Two-stage light gas guns (2SLGGs) can reliably accelerate projectiles to velocities between 1.5-8.0+ km/s and are used in hypervelocity impact, aerospace, and hypersonic research. 2SLGG operation involves a va- riety of physical phenomena including combustion, gas-compression, heat-transfer, and friction. Due to the wide range of operational parameters and experimental uncertainty, accurate muzzle velocity predictions can be a serious challenge. In this paper, a series of regression models for predicting muzzle velocity were fitted to and validated against 171 2SLGG launches (projectile velocities, 1.5-6.8 km/s) performed at the Texas A&M University Hypervelocity Impact Laboratory (TAMU HVIL). Most of the regression models analyzed had minimal accuracy improvement compared to basic linear regression. However, a neural network model (RMSE = 0.234 km/s) utilizing several methods to combat over-fitting, showed consistent improvement over linear regression (RMSE = 0.260 km/s) and Gaussian process regression (RMSE = 0.240 km/s). Regression model projectile velocity estimates were compared to results from the classical Piston Compression Light Gas Gun Performance (PCLGGP) and state-of-the-art LGGUN 2SLGG performance prediction codes. All of the regression models demonstrated significantly better predictive capabilities than the PCLGGP model (RMSE = 0.597 km/s), particularly at lower velocities. The regression model absolute errors from the 2SLGG experiments also compared very favorably to general absolute error estimates obtained using LGGUN. These results suggest that easy-to-implement, maintain, and scalable regression models may provide a viable alter- native to complex physics-based computational models for 2SLGG launch velocity predictions, particularly as the volume of available experimental data increases. Such regression models have the potential to markedly improve predictive capabilities, identify complex coupling between experimental parameters, and reduce uncertainty. Keywords: two-stage light gas gun (2SLGG), aeroballistic range, hypervelocity impact, regression, neural network, empirical model 1. Introduction and Motivation Hypersonic vehicles and spacecraft are subjected to extreme operating environments that can adversely 1 affect mission performance. For hypersonic vehicles travelling well above the speed of sound, impacts with 2 ∗Corresponding author Email address: TELacyJr@tamu.edu (Thomas E. Lacy, Jr.) Preprint submitted to Journal of Dynamic Behavior of Materials February 12, 2024 small, slow-moving atmospheric particles (rain, ice, dust, etc.) can be devastating. In space, micrometeoroid 3 orbital debris (MMOD) can impact spacecraft and planetary structures with relative velocities ranging from 4 2–70 km/s [1], leading to catastrophic system failure or possible loss of life. Although the impact physics 5 between the projectile and target are well-known at lower velocities, such knowledge does not directly transfer 6 to impacts at velocities exceeding roughly 2.5–3.0 km/s since the material response generally transitions from 7 strength-dominated (high-velocity) to shock/pressure-dominated (hypervelocity)1behavior [2]. Since the end 8 of WWII, numerous laboratories have been established worldwide to conduct hypervelocity impact (HVI) 9 research aimed at characterizing and mitigating HVIs. Many of these facilities employ a two-stage light gas 10 gun (2SLGG) to accelerate projectiles to hypervelocities. An extensive review of such 2SLGG facilities, their 11 capabilities, and research areas is presented in Rogers et al. [3]. 12 In general, 2SLGGs can be used to efficiently accelerate projectiles to velocities between 1.5-8.0 km/s. 13 Accurate and reliable prediction of 2SLGG muzzle velocities as a function of launch parameters can be 14 extremely challenging but is essential for the execution of tightly controlled launches. Without reasonably 15 accurate predictions of muzzle velocity, multiple launches may be necessary to achieve a desired projectile 16 velocity. Moreover, replicate experiments performed at a specific velocity (with an acceptable tolerance) can 17 be difficult to perform. This can dramatically increase experimental costs and turnaround times, as well as 18 limit the utility of single experiments where repeated shots are not feasible ( e.g., there are a limited number 19 of targets). Thus, incorporating robust and adaptable tools for predicting launch velocities is critical for the 20 execution of a viable test plan. The actual 2SLGG muzzle velocity depends on the complex coupling between 21 a multitude of operational parameters, including gunpowder type, gunpowder mass, piston mass, burst disk 22 rupture pressure, light gas initial pressure, projectile package mass, pump tube geometry, launch tube 23 geometry, frictional forces between components, etc. For example, the free volume of the powder chamber 24 can profoundly affect the powder burn efficiency, compression piston acceleration profile, and energy transfer 25 to the projectile. 