More details on various sensitivity analysis methods and their applications can be found in Tomlin et al. (1997); Turányi (1990).
An alternative approach to sensitivity analysis uses the rates of production of the species. After a simulation, the reaction rate of each reaction in the original solution is analyzed to find reactions that contribute to production/consumption of some of the key species. While this method can be effective for discovery of influential species, the method is not effective for general combustion system analysis across different thermodynamic conditions.
In addition to these two common techniques, various other reactionbased methods can be found in the literature that aim at identifying unimportant reactions in the chemical network Vajda et al. (1985); Maas and Pope (1992a, b); Lam and Goussis (1994); Lam (1993); Bhattacharjee et al. (2003). These methods are often presented as mechanism reduction approaches. However, in general, the computational cost of these techniques is high when applied to large scale mechanisms, as many perturbations of the rate parameters or Jacobian evaluation of functions are required.
In this paper, we propose a novel approach based on datadriven sparselearning methods to analyze reaction mechanisms and identify the most influential reactions. The datadriven approach simplifies complex nonlinear dynamical models to scalable linear models; the sparselearning methodology removes weak interdependencies with the reaction network and creates a sparse network. Unlike other methods in the literature, the proposed method does not require any understanding of the underlying chemical process, instead it learns from data generated from the chemical process. The proposed method is applied to the analysis of two mechanisms describing the chemistry of H2 fuel and C3H8 fuel, respectively. We also demonstrate how this new method can be used to derive a reduced version of the parent chemical mechanism without the need for additional simulations, making it considerably more efficient compared to other equivalent methods. By finding a set of influential reactions using species concentrations and reaction rates, the proposed method is not biased toward any specific combustion property Harirchi et al. (2017).
2 Methodology
2.1 Model Framework
The time evolution of a reactive system is commonly modeled using massaction kinetic equations Feinberg (1987); Gunawardena (2003); Gillespie (2007); Chellaboina et al. (2009); Anderson and Kurtz (2011), which, by applying Euler discretization method Ascher and Petzold (1998), can be written as
(1) 
where is an vector of concentrations of all the species at time , is the number of reactions whose stoichiometric coefficients are contained in the matrix , is the sampling time used for discretization, and denotes the discretization error and process noise. Finally, is the vector of the rates of reactions, which can be expressed as:
(2) 
where is the rate constant for the reaction, denotes the component of , and are the stoichiometric coefficients for the reactants of reaction and zero otherwise.
For each reaction, we introduce a selection (binary case) or weight (realvalued case) variable , which describes the significance of the reaction at time (larger values correspond to more influential reactions), and the selection/weight vector , which represents the weights of all the reactions at time .
Considering the binary case, where selects a subset of all reactions to describe the evolution of the system, an error is introduced in the concentration of the species that can be expressed for the species as
(3) 
where denotes the row of matrix and denotes the elementwise product. Based on these definitions, we can impose a constraint on the concentration error of each species at all times as
(4) 
(5) 
where is a normalization factor at time , defined as the sum of absolute changes in all concentrations at time , and is a tuning parameter that indicates the acceptable error tolerance of , e.g., enforces a maximum of 5% error.
The constraint introduced in Eq. 4 is effective against the addition of noise in the concentration evolution but is not very effective in limiting constant drifts in the species concentration. To correct for this issue, we added an additional constraint to limit the propagation of error over time. For the change in concentration of the species in time horizon , we have that
(6) 
where is the number of time samples in the time horizon and is a tuning parameter that limits the amount of concentration drift. For the choice of , two points should be considered:

Physical effect: larger choice of enforces the aggregated error in the concentrations of all species that is caused by removing some of the reactions to remain in the acceptable range for a larger time horizon. On the other hand, the choice of smaller , relaxes the propagated error to be in the same range , but for a smaller time horizon. Consequently, the larger results in a reduced mechanism with more reactions.

Computational cost: larger results in less number of optimization problems of larger size, and smaller choice of creates more optimization problems of smaller size. The effect of on the number of variables in the optimization is illustrated in Table 1.
2.2 SparseLearning Reaction Selection (SLRS) method
The task of finding the most influential reactions is a matter of determining the smallest subset of reactions that are active at any given time, such that the error in the concentrations induced by limiting the number of reactions remains in a userspecified tolerance range at all times.
