Artificial Neural Network Models for Octane Number and Octane Sensitivity: A Quantitative Structure Property Relationship Approach to Fuel Design

The present work employed a novel predictive modeling approach using artificial neural networks (ANNs) trained with quantitative structure-property relationship (QSPR) descriptors to forecast properties of single-component fuels. ANNs are shown to be strong tools in predicting properties of molecules that would otherwise need physical testing; physical testing is disadvantageous for assessing large sets of potential fuels as it requires samples of each molecule, analytical equipment, and time to run trials. On the contrary, the use of ANNs does not require physical samples or specialized apparatuses while significantly reducing the testing time. Predictive modeling also bears the advantage of the potential to discover new relationships between structure and property that may otherwise be overlooked during the physical trial and error testing.

The remaining sections of the Introduction discuss octane number and octane sensitivity (OS), which are the primary properties predicted in the current work, the use of QSPR descriptors for predicting fuel performance, and a brief literature review of previous efforts to predict various fuel properties. Then, the methodology is described before the results and discussion of the ANN performance and the most important descriptors for octane number and sensitivity predictions.

Octane number is representative of the auto-ignition resistance of a gasoline-like fuel in a spark-ignition engine; specifically, it is a measurement of the fuel's ability to resist knock (i.e., the occurrence of auto-ignition in the end gas in front of the deflagration flame or before the spark) [1]. Two distinct octane numbers are used to quantify knock resistance, research octane number (RON), and motor octane number (MON), and are powerful predictors of a fuel's engine performance [2]. Together, RON and MON represent the auto-ignition resistance of a fuel at varying engine operating conditions, where higher values indicate higher resistance to knock. Both RON and MON are defined by American Society for Testing and Materials (ASTM) standards, and require the use of a cooperative fuel research engine to measure [3,4]. Their experimental procedures reflect different engine conditions, including inlet temperatures and engine speeds (in revolutions per minute). More specifically, RON is tested at a lower temperature and engine speed while MON is tested at a higher temperature and engine speed. Each procedure requires two reference fuels to measure the RON/MON of a given compound, the most common being n-heptane and isooctane (2,2,4-trimethylpentane), which form a 0–100 scale. These experimental procedures require a considerable amount of time and approximately 500 ml of each reference fuel and of the compound to be measured, which may be especially difficult and/or expensive to obtain for brand new or newly proposed fuel molecules [5].

The OS of a compound is defined as the difference between its RON and MON (i.e., RON−MON), and is used to measure the difference in performance of a fuel at different engine conditions. The OS of a compound affects its ability to resist knock at varying pressures and temperatures [6,7]; specifically, that a lower sensitivity at low load/pressure and a higher sensitivity at high load/pressure both yield a higher resistance to knock [8]. Knock is generally seen as a barrier for researchers who work toward increasing the efficiency of spark-ignition engines, and a comprehensive understanding of auto-ignition and knock has yet to be achieved. It is therefore important that further studies of knock, including the nuanced role of OS, are performed to ultimately raise the efficiency of spark-ignition engines and the fuels they use. The present work shows that ANN models trained with QSPR descriptors do not only predict OS, but can provide insight into the molecular mechanics that give rise to desirable OS values. This insight can aid fuel design researchers in creating optimal target molecules without having to synthesize and test multitudes of molecules via trial and error. Promising next-generation biofuels, such as 2,5-dimethylfuran (DMF) and butanol, are being studied extensively as a result of their high OS in comparison to that of traditional fossil-derived fuels [9,10]; ANNs, as shown in the present work, can be used to sift through hundreds of potential molecular candidates to identify top OS performers.

Quantitative structure-property relationship descriptors are numerical variables describing various physical, chemical, and electromechanical aspects of a given compound. Thousands of QSPR descriptors exist and are used across a wide array of fields including ecology, chemistry, pharmacology, and biology [11–16]. Within the fuels community, QSPR descriptors are used to identify relationships between structural features and fuel properties [17,18]. These descriptors can be classified into groups such as constitutional indices, information indices, matrix-based descriptors, topological indices, and functional group counts. Specific examples of QSPR descriptors discussed in the present work include nCsp2, a constitutional index representing the number of sp² hybridized carbons in a molecule; a more complex descriptor analyzed in the present work is ChiA_B(s) representing the average Randic-like index from the Burden matrix weighted by I-state. Both of these descriptors are described in more detail in Sec. 3.2 discussing the top RON/MON descriptors. Evidently, QSPR descriptors offer fundamental insight into the physical and electronic components ultimately responsible for a molecule's reactivity. Thus, they surpass the use of physicochemical properties (boiling point, heat of vaporization, polarity, etc.) and functional groups when studying or predicting molecular fuel performance. QSPR descriptors were chosen to use within the current work because they are able to fully encapsulate a fuel's molecular structure which has been shown to greatly influence device performance [19]. Ultimately, QSPR descriptors provide a quantitative, information-dense profile of a molecule that can be used as an ANN input to predict derived properties such as octane number or sensitivity.