26 Several methods have been developed to predict 2SLGG performance and muzzle velocity. Charters et 27 al.[4] at the NASA Ames Research Center developed one of the earliest and simplest models, implemented in 28 the Piston Compression Light Gas Gun Performance (PCLGGP) software package ( i.e., “Charters’ code”). 29 Charters’ code has been widely adopted and is relatively easy to implement since it involves the simultaneous 30 solution of a set of linear algebraic equations. The PCLGGP model, however, makes a number of simplifying 31 assumptions that can limit accuracy (such as negligible friction, negligible heat loss, and no gas flow in the 32 pump tube). In contrast, the Simple Isentropic Compression model [5] makes fewer assumptions (accounting 33 for friction and simplified gas flow in the pump tube, as well as subsonic-to-sonic gas flow in the nozzle) but 34 is slightly more challenging to implement than the PCLGGP model since it requires the numerical solution 35 of a coupled set of nonlinear differential equations. The Richter-Von Neuman “ q-method” [6] can account 36 1For simplicity, the transition from high-velocity to hypervelocity is loosely considered to occur over the range 2.5-3.0 km/s [2]. 2 for supersonic flow and shock waves in the pump tube, resulting in improvements in the predicted launch 37 velocities. The recently updated LGGUN program developed by Bogdonaff et al. [7, 8] at NASA Ames 38 is arguably the most sophisticated 2SLGG prediction code. LGGUN is a quasi-one-dimensional Gudonov 39 code that is second-order accurate in time and third-order accurate in space. Validation studies using 40 LGGUN report high accuracy, but the code implementation and interpretation of results requires considerable 41 expertise on the part of the user. 42 The limitations and difficulties associated with numerical 2SLGG performance prediction tools have 43 motivated the development and implementation of a few empirical approaches. Fraunhofer EMI developed 44 a neural network model to predict 2SLGG muzzle velocities in an effort to improve gun performance [9, 45 10]. However, limited information was provided regarding the neural network model’s design and accuracy. 46 Shojaei et al. [11] evaluated a series of regression methods for predicting 2SLGG muzzle velocities, with a 47 focus on random forest regression [12]. In the current study, we dramatically extend the work of Shojaei et 48 al.[11] by considering a more robust set of regression models. Random forest regression was not included 49 since an initial screening of regression models suggested that it did not perform as well as basic linear 50 regression. Apart from Shojaei et al. [11] and the current study, most techniques for predicting 2SLGG 51 performance are physics-based numerical models with, at most, empirically derived parameters. To the best 52 of the authors’ knowledge, there are no other purely empirical 2SLGG prediction studies reported in the 53 literature. 54 Since the advent of 2SLGGs ( circa 1950) [13, 14], many computational resources and advanced regression 55 techniques have been developed which can be used to predict launcher performance. For instance, artificial 56 neural network regression [15], support vector regression [16], and Gaussian process regression with particular 57 kernels [17] are often used for nonlinear regression since they serve as “universal approximators” [18, 19] able 58 to approximate any deterministic relationship between inputs and outputs. The different regression methods 59 vary widely in their implementation, accuracy, and unique strengths. 60 In this study, a series of regression methods (linear, LASSO, ridge, elastic-net, neural network, Gaussian 61 process, and support vector) are used to predict 2SLGG performance. Specification of each regression method 62 is first presented followed by a discussion of regression results. Concluding remarks are provided and future 63 improvements to the approach are suggested. 64 2. Methodology: HVI Experiments and Regression Models for Predicting 2SLGG Muzzle 65 Velocity 66 All regression models were developed using performance data from the powder-driven 12.7 mm bore 67 2SLGG used in the Texas A&M University Hypervelocity Impact Lab (TAMU HVIL) [2]. A brief overview 68 of the 2SLGG operation and the regression models are included in the following discussion. 69 3 2.1. 2SLGG Operation and Experiments 70 In essence, a 2SLGG harnesses the energy generated by a single-stage launch system to compress a 71 light working gas, which ultimately drives a projectile. Typically, 2SLGGs are comprised of seven primary 72 structural and consumable elements: (1) a pressure breech, (2) pump tube, (3) central breech, (4) launch 73 tube, (5) deformable compression piston, (6) burst disk (a.k.a. petal valve), and (7) projectile or projectile 74 package (projectile + sabot). The pressure breech, pump tube, central breech, and launch tube are coaxially 75 arranged and connected with rigid coupling mechanisms and interfacing O-rings to ensure gas-tight seals [3]. 