To solve this problem, we formulated the following mathematical approach to datadriven sparselearning reaction selection. For each time batch of size , we solve the following integer linear programming (ILP) problem:
()  
where are binaryvalued optimization variables. Note that solving problem () delivers the minimum number of reactions such that the error tolerance constraints on individual concentrations (Eq. 4) and on error propagation (Eq. 6) are satisfied.
To further reduce the complexity of our datadriven sparselearning approach, we can use convex relaxation methods Nocedal and Wright (2006)
by replacing the binary variable constraint with its convex
hull, i.e., for all . The relaxed problem has the following form:()  
This relaxation of () yields a linear programming problem, which can be solved efficiently in polynomial time, and therefore makes our approach promising for largescale chemical reaction networks at the cost of losing optimality guarantees. Moreover, the solution of () is a realvalued vector, which is more informative but not immediately suitable for the selection of a subset of reactions. In order to project this solution onto a binary vector, we set a threshold as follows
(7) 
where is a threshold value and is in the form of a selection vector similar to the solution of problem ().
The next step is to find the reduced mechanism not just for the time horizon , but for the entire sampled interval , where is the last time sample. In order to do this, we solve problem () for time intervals , where is the largest integer multiple of that is smaller than or equal to , as described in Algorithm 1. The output of Algorithm 1 is a matrix , with columns indicating the selected reactions at each time instance under specific initial conditions, namely temperature , equivalence ratio , and pressure (i.e., ).
2.3 Complexity Analysis and Computational Cost
The original problem () is a standard ILP problem, and thus can be solved using stateoftheart ILP solvers such as Gurobi Gurobi Optimization (2016) and CPLEX CPLEX (2009). Even though these solvers can solve problems with large numbers of integer variables relatively fast by employing branch and bounding algorithms, the worstcase complexity of ILP is exponential in the number of integer variables. On the other hand, the relaxed problem () is a linear programming (LP) problem, which can be solved in polynomial time. Table 1 lists the factors (i.e., variables and constraints) that affect the performance of the two approaches.
Real  Integer  Linear  Integral  

variables  variables  constraints  constraints  
0  
0  0 
The performance of the integer programming solution and the relaxed linear programming solution are compared in Tab. 2. These runtime values are calculated by taking the average of the times it takes to run Algorithm 1 (with the corresponding formulation) for different values of . The results show that solving the relaxed formulation () significantly reduces the computational cost to less than th of the integer formulation. It should be noted that both methodologies can easily be accelerated via parallelization, as each initial condition can be analyzed independently. Since the original problem () is not computationally scalable, the rest of this paper focuses on the relaxed problem.
Average (core s)  Total (core hr)  

2273  136.38  
394  23.64 
The total analysis time for 48 H2/air system simulations using sparse learning reaction selection approach is 2.99 seconds. All the calculations are performed on a Linux based machine with a 2.1 GHz processor and 8 GB of RAM.
3 Analysis of H2/air system
As the first system of investigation, we used the proposed relaxed SLRS approach to study the combustion of H2 in air. Hydrogen combustion is chosen because it yields one of the smallest reaction networks in combustion (8 species and 62 reactions counting both forward and reverse processes Hong et al. (2011)) and is a wellstudied system.
Constantvolume, homogeneous reactor simulations, performed with Ansys Chemkin ANSYS (2016) and a H2 oxidation mechanism by Hong et al. Hong et al. (2011), were used to create the reference chemical reaction network for two different initial conditions ( atm, K, ). For each condition, we identified the influential reactions using SLRS parameters of , , and , and the results of five sampling regions are shown in Fig. 1.