QSPR descriptors were selected over other molecular ANN inputs as they can capture fundamental chemical properties and relationships which superficial inputs, such as functional groups, may overlook [20]. For example, an alternative input to QSPR descriptors may be fragments of molecules that contain unique functional groups like double bonds which characterize alkenes. Alkenes often have higher RON/MON due to the heightened stability offered by their double bonds under high temperatures. While both functional group input and QSPR descriptors will be able to highlight this trend, QSPR descriptors offer a more fundamental explanation. As shown in the results of the present work, the number and placement of double bonds also have an effect on the RON/MON. The SssCH2 descriptor shows that double bonds located inside a carbon chain, rather than on the terminal ends, have increased stability which contributes to higher octane number. Using functional groups as inputs would fail to recognize the impact of double bond placement whereas QSPR descriptors are able to capture the chemical and reactive significance.

Limitations to QSPR descriptors include the ease of interpretation of complex descriptors and equivalent values for two different molecules. While some descriptors are simple, others are more complex. For example, the AVS_B(s) descriptor defines the average vertex sum from the Burden matrix weighted by I-state. The individual components which constitute this descriptor (vertex sum, Burden matrix, and I-state) must be understood as isolated variables before they can be interpreted as an aggregate descriptor. This may be a barrier for combustion scientists who do not specialize in chemistry or QSPR descriptors. Complex descriptors may pose obstacles in designing next-generation alternative fuels as it can be difficult to synthesize a target molecule which meets the specifications of multiple intricate descriptors. However, considering multiple QSPR descriptors when analyzing the relationship between structure and property can aid in creating a fuller picture with less obscurity. Additionally, it is possible that two different molecules share a same QSPR descriptor value. While this may seem like a drawback when analyzing groups of potential fuel molecules, the equivalent values can be deciphered based on the molecule's structure. Further, other descriptors can be used to create more complete profiles in which each molecule is uniquely depicted.

A diverse set of methods exists for predicting numerous combustion-related properties of hydrocarbons and oxygenated compounds [21–24]. For example, Li et al. [25] tested 23 different regression approaches to predict cetane number (CN), RON, and MON. Kessler et al. [26] predicted yield sooting index using artificial neural networks, graph neural networks, and multivariate equations, comparing the accuracy of these methods. In addition, unified sooting tendency databases have been developed [27]. Results from the comparative study show that while graph neural networks and multivariate equations may offer a higher degree of interpretability, ANNs outperform both with respect to prediction accuracy [26]. Additional studies have shown that ANNs trained using chemoinformatic descriptors can accurately predict the CN and flash point of alternative fuel compounds [17]. Furthermore, ANNs trained with experimental data and QSPR descriptors have been shown to accurately predict the CN of a variety of molecular classes including furanic compounds derived from renewable resources [28]; predictive calculations using QSPRs have also successfully predicted CN of fuels [18]. ANNs have demonstrated success in predicting the RON/MON of gasolines and gasoline blends based on chromatographic analysis and volumetric content, respectively [29,30]. Moreover, the RON of neat hydrocarbons can be predicted by inputting the FTIR spectra of fuel mixtures into a multivariate analysis process using the principle component regression method [31]. ANNs are advantageously capable of generalizing predictions for data not observed during training based on relationships it observes between multidimensional (multivariate) input/target training data [32,33].

The present work utilizes ANNs trained with experimental RON/MON/OS and QSPR descriptors to create predictive models for each property. QSPR descriptors are employed due to the extensive range of attributes they describe for a given compound, allowing the ANN to form complex correlations between multiple compounds and experimental RON/MON/OS. Additionally, predicted values of RON/MON are used to calculate the derived octane sensitivity (dOS) (RON_pred−MON_pred), which is compared to using an ANN to directly predict OS. Furthermore, individual relationships between QSPR descriptors and RON/MON/OS are illustrated, highlighting key structural components of compounds that relate to these properties.

ANNs applied to the prediction of octane sensitivity are new to the combustion field with only one other published work reporting successful results [47]. However, there are stark differences in the approach and design between the present work and that found in Lehn et al. [47]. One key difference is that Lehn et al. used a hybrid approach for selecting input parameters by using chemical and physical knowledge such as Joback molecular groups. The present work does not use a hybrid approach; thus, it is shown that comparable ANN prediction accuracy of OS can be obtained even without detailed molecular-input features or a priori knowledge of influential structures. Lehn et al. showed that direct predictions of OS outperformed dOS calculated from predicted RON and MON values. Furthermore, in contrast to employing a sensitivity analysis to highlight the structural impacts on targeted quantities as in Lehn et al., the present work uses a correlation analysis between top contributing descriptors to illustrate interconnection between structure, top descriptors, and target properties. The presented work thus proves that a variety of approaches for ANN construction and data analysis are not only possible, but also accurate, for OS predictions.