76 Initially, the piston occupies the uprange end of the pump tube, and the projectile is situated at the uprange 77 end of the launch tube, just downrange of the burst disk (Figure 1a). A predetermined quantity of a low- 78 molecular-weight light working gas ( e.g., hydrogen or helium) is injected into the pump tube. As an aside, 79 gunpowder combustion generates the piston driver gas in most guns (˜80%), but a few guns use cold-gas (He, 80 N2,etc.) stored in high-pressure reservoirs, released through a fast-acting valve [3]. While cold-gas-driven 81 2SLGGs are not considered in this study, their inclusion in the future would be relatively straightforward. 82 For the HVIL gun and most other powder-driven 2SLGGs, the main (secondary) powder charge is ignited in 83 the pressure breech by a smaller, faster-burning primary charge. High-pressure combustion gases generated 84 in the powder/pressure breech are used to drive a deformable ( e.g. polyethylene) piston downrange within 85 the pump tube (Figure 1b); this compression process rapidly elevates the pressure and temperature of the 86 working gas. When a specific burst pressure is exceeded, the burst disk ruptures, exposing the projectile to 87 the compressed working gas, which propels it down the launch tube (Figure 1c). Eventually, the projectile 88 exits the muzzle at velocities up to ∼8 km/s (Figure 1d) [3]. Most 2SLGGs are accompanied by coaxial 89 range tankage that collectively form an aeroballistic range. Tankage assemblies usually include one or more 90 enclosed cylindrical tanks that contain the projectile during its free flight and/or impact. Range tankage 91 configurations and applications are largely outside the scope of this work but are summarized in Rogers et 92 al.[3]. For reference, key design parameters and capabilities of the HVIL 2SLGG are given in Table 1. 93 After the projectile package leaves the muzzle, the sabot generally must be stripped from the projectile 94 by some means. For smooth bore guns, such as that in the HVIL, sabots are separated from the projectile 95 viaaerodynamic forces within one or more of the range tanks [3, 20, 21]. The tankage internal pressure 96 can be varied to induce different degrees of sabot separation. Of course, this separation process can also 97 decelerate the projectile. Hence, the range tankage “backfill” pressure, in addition to primary and secondary 98 powder types/masses, piston mass, piston release pressure, working gas type and initial pressure, burst disk 99 rupture pressure, and projectile package mass can be readily varied to tune 2SLGG launch conditions and 100 performance. Accurately predicting the muzzle velocity given some or all of these variables is the primary 101 goal of this work. The fundamental operation of most unaltered 2SLGGs is very similar [3], allowing this 102 study’s findings, interpretations, and predictive methods to be easily adapted to other gun configurations. 103 4 Figure 1: An illustrative guide to the operating principle of a 2SLGG, detailing (a) the moment prior to projectile launch, (b) the phase of working gas compression, (c) the instant the burst disk ruptures and acceleration of the projectile commences, and (d) immediately following the projectile’s exit from the muzzle as it enters the aeroballistic range tankage. Figure reprinted with modification from Rogers et al. [3]. 2.2. Variable/Feature Selection 104 In total, ten different variables (features) were used for predicting 2SLGG muzzle velocity. Most of the 105 variables were continuous: projectile package mass, primary powder mass, secondary powder mass, pump- 106 tube fill (working gas) pressure, and tankage (backfill) pressure. The steel burst disk rupture pressure was 107 varied by changing the disk’s score depth. This score depth was treated as quasi-discrete since only two depths 108 were considered. The primary powder type and mass were not the only variables that influenced the pressure 109 profile in the powder breech (and thus piston acceleration in the pump tube): changing the internal volume 110 of the powder breech for a given powder mass resulted in more efficient powder combustion. Increasing the 111 piston release pressure had a similar effect. The powder breech internal volume was varied by incorporating 112 volume reducers. The number of volume reducers was treated as a discrete variable. Changes in piston 113 release pressure were somewhat controlled by discrete variations in the piston frictional fit within the pump 114 tube. The piston release pressure was treated as a subjective discrete variable characterized in terms of how 115 “tightly” the piston fit into the uprange end of the pump tube; “low,” “med,” or “high” levels of tightness 116 5 corresponded with parameter values of 0.0, 0.5, and 1.0, respectively. Finally, five categorical features were 117 used to represent the five different gun powders employed in previous experiments: H4831 [22], H4831SC [22], 118 IMR 4831 [23], 50BMG [22], and SW50BMG [24]. The 50BMG and SW50BMG powder burn data needed for 119 PCLGGP predictions were not available at the time of this publication. Hence, to compare with PCLGGP 120 predictions, data using these powders was omitted. However, the full-scale laboratory implementation of the 121 regression model for 2SLGG muzzle velocity is trained on more of the data which includes all powder types. 