Condition  Influential Reactions 

P1  H + O2 ( + M) ¿ HO2 ( + M) 
(all )  H + O2 ( + O2) ¿ HO2 ( + O2) 
H2 + HO2 ¿ H + H2O2  
H2 + O2 ¿ H + HO2  
H2 + OH ¿ H + H2O  
P2  H + O2 ( + M) ¿ HO2 ( + M) 
(all )  H2 + HO2 ¿ H + H2O2 
H2 + OH ¿ H + H2O  
P3  H + O2 ( + M) ¿ HO2 ( + M) 
(all )  H2 + OH ¿ H + H2O 
HO2 + HO2 ¿ O2 + H2O2  
P4  H + O2 ( + M) ¿ HO2 ( + M) 
(all )  H2 + OH ¿ H + H2O 
H2O2 + H ¿ H2O + OH  
HO2 + HO2 ¿ O2 + H2O2  
P5  H + HO2 ¿ OH + OH 
(all )  H + HO2 ¿ H2 + O2 
H + O2 ¿ O + OH  
H2 + O ¿ H + OH (DUP)  
H2 + OH ¿ H + H2O  
H2O2 ( + M) ¿ 2OH ( + M)  
H2O2 + H ¿ H2O + OH  
OH + HO2 ¿ H2O + O2  
P5  2H ( + M) ¿ H2 ( + M) 
(only )  2H + H2 ¿ 2H2 
H + HO2 ¿ H2O + O  
H + OH ( + M) ¿ H2O ( + M)  
H2O + O ¿ OH + OH  
H2O2 + H ¿ HO2 + H2  
O + H ( + M) ¿ OH ( + M)  
O + HO2 ¿ OH + O2  
OH + H + H2O ¿ 2H2O 
The analysis of the results, listed in Table 3, shows that the set of influential reactions prior to the ignition time is identical for both stoichiometric and fuelrich cases. Initially at P1, the algorithm identifies reactions that consume the fuel as well as threebody reactions generating HO2 as influential. As the ignition process progresses (from P2 to P3), reactions associated with HO2 production/consumption become prominent, with a shift in the H2O2 production pathway from H2 + HO2 to recombination of two HO2 radicals, likely due to increased concentration of HO2 radicals. As the system is approaching ignition (P4), the relaxed model introduces an additional influential reaction pathway that leads to the formation of OH.
Once ignition is reached (P5), we observe a difference between the two systems in influential reactions. Interestingly, influential reactions from the stoichiometric case are a subset of the ones from the fuelrich case. In both cases, the algorithm selects chain branching reactions (e.g., H +O2 ¿ O + OH, H2 + O ¿ H + OH) that are wellknown factors for onset of hightemperature ignition as well as reactions that are associated with HO2 and water formation. In addition to these reactions, under fuelrich conditions, nine more reactions are identified including three H2 formation reactions from H radical, which describe the dynamic equilibrium between H radical and remaining H2.
The time evolution of the influential reactions identified by the proposed algorithm described above agrees well with the general understandings of the ignition of hydrogen as discussed in the literature Law (2010); Hong et al. (2011). For example, when the temperature is lower than 900 K (P1 to P4), the proposed algorithm correctly identifies the preferential pathway to produce HO2 from H + O2 instead of the chain branching pathway that generates O + OH, which is important only at higher temperatures. Moreover, the proposed algorithm correctly captures the dominance of fuel consumption reactions during the very early phase, as well as the reactions that lead to ignition at the ignition time (H2O2 dissociation, chain propagation/branching reactions for HO2).
As the proposed sparselearning approach is designed to identify the influential reactions of a combustion process, it can be leveraged to generate a reduced mechanism, i.e., a coarser reaction network of the analyzed mechanism. We emphasize that, while the reduction of the H2 mechanism is not intended for a realworld application, our results highlight the algorithm’s capability to generate reduced mechanisms.
To this end, we generated and analyzed 48 homogeneous reactor constantvolume simulations of H2/air combustion with initial conditions between 5–20 atm, 800–1100 K, and equivalence ratios between 0.5–2. From these simulations, we selected the union of the influential reactions identified for each initial condition at each time, obtaining a set of 31 reactions. The ignition delay times in constantvolume simulations for the reduced mechanism were compared against those obtained using the full mechanism.
Figure 2 shows the percentage deviation in ignition delay time from the full H2 mechanism for selected conditions. The results which are representative of all the tested conditions, show that the differences in ignition delay time are well below 2% for the reduced mechanism over a wide range of conditions, despite the significantly smaller number of reactions compared to the full mechanism (31 vs 62 reactions). The maximum deviation occurs when the initial temperature is above 1000 K, while lowtemperature ignition delay times are nearly identical to the full mechanism. Overall, the excellent reproducibility of the ignition delay time (less than 2% over all the conditions) indicates that the proposed method is a very effective approach to build a reduced mechanism that performs well in a wide range of thermodynamic conditions. Moreover, it demonstrates that the ignition delay time, which is one of the most important combustion characteristics in practical energy conversion devices, can be preserved with the proposed method even if it is not explicitly taken into account directly during the reduction process.