Experimental data for RON and MON were collected from several sources [1,34–41], yielding 344 unique compounds each with an experimental value for RON and/or MON. Only experimentally measured RON/MON values were included in the ANN training dataset, and reported data for predicted values of RON/MON were not utilized during the ANN training procedure. Following ASTM standard test methods D2699 (RON) and D2700 (MON), experimental measurements for ON between 80 and 90 have a precision to ± 0.1 octane units; ON measurements below 80 and above 90 do not have precision values as the data are not available [3,4]. Furthermore, reported data for compounds well outside the appropriate range for RON/MON were not considered during the procedure; specifically, n-dodecane with RON/MON values of −40, and n-undecane with RON/MON values of −35 were excluded. Upon removing compounds with predicted RON/MON values and the significant outliers, the data set consisted of 278 unique compounds. Their OS were calculated given their experimental RON/MON values (RON_exp−MON_exp).

The 278 compounds in the data set were split into three subsets, denoted as the training set, validation set, and test set comprising 80%, 10%, and 10% of the total data set, respectively. Compounds for each subset were chosen such that each subset contained a proportionally equal number of compounds based on the range of experimental OS values. Each property (RON/MON/OS) utilized these subsets for ANN training, and each subset remained constant to ensure an adequate comparison of ANN accuracy, specifically the ability of the ANNs to generalize predictions for data not observed during training (test set predictions). Simple molecular-input line-entry system (SMILES) strings were produced or aggregated for all 278 compounds. The alvaDesc software package (version 1.0.22) was used to generate 5305 QSPR descriptors for each compound using the SMILES strings, forming unique sets of quantitative values for each compound [41]. QSPR descriptors and known experimental octane number and sensitivity for each compound represent the input and target data used by the ANNs during training.

Training an ANN for all 5305 available QSPR descriptors would be computationally difficult, and many of the descriptors contain little information relevant to a molecule's RON/MON/OS. Therefore, random forest regression from the Scikit-learnpython package was utilized to rank each QSPR descriptor by its correlation to RON, MON, and OS, measured by a random forest regression-derived value, importance [42]. A higher value of importance implies a stronger correlation between a given QSPR descriptor and RON/MON/OS [43]. The sum of all 5305 QSPR descriptor importance values for each property is equal to one. Regression and the subsequent ranking of QSPR descriptors, were performed with respect to the training and validation subsets. The 250 most important QSPR descriptors for RON, MON, and OS, as measured by the random forest importance metric, were chosen as input variables for each property's ANN to balance accuracy and training time. Figure 1 shows the importance values of the 250 most important QSPR descriptors for RON, MON, and OS. Including fewer QSPR descriptors decreases ANN accuracy, while including more QSPR descriptors increases training time with relatively insignificant improvements in ANN accuracy. Appendix  A. Tables 5–7 list the ten most important descriptors for MON, RON, and OS, respectively.

Artificial neural networks were constructed using ECNet, an open-source python package compiled specifically for constructing predictive models for fuel properties [44]. The ANN architecture for RON, MON, and OS consisted of 250 input neurons (one for each QSPR descriptor), two hidden layers with 128 neurons each, and an output layer with one neuron (corresponding to RON, MON, or OS). Two hidden layers are used to better capture the non-linear relationship between input parameters and the target property. The rectified linear unit activation function was used at each layer with the exception of the output layer which utilized a linear scaling function. The rectified linear unit activation function (1) is not constrained between [0, 1] or [−1, 1] like sigmoid or hyperbolic tangent and (2) can inherently act as a dropout neuron, helping the network with training [45]. The Adam optimization function in conjunction with the mean squared error loss function was used for regression with hyper-parameter values of 0.9 for β1, 0.999 for β2, 1.0 × 10⁻⁸ for ɛ, 0.001 for learning rate, and 0.0 for learning rate decay [46]. ANNs for each property regressed with respect to the training subset. After each training iteration (epoch), the mean squared error of the validation subset's predictions was evaluated. Training was terminated once performance ceased to improve for the validation subset to prevent overfitting. Performance of the ANN was determined given the root-mean-square error (RMSE) of predictions for the test set, providing a metric for how well the ANNs are able to generalize predictions for data not observed during training. Twenty-five ANNs were constructed for each property to average the error metrics and ensure consistency in results.

In addition to constructing ANNs for OS, the ANNs constructed to predict RON and MON were used to derive OS from RON/MON predictions, denoted as dOS (calculated as RON_pred−MON_pred). Twenty-five calculations of dOS were performed for each subset as a result of the 25 ANNs trained for RON and MON. The RMSEs of each subset were compared to the subset RMSEs resulting from ANNs trained directly with experimental OS data.