122 2.3. Model Validation and Hyperparameter Tuning 123 A regression model is only useful so far as it can accurately predict new data. Since models can overfit 124 data (accurately predicting given data but mispredicting new data) the only reliable metric for a regression 125 model’s performance ( i.e.,accuracy) is how well it predicts “unseen” data that was not used to fit the model. 126 In practice this means that some of the data must be withheld during the training process to validate model 127 performance afterwards. A common method is holdout validation which involves partitioning the data into 128 a training set (for fitting the model) and a testing set (for validating model performance) [25]. Determining 129 how much data to withhold for testing presents a nuanced challenge, particularly for smaller datasets. The 130 training set needs enough data to construct a robust model while the testing set needs enough data to give 131 a reliable and representative estimate of model performance. [25] 132 During training, parameters (analogous to model coefficients) are fit to match the training data through 133 some sort of optimization process ( e.g., gradient descent [26]). However, some regression models contain both 134 parameters (optimized during the training process) and hyperparameters which affect the training process 135 and must be decided beforehand. Examples of hyperparameters include the learning rate, number of training 136 iterations, penalty terms, and the number of layers in a neural network. These hyperparameters can have a 137 significant impact on a model’s performance but must be determined outside of the model’s typical training 138 Table 1: Key design and operational parameters for the TAMU HVIL 2SLGG [2]. Regression model parameters are indicated with an asterisk (*). Parameter Value 2SLGG total length (m) 13.70 Pump tube inner diameter (mm) 44.00 Pump tube length (m) 3.70 Launch tube inner diameter (mm) 12.70 Launch tube length (m) 3.70 Single-projectile diameter range (mm) 2.0–12.7 Achievable projectile velocity range (km/s) 1.5–8.0 Maximum rated kinetic energy (kJ) 80.00 Projectile package mass*(g) 2.0–6.1 Primary powder mass*(g) 1.0–2.0 Secondary powder mass*(g) 50–165 Pump tube fill (working gas) pressure*(MPa) 0.96–1.93 Tankage (backfill) pressure*(kPa) 0.0–66.0 Piston Mass*(g) 350-800 Burst disk score depth*(mm) 0.36, 0.51 Piston Fit Tightness*low (0.0), medium (0.5), high (1.0) Powder chamber volume*(cm3) 172.6, 265.5, 424.8 Secondary powder type*H4831, H4831SC, IMR 4831, 50BMG, and SW50BMG 6 process. To create better models, hyperparameters are often tuned algorithmically to achieve the best model 139 performance on some unseen data set (tuning data). Like the testing set, the tuning set gives an estimate of 140 how well the model predicts unseen data with specific hyperparameters. However, the tuning set and testing 141 set must be disjoint to avoid “peeking:” including testing data in the hyperparameter tuning process. Once 142 the testing data has been peeked it is no longer truly unseen data which risks overfitting and can bias the final 143 results. A simple method for segmenting the testing, tuning, and training sets is to first partition the data 144 into a training and testing set and then further partition the training set into a tuning set and final training 145 set as demonstrated in Figure 2. Since hyperparameter tuning shrinks the training set, it makes training 146 robust models on limited data more difficult. For this reason, using the final hyperparameters, the model is 147 typically retrained on the combined training and tuning data before validating the model’s performance on 148 the testing data. 149 To alleviate the problems with holdout validation on small data sets, cross-validation can be used in- 150 stead [25]. In k-fold cross-validation, the data is partitioned into ksubsets (usually 5-10). For each subset, 151 a model is trained on the other k−1 subsets combined and tested on the original subset. If this is done for 152 each subset, the aggregate results give a measure of model performance based on all data while still using 153 (k−1)/kof the data to train each model. This gives a more reliable and representative measure of model 154 performance while still using sufficient data to train robust models. A drawback of cross-validation is its 155 computational cost, as it requires the model to be trained ktimes instead of just once. However, when 156 several models are trained on slightly different data (as in each fold), the predictions can be averaged to give 157 a result less prone to overfitting. This is referred to as k-fold averaging [27] and is one of many ensemble 158 Figure 2: Illustration of data partitioning for hyperparameter tuning and model validation using basic holdout validation. 