4 Analysis of Propane Combustion
As a second application of the proposed sparselearning approach, we analyzed the network of reactions describing the chemistry of propane. We used the full mechanism by Petersen et al. Petersen et al. (2007), which includes 117 species and 1270 reactions. Similar to the hydrogen case described in the previous section, we used a 0D reactor for the simulation of a stoichiometric propane/air mixture at 20 atm and 700 K. This system was chosen to identify the reactions that are responsible for the twostage ignition behavior, a distinctive lowtemperature ignition characteristics of alkanes.
Fig. 3 reports the calculated heat release rate and temperature profile for this system. The twostage behavior is highlighted by the peak of heat release rate and subsequent decrease (0.03 – 0.035 s) before the ignition occurs ( 0.0372 s). This trend is generally associated with the competition between different lowtemperature pathways (see Fig. 4 for a scheme of the major lowtemperature pathways in alkane mechanisms Curran et al. (1998, 2002); Westbrook et al. (2011); Sarathy et al. (2011)).
At low temperatures, once olefin peroxides (QOOH) are initially formed, the pathway to ketohydroperoxides via second O2 addition (QOOH ¿ O2QOOH ¿ ketohydroperoxide) is favored. However, as temperature increases during the ignition process, other pathways (i.e., QOOH ¿ Q or carbonyl or cylic ether) become increasingly relevant and dominate the lowtemperature reactivity. As olefins (Q) and carbonyl radicals are more stable than ketohydroperoxides in such conditions, the charge reactivity and heat release rate slightly decrease as shown above.
To test if this behavior is captured by the proposed algorithm, we analyzed the system (, , ) at two different times, namely, before (P1) and after (P2) the peak heat release rate point as shown in Fig. 3. Out of 1270 reactions in the mechanism, the algorithm identified 68 influential reactions at P1 and 49 influential reactions at P2, with most of the differences between the two sets occurring in the lowtemperature chemistry reactions, which are listed in Table 4. The algorithm correctly identifies reactions in both groups of pathways as influential at P1, while at P2 the pathway to ketohydroperoxide is omitted and only the reactions that lead to Q or carbonyl production (less reactive pathway) are selected as influential.
Condition  Influential reactions  Class 

P1  C3H6OOH12 ¿ C3H6 + HO2  24 
C3H6OOH21 ¿ C3H6 + HO2  24  
C3H6OOH13 ¿ C3H6O13 + OH  25  
C3H6OOH21 + O2 ¿ C3H6OOH21O2  26  
C3H6OOH13 + O2 ¿ C3H6OOH13O2  26  
C3H6OOH12 + O2 ¿ C3H6OOH12O2  26  
C3H6OOH13O2 ¿ C3KET13 + OH  27  
C3H6OOH21O2 ¿ C3KET21 + OH  27  
C3H6OOH12O2 ¿ C3KET12 + OH  27  
C3KET13 ¿ CH2O + CH2CHO + OH  28  
C3KET21 ¿ CH2O + CH3CO + OH  28  
C3KET12 ¿ CH3CHO + HCO + OH  28  
P2  C3H6OOH21 ¿ C3H6 + HO2  24 
C3H6OOH12 ¿ C3H6 + HO2  24  
C3H6OOH13 ¿ C3H6O13 + OH  25 
Since our approach was able to capture the lowtemperature behavior of propane combustion, we analyzed the differences between low (, 700 K, 20 atm, simulated above) and hightemperature (, 1500 K, 20 atm) combustion using the same parameters as before (, ). To simplify the comparison, we added an additional constraint to the optimization problem to accumulate the selection of influential reactions over the duration of each simulation.