After using the random-forest-derived importance measure to rank the 250 most important descriptors, the alvaDesc correlation analysis tool was used on the seven unique descriptors amongst the top three most important for either RON, MON, or OS. This tool provides values that are unique to the set of molecules used in the ANN input data set. The correlation values range from −1 to 1 where −1 is perfectly inversely correlated and 1 is perfectly positively correlated; a correlation value of zero suggests no correlation. A matrix of the correlation values is found in Appendix  B. Table 8 and the values are used throughout the discussion to illustrate the interconnection between certain structural features, the top QSPR descriptors, and ultimately octane number and sensitivity. This correlation value is important to consider as certain structural features may simultaneously contribute to multiple different descriptors.

The performance of the ANNs and the chemical analysis of top QSPR descriptors are examined in this section. The correlation values between the descriptors themselves are also discussed within the chemical analysis primarily to emphasize structure, descriptor, and property interconnections. The top descriptors for RON and MON (GATS2m, SssCH2, and ChiA_B(s)) are discussed together, given their overlap. The last portion of this section analyzes the top three OS descriptors, namely AVS_B(s), nCsp2, and SIC1. The analyses of the top descriptors serve to identify and explain desirable structural features as guidance for the synthesis of high-quality next-generation fuels.

While it is immediately evident that ANNs constructed from multiple QSPR descriptors provide powerful predictive capabilities in comparison to simple correlations with single QSPR descriptors, additional understanding is gained by analyzing individual, important descriptors with respect to RON, MON, and OS. Furthermore, this approach directly addresses a historical drawback of ANNs—namely, that it can be very difficult to extract meaningful data from the neural network weights and thus acts as a “black box” tool, which negates potential insight gained from the underlying mathematical and statistical frameworks of the ANN.

Figure 2 shows parity plots for training, validation, and test set predictions of individual strong performing ANNs for RON, MON, dOS, and OS. The test set RMSE for RON, MON, dOS, and OS of these handpicked models were found to be 7.3, 5.9, 7.9, and 6.3, respectively. Center dashed lines represent 1:1 parity, and outside dashed lines represent ± the test set's RMSE. It is seen that the test set RMSE for OS is lower than the test set RMSE for dOS, indicating that predicting OS directly is more beneficial than deriving OS from RON and MON predictions. However, it is apparent that both methods for predicting OS are relatively similar in accuracy, highlighting the viability of both methods. Additionally, percent error is calculated by dividing the test set RMSEs by the range of each property in order to show full-scale error; the property ranges, determined by subtracting the maximum and minimum measured value from the experimental data, are 137, 146, 58.3, and 58.3 for RON, MON, dOS, and OS, respectively. The resulting errors are 6.4%, 5.2%, 15.3%, and 9.9% for RON, MON, dOS, and OS, respectively.

The average and standard deviation of both RMSE and R² over all 25 ANNs for each property were calculated and are reported in Table 1, showing how much the test set error varied across the 25 ANNs generated for each property. The largest deviations are found within the RMSE and R² of dOS, confirming that it is more precise to directly predict OS rather than deriving it from predicted RON and MON, consistent with the earlier results of Lehn et al. [47]. The averaged R² for dOS is a negative value, indicating an inverse relationship between predicted and experimental values of OS across 25 ANNs; furthermore, there is a very high standard deviation associated with the dOS models which suggest that the ANN varies in its ability to predict consistent values for the test data. While these averaged metrics may seem unfavorable, stronger or more adequate models are still possible within these highly deviating models, like the one shown in Fig. 2(c). Using the same previously defined property value ranges from the experimental data, the dOS error across 25 ANN test sets was 15.3% while OS was only 9.9%. However, the MON models had the least error with an RMSE of 7.6 ± 0.7 out of the 146 value range, resulting in a 5.2% error. Furthermore, the MON model fits the observed data with an R² fit of 0.7688 ± 0.0434. The dOS predictions have the poorest fit to observed data with an R² of −0.2256 ± 0.5673. Contrarily, the RON predictions fit the observed data most strongly with R² equal to 0.8105 ± 0.0744, noting a much lower standard deviation compared to dOS and a 6.4% full-scale error. Additionally, while these variance metrics show typical performance of trained ANNs, it is possible to achieve a better-performing ANN by up to two standard deviations lower than these reported averages, such as those ANNs outlined in Fig. 2.

The relationship between the top descriptors and octane number, RON and MON, are shown in Figs. 3–5. The descriptors analyzed are GATS2m, SssCH2, and ChiA_B(s), which jointly characterize molecules based on their structural uniformity and connectivity.

Figures 3(a) and 3(b) show the relationships between RON/MON and GATS2m, an auto-correlation index descriptor. GATS2m quantifies structural uniformity across segments of two bonds at a time. When auto-correlation indices are used with respect to compound structure, they indicate repeating patterns within a given compound [48]. Furthermore, a higher value of the auto-correlation index implies a more significant resemblance in neighboring atoms [49]. Structural invariability within molecules can be observed in straight-chain hydrocarbons, homoatomic structures, and uniformly hybridized molecules. Alternatively, structural resemblance, or GATS2m, is lower in molecules bearing heteroatoms, high degree of branching, and unsaturated carbons. The loosely inverse relationship between the GATS2m value and the RON/MON value is easily observed when comparing the reference fuels used for octane number, isooctane, and n-heptane. These reference fuels are used to describe upper and lower limits of knock resistance and the main structural difference is found in their degree of branching. Isooctane, a branched hydrocarbon composed of 8-sp³ carbons, has an ON value of 100 and GATS2m value of 1.03. In contrast, n-heptane, a non-branched hydrocarbon composed of 7-sp³ carbons, has an ON of 0 and a GATS2m value of 1.39. The degree of branching of a carbon atom dictates the stability of radical intermediates and products, where the stability of carbon radicals increases from primary to tertiary carbons [50]. Branched alkanes, like isooctane, offer more stable, tertiary carbons when they are involved in radical reactions, thus slowing down reaction rates. Linear alkanes, like n-heptane, cannot offer tertiary carbons to radicals; this lack of stability during radical reactions is the cause of increased reactivity, resulting in a higher auto-ignition propensity. In addition to the degree of branching, the previously discussed factors which influence structural variability also influence the ON value. For example, the presence of oxygen atoms or double/triple bonds permits electron delocalization and subsequently enhanced stability. Therefore, the inverse relationship between the GATS2m value and the octane number of a molecule originates in multiple structural features which give rise to relatively stable radical intermediates, causing reaction rates and heat release to slow.

The SssCH2 descriptor further characterizes structural variability within molecules, focusing on the presence of methylene, or CH₂, groups. Figures 4(a) and 4(b) show the loosely inverse relationship between SssCH2 and RON and MON, respectively. The SssCH2 descriptor defines the sum (S) of ssCH2 atom E-states within a molecule. The ssCH2 atom type is a methylene group, hence the “ss” notation signifying two single bonds to the carbon atom. The E-state of any atom type is the summation of an assigned intrinsic atomic value, called the I-state, and a perturbation effect, which accounts for the electronic influences from the surrounding atoms of the molecule [15]. Thus, E-state values are calculated values while I-state values are constant and assigned [12]. For example, the ssCH2 atom type has an I-state value of 1.50, while the ssssC I-state is 1.25, sssCH is 1.33, and sCH3 is 2.00. The summation of ssCH2 E-states is not equivalent to the number of ssCH2 atom types within a molecule since E-states differ from atom to atom. While the atom type count and E-state summation are not equivalent, they are still related. As an example, adding more ssCH2 atom types to a molecule can still increase the SssCH2 value as seen with hexane (C₆H₁₄, SssCH2: 5.536) and heptane (C₇H₁₆, SssCH2: 7.008).

Like GATS2m, the loosely inverse relationship between RON/MON and SssCH2 can be understood through analyzing the variety of structural factors which simultaneously contribute to SssCH2 and reactivity. Table 2 shows the impact of structural modifications, using heptane as a reference molecule, and how they directly impact reactivity (RON) and SssCH2. For most molecules, an increase in SssCH2 corresponds to a decrease in RON, however, as seen for cycloheptane in Table 2, there are exceptions where an increase in SssCH2 is accompanied by an increase in RON. In this specific case, linear alkanes of similar carbon lengths are unable to participate in certain stabilizing low-temperature oxidation reactions which are innate to cyclic alkanes [51]. Specifically, non-branched alkane rings participate in ring-opening reactions to create highly stable conjugate olefins within the range of low-temperature oxidation. The linear analogs of these cyclic compounds are unable to reproduce such stable olefin products primarily because of their open-chain structure which lacks the steric geometry unique to cyclic analogs.

In summary, it is seen that the SssCH2 does not directly dictate the ON of molecules, but it indicates the presence of a variety of structural features (aromaticity, alkenes, cycles, branching) which influence ON through differing mechanisms and magnitudes, dependent on operating conditions such as temperature and engine speed. Generally, decreasing the SssCH2 leads to enhanced stability and an increase in ON. However, the direct cause for the increase in ON within these molecules is not the decreased SssCH2 itself, but the unique stability which each of these structural features create.

Lastly, the ChiA_B(s) descriptor is representative of the overall connectivity within a molecule. Figure 5(a) shows the relationship between ChiA_B(s) and RON. The ChiA_B(s) descriptor represents the average Randić-like index from the Burden matrix weighted by I-State. The Randić index is a topological index which expresses connectivity of a molecule through the summation of the degree of adjacent vertices. Given its wide application in pharmacology, biology, ecology, and chemistry, there are a myriad of works studying the Randić index, including step-by-step explanations of the formula and its relationship to other topological indices [13,14,52]. Within the chemical field, the Randić index is often recognized for its strong linear correlation with the boiling point and chromatographic retention time of alkanes.