Figure 5 shows the relevance of each reaction in both conditions, as increasing values of correspond to more influential reactions. The reactions are labeled by using an incremental number obtained by listing forward and reverse reactions in the same order they are presented in the original mechanism. For example, the first reaction in the parent mechanism is H + O2 ¡=¿ O + OH, which becomes reaction 1 (H + O2 ¿ O + OH) and 2 (O + OH ¿ H + O2) in the abscissa of Figs 5 and Figs 6. From the comparison of the two plots, we find that the algorithm identifies more influential reactions () for the lowtemperature system (420) than for the hightemperature one (350). This trend is expected, as the former system needs to include reactions from the lowtemperature regime in addition to the hightemperature reactive pathways, which is in agreement with other works presented in the literature Niemeyer et al. (2010); Ranzi et al. (2012). The detailed comparison between the two sets of reactions can also provide useful insights on the lowtemperature chemistry. For example, most of the reactions between 647–664, which describe hydrogen abstractions from C2H3CHO by various radicals, are identified as influential only for the 700 K case. The analysis of the mechanism indicates that C2H3CHO is predominantly associated with the lowtemperature chemistry including RO2 and O2QOOH elimination pathways, e.g., QOOH ¿ Q ¿ smaller. Similarly, reactions between 1301–1326, which represent decomposition pathway of C3 carbonyl compounds (C3H6O12, C3H6O13), are irrelevant for the hightemperature system while some of them are influential for the 700 K case. As these C3 carbonyls are mostly created by QOOH chemistry, pathways that are wellknown to be associated with lowtemperature conditions, we find again that the algorithm selection matches our understanding of this kinetic mechanism.
To create a list of influential reactions a cutoff () for (see Eq. 7) is required. However, directly comparing the values of in their multireaction context is more informative. For example, if we analyze in detail the reactions between 1235–1288 (shown in Fig. 6) that represent several lowtemperature reactions (e.g., RO2 ¿ QOOH, QOOH destruction pathways, QOOH + O2 ¿ O2QOOH ¿ KET ¿ smaller, O2QOOH ¿ smaller), we can see that while almost all the reactions have , their relevance () changes in the two systems. In this way, it is straightforward to identify the reactions that are relevant in both regimes (e.g., 1245) or reactions that are prevalent in a specific regime (e.g., 1259 or 1267), particularly for complex and larger reaction networks, for which the assessments of relative importance among different pathways for specific combustion behavior is very challenging.
Using a similar approach as for the H2 mechanism study, we assembled a reduced mechanism for the propane chemistry. To formulate a reduced mechanism that works in both low/hightemperature regime, we analyzed influential reactions identified from 216 conditions (700–1500 K, 1–50 atm, and equivalence ratios of 0.5–2) from homogeneous reactor simulations. The performances of the reduced version of the propane mechanism, which consists of 111 species and 691 reactions, were again tested by comparing ignition delay times with the ones predicted by the parent mechanism. The results, of which a representative sample is shown in Fig. 7, show that ignition delay times predicted by the reduced mechanism are within 40% of those by the parent propane mechanism and that the NTC (negative temperature coefficient) behavior predicted by the parent mechanism is preserved.
5 Conclusions
In this work, we present a new method to analyze complex chemical reaction networks by employing a datadriven sparselearning approach. Using concentration and reaction rates at a given time, the proposed optimization approach is able to identify the most influential reactions in the network, that is the smallest subset of reactions that is able to approximate the concentration of the species to within a prescribed error tolerance using a convex relaxation method to solve the underlying optimization problem. The proposed approach has the following advantages: guaranteed concentration approximation accuracy over all time points; low computational cost; and versatility due to its applicability to general chemical reaction networks.
We tested our approach on reaction networks generated by the combustion of two fuels, hydrogen and propane. When applied to the H2 combustion, our method was able to identify the key reactions that mark the different phases of the ignition process. Moreover, a reduced mechanism of the H2 oxidation is built by collecting the influential reactions at all times in a wide range of thermodynamic conditions (5–20 atm, 800–1100 K, 0.5–2), displayed a deviation from ignition delay time of less than 2%, while using only half of the reactions.
The analysis of the C3H8 combustion mechanism showed that our method can identify the changes in lowtemperature pathways and capture the propane’s characteristic twostage ignition behavior, as well as the differences in relevant reactions between low and hightemperature ignition conditions. Similarly to hydrogen combustion, we built a reduced mechanism consisting of 111 species (reduced by 5.1%) and 691 reactions (reduced by 45.6%), by analyzing the combustions of 216 different systems. The ignition delay times obtained with the reduced mechanism are within 40% deviation of the original mechanism in a wide range of conditions (700–1500 K, 1–50 atm, 0.52).