However, “Randić-like” indices are generalizations of the original Randić connectivity index as they have been mathematically altered. Each modification to the original index adds a level of complexity during interpretation. In this case, a molecule's hydrogen-depleted Burden matrix weighted by I-state is summed instead of summing the degree of the vertices. Lastly, this term is averaged by dividing by the number of bonds in the molecule. Evidently, molecules with more bonds will have a lower ChiA_B(s) value, given that the denominator is larger than those for smaller molecules.

The Burden matrix is a symmetrical square matrix where the outer diagonal values are numerical representations of bond type; the diagonal values are typically zero, but, in this case, are weighted by the I-state, or intrinsic state value of certain atom types. Thus, this descriptor simultaneously reflects multiple features of a molecule such as the degree of connectivity, the intrinsic states of each atom, and the number of bonds within the whole molecule.

This descriptor is clearly much more complex than the original Randić index, leading to seemingly vague relationships between structure and the descriptor at first glance. Furthermore, an oddly shaped relationship between the descriptor and RON is formed despite its high rank of importance. However, such obscurity can be clarified by looking at ChiA_B(s) and its correlation to other top descriptors and their unique relationship with RON. The ChiA_B(s) descriptor shares moderate linear correlation with the other top descriptors for RON; ChiA_B(s) has a 0.7071 and 0.8555 correlation with SssCH2 and GATS2m, respectively. As GATS2m decreases, the structural uniformity begins to fade due to the presence of structural features such as aromatic systems, oxygen atoms, and triple bonds. A similar appearance of structural features occurs as ChiA_B(s) decreases in value; oxygen, double bond, and triple bond atom types bear larger I-state values, therefore enlarging the numerator of the connective index.

In summary, it is shown that smaller ChiA_B(s) values, and generally larger RON values, are produced by molecules bearing higher degrees of branching, shorter straight chains, oxygen atoms, alkenes, alkynes, and aromatic systems. As mentioned in the previous GATS2m and SssCH2 discussions, such structural features enhance the stability of intermediates and combustion products, suppressing radical reactions which dominantly occur at lower engine temperatures and are instrumental in the buildup toward ignition. Therefore, auto-ignition of such molecules is hindered until higher temperatures are reached where decomposition occurs.

Octane sensitivity had a different set of important QSPR descriptors. The top three descriptors for OS characterized molecules based on carbon hybridization and structural diversity. Figure 6 contains the plots for the relationship between OS and AVS_B(s), SIC1, and nCsp2.

The descriptor of top importance for octane sensitivity is AVS_B(s), which represents the average vertex sum from the Burden matrix weighted by I-State. Figure 6(a) shows the relationship between AVS_B(s) and OS. Similar to the ChiA_B(s) descriptor, AVS_B(s) is a numerical representation of multiple structural components: connectivity, bond order, and I-state value of all atoms within the molecule. The highest valued AVS_B(s) compounds are those containing atoms with large I-state values; carbonyl, hydroxy, and ether oxygens have a 7.00, 6.00, and 3.50 I-state value, respectively [15]. This is significantly higher than the I-states of other carbon-based atom types found in saturated hydrocarbon fuels, such as methyl (2.00) and methylene (1.50) carbons. Molecules bearing aromatic systems or double bonded carbons are also amongst the highest AVS_B(s) values since terminal and internal alkene carbons have I-state values of 3.00 and 2.00, respectively. Compounds which are oxygenated and bear double bonds or aromatic systems, are known to withstand auto-ignition at lower temperatures since they participate in stabilized radical reactions. Electronegative groups like double bonds draw electron densities closer to themselves, making alpha hydrogens, those directly adjacent to the electronegative group, more abstractable [54]. These functional groups can then stabilize the newly formed radical intermediates and products. For example, alkenes stabilize allylic radicals through electron delocalization and terminal hydroxy groups can form highly stable oxygenated products such as aldehydes [55]. This hinders decomposition, and therefore auto-ignition, since heat producing chain reactions are delayed through retarded reaction rates, until higher temperatures are reached. It is at these higher temperatures that decomposition reactions experience increased reaction rates, creating a large sensitivity value [56].

Figure 6(c) shows the relationship between OS and nCsp2, a basic constitutional index which measures the number of sp² hybridized carbon atoms in each compound. An sp² hybridized carbon atom will (1) have the ability to form double bonds with another adjacent sp² carbon, (2) have bond angles of 120 deg, or trigonal planar structures, and (3) create resonant structures through electron delocalization and conjugated double bonds. Figure 6(d) illustrates the distribution of OS at varying values of nCsp2. For each distribution, top and bottom bars represent the minimum and maximum values of OS, the center bar illustrates the median value of OS, and the width of distribution shows the overall distribution of OS. It is observed that compounds with 2 and 4 sp² hybridized carbon atoms tend to have higher values of OS. Molecules with 2 or 4 sp² hybridized carbons, also known as alkenes, change the kinetic rates of reactions during combustion at low (RON) and high (MON) temperatures when compared to their sp³ carbon analogs. Therefore, their high sensitivity tendencies, or their large difference between RON and MON, stems from the reactions that contribute the most to ignition under low versus high temperature conditions. At low temperatures, alkenes participate mostly in peroxy chemistry that involves oxidation, chain branching, and radical intermediates, also called QOOH. Due to the resonant structures created through electron delocalization, alkenes increase stability in the short-lived and highly reactive intermediates and products of such reactions. Such stability slows reaction rates which give rise to higher RON, indicative of less reactivity at low temperatures. On the contrary, different reactions dominate at higher temperatures due to the negative temperature coefficient (NTC) behavior of alkenes and their peroxy reactions [56]. As temperatures increase, the dominant peroxy chemistry begins to slow and is replaced by reactions relating to decomposition and ignition, producing a lower MON value and, thus, a larger OS value.

Figure 6(b) shows the relationship between OS and SIC1, the structural information content index of first-order neighborhood symmetry. This descriptor is an information index whose foundation resides in bond fragments, molecular symmetry, and information theory. This descriptor specifically measures structural information provided by a molecule across fragment lengths of 1 bond such as C–C or C–H. This descriptor applies Shannon's information theory to hydrogen inclusive molecular graphs. Shannon's information theory, and SIC (structural information content index), adhere to the next three statements:

• An event with 100% probability is not surprising at all and yields no information.

• Less probable events are more surprising, thus yielding more information.

• If N independent events are measured separately, then the total amount of information between the events is equal to the sum of the information yielded by the N individual events.

“Events,” in this context, are analogous to the first-order fragment types, or fragments of one bond length. Information is measured in entropic units called bits, equivalent to a measure of surprise, and the use of Shannon's entropy equation facilitates the calculation of bits provided by a molecule's fragments [16]. If a molecule has a large bit value, then it is denser with information when compared to a lesser bit value. Symmetry within a molecule is seen to reduce the bit value; this is because chemically equivalent fragments contribute to a loss of “surprise” as the molecule becomes more chemically uniform, or easily predictable; the second principle of Shannon's information theory is followed here, where events are equivalent to first-order fragment types.

Table 3 illustrates how structural diversity influences both SIC1 and OS. A difference in diversity can be seen when comparing the first-order fragments belonging to the highest and lowest SIC1 molecules in Table 3.

A molecule with high information content, given its positive trend with OS, has variation that indicates the presence of multiple features which are known to increase octane sensitivity. For example, introducing oxygen, as an alcohol (OH) or carbonyl (C = O), to a hydrocarbon can significantly increase the SIC1 since these new fragments add diversity to a molecule otherwise composed of only C–C and C–H bonds. Similarly, adding sp² carbons also increases the SIC by facilitating new C = C and C = O fragments. These fragments are known to increase octane sensitivity as shown with previous OS descriptors discussed [54,55].

The top OS descriptors each belong to different classes of indices, yet they are moderately linearly correlated given their relationships to structural features that influence OS. SIC1 is somewhat linearly correlated (0.6614) with nCsp2 since an increase of sp² carbons will increase the number of unique fragments within a molecule compared to its sp³ hybridized analog. It is noteworthy to recall the impact of alkenes and oxygen atoms on fuel stability and sensitivity. An even stronger correlation is found between AVSB(s) and SIC1 (0.7276); as the average vertex sum increases, molecules yield more bits of information since their structure becomes more complex across neighborhood symmetry of order one.

While there are multiple descriptors of high importance when determining RON, MON, and OS, it is seen that the top descriptors are structurally interconnected, repeatedly emphasizing certain structural features which are known to affect fuel performance. As shown in this analysis, knock resistance is heavily dependent on degree of connectivity, double bonds, aromatic systems, structural diversity, and the presence of oxygen atoms within fuel compounds. Such features and analyses should serve as guidelines when searching for desirable alternative fuel sources.

The ANNs trained in the present work can further be validated through their ability to recognize top performing next-generation fuels currently being investigated by the scientific community. Thus, ANNs will also be able to identify other fuels that possess unexplored molecular structures bearing optimal QSPR descriptor values. Using furan as a reference molecule, the following analysis shows how the ANN is able to clearly extract high-performing molecules and numerically explain how single-bond changes to a molecule impacts its OS. The molecules analyzed are restricted to furanic derivatives for purposes of clarity.