This study showcases the potential of the proposed datadriven approach to analyze very complex reaction networks and to perform mechanism reduction in a computationallyefficient manner.
Acknowledgments
This research has been funded by the US Army Research Office grants W911NF1510241 and W911NF1410359. PE and AV thank the College of Engineering at the University of Michigan for partially supporting this work.
References
 Law (2007) C. K. Law, Combustion at a crossroads: Status and prospects, Proceedings of the Combustion Institute 31 (2007) 1–29.
 Tomlin et al. (1997) A. S. Tomlin, T. Turányi, M. J. Pilling, Chapter 4 Mathematical tools for the construction, investigation and reduction of combustion mechanisms, in: M. J. Pilling (Ed.), Comprehensive Chemical Kinetics, volume 35 of LowTemperature Combustion and Autoignition, Elsevier, New York, 1997, pp. 293–437.
 Turányi (1990) T. Turányi, Sensitivity analysis of complex kinetic systems. Tools and applications, Journal of Mathematical Chemistry 5 (1990) 203–248.
 Lu and Law (2009) T. Lu, C. K. Law, Toward accommodating realistic fuel chemistry in largescale computations, Progress in Energy and Combustion Science 35 (2009) 192–215.
 Vajda et al. (1985) S. Vajda, P. Valko, T. Turányi, Principal component analysis of kinetic models, International Journal of Chemical Kinetics 17 (1985) 55–81.
 Maas and Pope (1992a) U. Maas, S. B. Pope, Implementation of simplified chemical kinetics based on intrinsic lowdimensional manifolds, Symposium (International) on Combustion 24 (1992a) 103–112.
 Maas and Pope (1992b) U. Maas, S. B. Pope, Simplifying chemical kinetics: Intrinsic lowdimensional manifolds in composition space, Combustion and Flame 88 (1992b) 239–264.
 Lam and Goussis (1994) S. H. Lam, D. A. Goussis, The CSP method for simplifying kinetics, International Journal of Chemical Kinetics 26 (1994) 461–486.
 Lam (1993) S. H. Lam, Using CSP to Understand Complex Chemical Kinetics, Combustion Science and Technology 89 (1993) 375–404.
 Bhattacharjee et al. (2003) B. Bhattacharjee, D. A. Schwer, P. I. Barton, W. H. Green, Optimallyreduced kinetic models: Reaction elimination in largescale kinetic mechanisms, Combustion and Flame 135 (2003) 191–208.
 Lu and Law (2005) T. Lu, C. K. Law, A directed relation graph method for mechanism reduction, Proceedings of the Combustion Institute 30 (2005) 1333–1341.
 Lu and Law (2008) T. Lu, C. Law, Strategies for mechanism reduction for large hydrocarbons: Nheptane, Combustion and Flame 154 (2008) 153–163.
 Pepiotdesjardins and Pitsch (2008) P. Pepiotdesjardins, H. Pitsch, An efficient errorpropagationbased reduction method for large chemical kinetic mechanisms, Combustion and Flame 154 (2008) 67–81.
 Niemeyer et al. (2010) K. E. Niemeyer, C.J. Sung, M. P. Raju, Skeletal mechanism generation for surrogate fuels using directed relation graph with error propagation and sensitivity analysis, Combustion and Flame 157 (2010) 1760–1770.
 Chang et al. (1987) W. D. Chang, S. B. Karra, S. M. Senkan, A computational study of chlorine inhibition of CO flames, Combustion and Flame 69 (1987) 113–122.
 Lifshitz and Frenklach (1980) A. Lifshitz, M. Frenklach, Oxidation of cyanogen. ii. the mechanism of the oxidation, International Journal of Chemical Kinetics 12 (1980) 159–168.
 Fridlyand et al. (2017) A. Fridlyand, M. S. Johnson, S. S. Goldsborough, R. H. West, M. J. McNenly, M. Mehl, W. J. Pitz, The role of correlations in uncertainty quantification of transportation relevant fuel models, Combustion and Flame 180 (2017) 239–249.