Using furan as a base reference molecule, the addition of methyl groups to obtain 2-methylfuran (2-MF) and 2,5-DMF is discussed in relation to their OS and top QSPR descriptors (AVS_B(s), nCsp2, and SIC1); the values are summarized in Table 4. Furanic compounds have adequate fuel properties and can be sourced from biological feedstock such as lignocellulosic-derived sugars [57]. Furthermore, 2-MF and 2,5-DMF have OS values of 17 and 13.2, respectively, both have a higher energy density than ethanol, and limited solubility in water [57]. The following analysis of the top QSPR descriptors of furanic compounds (furan, 2-MF, 2,5-DMF) validates the utility of the ANN models to extract complex relationships between molecular structure and OS, specifically its ability to highlight promising potential molecules that are currently being studied as next-generation biofuels. This is especially important as a tool for fuel design as undiscovered trends in structure and reactivity are recognized by the ANN.

All three furans have an nCsp2 value of four, placing them in the highest OS grouping in Figs. 6(c) and 6(d), confirming the known relationship between conjugated double bonds and their reactivity at low-temperature oxidation. Had the NTC of double bonds in low-temperature oxidation not been known, the ANN would evidently hint at this phenomenon which could be used as guidance for reaction kineticists and fuel designers alike.

Adding a single methyl to furan increases the diversity of bond type within the molecule, resulting in a rise of SIC1 from 0.57 to 0.73, outlined in Table 3 as the highest SIC1 of the data set. The addition of a second methyl group to create 2,5-DMF, however, only increases the diversity by 0.02 since adding two of the same bond type increases diversity to a lesser extent than introducing a single new bond type. The ANN models confirm that structural diversity within a molecule permits for variation of reactivity at different conditions as suggested by the relation between increasing SIC1 and increasing OS.

The addition of each methyl group reduces the AVS_B(s) of these furanic compounds. As previously discussed, the AVS_B(s) descriptor is an averaged connectivity matrix weighted by I-State, encapsulating the significance of the I-state value of oxygen atoms present in compounds like furans. The addition of methyl groups decreases AVS_B(s) since the weight of the oxygen I-state (3.50) becomes less significant when averaging a larger sized matrix; furan's AVS_B(s) matrix is 9 × 9 and the addition of each methyl group expands the matrix to 12 × 12 (2-MF) and to 15 × 15 (2,5-DMF), diluting the 3.50 I-state value of their ether oxygen. The ANN models exemplify an intricate balance between the reactivity impact of bond order and I-state in a molecule, which warrants further investigation as to how the relationship correlates to OS. Studying the molecules and their respective OS solely using the intuition or physical properties, like boiling point, would not yield nearly as much information as the ANN is able to do using the AVS_B(s) matrix. Such information is significant as it can be applied to newly proposed fuel molecules.

In conclusion, the power of ANNs to extract hidden relationships between molecular structure and reactivity at varying conditions is a key tool in the design of optimal fuels. The ANNs trained in the present study are able to highlight both known phenomenon, such as the NTC of alkenes, and unknown phenomenon, such as the balance between I-state and connectivity, that gives rise to OS.

Predictive models using ANNs have been shown to be an accurate, sustainable, timely, and economically-viable approach for predicting fuel properties of next-generation fuels. Based on the present work focused on the RON, MON, and OS, the following specific conclusions can be made:

• ANNs are suitable for predicting RON and MON with overall test set errors of 6.4% and 5.2%, respectively.

• Predictive models using ANNs are also capable of predicting OS and dOS with overall test set percent error of 9.9% and 15.3%, respectively, showing that models perform better at predicting OS directly rather than by deriving OS from RON/MON predictions.

• The top descriptors for RON/MON are GATS2m, SssCH2, and ChiA_B(s), while those for OS are AVS_B(s), nCsp2, and SIC1.

• Knock resistance and octane sensitivity are heavily influenced by molecular connectivity, symmetry, double bonds, the presence of oxygen atoms, and aromatic systems.

• The ANNs in the present work are able to identify high-performing molecular fuels which are currently being studied in the literature today as viable fuels, thus highlighting the utility of ANNs to discover new fuel molecules.

• ANNs can pinpoint exact origins of reactivity within a molecule's simple and complex structural features, a useful tool for fuel design.

As such, in addition to the prediction power of artificial neural networks, the information contained with influential QSPR descriptors used as input parameters is useful in understanding the role of chemical structure and fuel chemistry. The approach presented here provides a foundation and direction for future chemical analysis of QSPR descriptors to aid in the development of high-performance alternative fuels.

This material is based upon work supported by the U.S. Department of Energy (DOE) Office of Energy Efficiency and Renewable Energy (EERE) Bioenergy Technologies Office (BETO) under award DE-EE0008479 as part of the CoOptimization of Fuels & Engines (Co-Optima) project. Additionally, this project would not have been possible without the support of alvaScience and their alvaDesc software package.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Tables 5–7 

Table 8