 Harirchi et al. (2017) F. Harirchi, O. A. Khalil, S. Liu, P. Elvati, A. Violi, A. O. Hero, A datadriven sparselearning approach to model reduction in chemical reaction networks, NIPS 2017 Workshop on Advances in Modeling and Learning Interactions from Complex Data, 2017.
 Feinberg (1987) M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors– i. the deficiency zero and deficiency one theorems, Chemical Engineering Science 42 (1987) 2229–2268.
 Gunawardena (2003) J. Gunawardena, Chemical reaction network theory for insilico biologists, Notes available for download at http://vcp.med.harvard.edu/papers/crnt.pdf (2003).
 Gillespie (2007) D. T. Gillespie, Stochastic simulation of chemical kinetics, Annu. Rev. Phys. Chem. 58 (2007) 35–55.
 Chellaboina et al. (2009) V. Chellaboina, S. P. Bhat, W. M. Haddad, D. S. Bernstein, Modeling and analysis of massaction kinetics, IEEE Control Systems 29 (2009) 60–78.

Anderson and Kurtz (2011)
D. F. Anderson, T. G. Kurtz,
Continuous time markov chain models for chemical reaction networks,
in: Design and analysis of biomolecular circuits, Springer, 2011, pp. 3–42. 
Ascher and Petzold (1998)
U. M. Ascher, L. R. Petzold, Computer methods for ordinary differential equations and differentialalgebraic equations, volume 61, SIAM, Philadelphia, USA, 1998.
 Nocedal and Wright (2006) J. Nocedal, S. J. Wright, Numerical Optimization, second ed., Springer, New York, USA, 2006.
 Gurobi Optimization (2016) I. Gurobi Optimization, Gurobi optimizer reference manual, 2016. URL: http://www.gurobi.com.
 CPLEX (2009) I. I. CPLEX, User’s manual for CPLEX, Int. Bus. Mach. Corp. 46 (2009) 157.
 Hong et al. (2011) Z. Hong, D. F. Davidson, R. K. Hanson, An improved H2/O2 mechanism based on recent shock tube/laser absorption measurements, Combustion and Flame 158 (2011) 633–644.
 ANSYS (2016) ANSYS, CHEMKIN version 18.0, 2016.
 Law (2010) C. K. Law, Combustion physics, Cambridge university press, Cambridge, UK, 2010.
 Petersen et al. (2007) E. L. Petersen, D. M. Kalitan, S. Simmons, G. Bourque, H. J. Curran, J. M. Simmie, Methane/propane oxidation at high pressures: Experimental and detailed chemical kinetic modeling, Proceedings of the combustion institute 31 (2007) 447–454.
 Curran et al. (1998) H. J. Curran, P. Gaffuri, W. J. Pitz, C. K. Westbrook, A comprehensive modeling study of nheptane oxidation, Combustion and flame 114 (1998) 149–177.
 Curran et al. (2002) H. J. Curran, P. Gaffuri, W. J. Pitz, C. K. Westbrook, A comprehensive modeling study of isooctane oxidation, Combustion and flame 129 (2002) 253–280.
 Westbrook et al. (2011) C. Westbrook, W. Pitz, M. Mehl, H. Curran, Detailed chemical kinetic reaction mechanisms for primary reference fuels for diesel cetane number and sparkignition octane number, Proceedings of the Combustion Institute 33 (2011) 185–192.
 Sarathy et al. (2011) S. M. Sarathy, C. K. Westbrook, M. Mehl, W. J. Pitz, C. Togbe, P. Dagaut, H. Wang, M. A. Oehlschlaeger, U. Niemann, K. Seshadri, et al., Comprehensive chemical kinetic modeling of the oxidation of 2methylalkanes from c7 to c20, Combustion and flame 158 (2011) 2338–2357.
 Ranzi et al. (2012) E. Ranzi, A. Frassoldati, R. Grana, A. Cuoci, T. Faravelli, A. Kelley, C. Law, Hierarchical and comparative kinetic modeling of laminar flame speeds of hydrocarbon and oxygenated fuels, Progress in Energy and Combustion Science 38 (2012) 468–501.
Comments
There are no comments yet.