7 methods [28] used to combat overfitting in regression models. 159 Adding hyperparameter tuning to k-fold cross-validation can be accomplished in two different ways. The 160 simplest option is to partition the training set into a single training and tuning set for each fold like in 161 holdout validation. Another option is to split the training set into several subsets to perform nested cross- 162 validation [29] which uses all the training data to optimize hyperparameters in the same way non-nested cross 163 validation uses all the data to validate model performance. The recursive data partitioning involved in nested 164 cross-validation is demonstrated in Figure 3 where an outer loop (a) is used to validate model performance and 165 within each of these outer loops another inner loop (b) is used to optimize the hyperparameters. Nested cross- 166 validation often leads to higher computational costs without added benefit [29], but was deemed necessary 167 in this study to utilize k-fold averaging. 168 In this study, the regression model performance varied dramatically depending on which data points were 169 used in the test set (some being easier or harder to predict than others) due to the limited number of data 170 points (171, see Appendix A) and large number of variables (10). As such, five-fold cross-validation was used 171 to give a more reliable estimate of model performance. For regression models needing hyperparameter tuning, 172 Figure 3: Pictorial demonstration of recursive data partitioning in nested cross-validation with five-fold cross-validation in the outer loop (a) and three-fold cross-validation in the inner loop (b). 8 hyperparameters were optimized using the Tree-Structured Parzen Estimator method [30] and nested cross- 173 validation (five outer and inner folds) was employed to leverage k-fold averaging. This approach minimized 174 overfitting by averaging predictions from all models in the inner loop, rather than retraining a single model 175 with the combined training and tuning data. 176 2.4. Regression Methods 177 The most basic form of regression is linear least-squares regression, which determines the weights (co- 178 efficients) that minimize the squared residual error. LASSO [31], ridge [32], and elastic-net [33] regression 179 function nearly identically to linear regression but add a penalty term (abbreviated as P) to the squared 180 error that is related to the magnitude of the weights {wi}: 181 LASSO: P=λLX |wi|, (1) ridge: P=λRX w2 i, (2) elastic-net : P=λLX |wi|+λRX w2 i, (3) where λLandλRare hyperparameters for scaling the penalty term. This encourages the regression model to 182 assign zero weights to variables that only marginally improve results which reduces over-fitting. A penalty 183 term of zero ( λL=λR= 0) makes all three methods identical to basic linear regression. 184 These linear regression methods can be expanded to higher order polynomial regression ( e.g., quadratic) 185 by using a higher order expansion of the independent variables. For example, a linear model with independent 186 variables x1, x2and dependent variable ycan instead be expressed using a quadratic form: 187 z1=x1, z2=x2, z3=x1x2, z4= (x1)2, z5= (x2)2. (4) Then the relationship f(z1, z2, z3, z4, z5) =ycan be fit using linear regression (just with more variables) 188 and is equivalent to a quadratic regression of the relationship g(x1, x2) =y. This is less useful for more so- 189 phisticated regression methods which can intrinsically derive higher order relationships ( e.g., neural network 190 regression) but is essential for modeling more complex phenomena with linear regression methods. 191 Support vector regression [16] is more complicated than the LASSO, ridge, and elastic-net methods and 192 seeks to find a curve that represents the data with maximal error ( ϵ) between the curve and experimental 193 values. Anything within ±ϵis treated the same whether the error is 0 or 0 .99ϵ. This helps create a curve that 194 fits the data within acceptable margins while avoiding over-fitting. Since ϵis a user-specified hyperparameter, 195 the problem can often be over-constrained at which point some “slack” beyond ϵis allowed but minimized. 196 Gaussian process regression [17] is based on Bayesian statistics where prior “beliefs” (assumptions on 197 9 probability) are updated based on new information. Gaussian process regression starts by assuming some 198 prior distribution of functions and then updates this distribution based on given data. Instead of assuming 199 a specific function for predicted model output, Gaussian process regression gives a distribution of functions 200 with common properties ( e.g., differentiability, periodicity, and how close two points need to be to affect 201 each other). In a simple case with no experimental error, giving the algorithm several data points updates 202 the prior distribution of functions to a new (posterior) distribution where all functions pass through the 203 given data points. From this new distribution, the expected value (or mean) at a specified point is the 204 prediction of the model. Noisy inputs can be easily accounted for by adding a hyperparameter for the 205 variance (or experimental error/noise) of the training data ( σ2 n) to the model. Unlike other regression 206 methods, Gaussian process regression can give an estimate of the variance ( i.e., uncertainty) of a prediction 207 using the variance of the posterior distribution at the specified point. However, Gaussian process regression 208 can become computationally expensive on larger data sets due to the inversion of an N×Nmatrix, where 209 Nis the number of data points. In addition, difficulties may arise from an excessive number of features 210 such as in image recognition. For this study, the Gaussian process regression model employs a radial basis 211 function (RBF) kernel [34] with added noise and is allowed to update its hyperparameters and restart the 212 process up to 200 times on each outer fold of the training data. Unlike most other models, it does not update 213 these hyperparameters using nested cross validation nor the Tree-Structured Parzen Estimator method [30] 214 but instead an internal process which maximizes the log-marginal likelihood [34]. The model’s accuracy was 215 determined using basic (non-nested) five-fold cross validation. 216 Neural network models are based on the structure of neural networks found in the brain [25]. They 217 consist of layers of interconnected “neurons” which can process and transmit information. Each neuron 218 receives the inputs from the previous layer and performs a weighted sum on these inputs, applies a bias, 219 and passes it through a nonlinear activation function to produce an output for the next layer of neurons. 220 Adjusting these weights and biases (through training the model) allows a sufficiently large neural network 221 to approximate arbitrary functional relationships [18]. However, since neural networks use many weights 222 and biases to fit the data, they are prone to over-fitting and lack interpretability. Many neural network 223 architectures exist in the literature with a variety of strengths and weaknesses. After testing many different 224 architectures and techniques, a simplified DenseNet architecture [15, 35] was used in this study due to its 225 feature-reuse capabilities ( i.e., inputs are not “lost” as the depth of the neural network increases). The neural 226 network model also employs dropout regularization [36] and k-fold averaging [27] to reduce overfitting. 227 3. 2SLGG Muzzle Velocity Prediction Results and Discussion 228 Linear least-squares (basic, LASSO, ridge, and elastic-net), support vector, Gaussian process, and artifi- 229 cial neural network regression were used to empirically model the 2SLGG’s muzzle velocity. The independent 230 variables (parameters) used were (1) projectile package mass, (2) primary powder mass, (3) secondary pow- 231 der mass, (4) pump tube fill (working gas) pressure, (5) tankage (backfill) pressure, (6) piston mass, (7) 232 10 burst disk score depth, (8) piston fit tightness, (9) number of volume reducers in the powder breech, and 233 (10) gunpowder type. The results of each regression model and their comparison to those from two numeri- 234 cal models (PCLGGP and LGGUN) are presented in this section. The models are compared based on the 235 root-mean-squared error (RMSE): 236 RMSE =vuut1 nnX i=1|f(⃗ xi)−yi|2, (5) where nis the number of ( ⃗ xi, yi) input(s)/output pairs to the regression model f(⃗ xi). RMSE (Eq. 5) 237 approximates the standard deviation of the errors (lower is better). All predictions and error measurements 238 are based on data points withheld during the training process. 239 3.1. Regression Results for 2SLGG Muzzle Velocity Predictions 240 The predicted projectile velocity obtained using the most basic model (linear regression) was compared to 241 that from PCLGGP (Charters’ code) over the given 2SLGG velocity range 1.5-6.8 km/s. The linear model 242 (RMSE = 0.260 km/s) matched the measured muzzle velocity values (diagonal dashed line) significantly 243 better than the PCLGGP model predictions (RMSE = 0.597 km/s) especially at lower muzzle velocities 244 (Figure 4a). The plot clearly shows that PCLGGP consistently over-predicted the actual values at lower 245 launch velocities ( i.e.,<5.0 km/s) but yielded more accurate results for velocities >5 km/s. Since Char- 246 ter’s code does not account for the effect of piston and projectile friction, the PCLGGP model has been 247 reported to consistently over-predict muzzle velocities by 10–20% [37]. Since these effects become less sig- 248 nificant for more energetic shots, the PCLGGP model error will tend to decrease with increasing muzzle 249 velocity. The difference between PCLGGP and linear regression is clearer in Figure 4b, which shows the 250 velocity normalized absolute error ( ϵ=|vmeasured −vpredicted |/vmeasured ) compared to the measured veloc- 251 ity. PCLGGP’s relative error exceeded 70% at lower velocities ( ∼2 km/s) but significantly improved with 252 increasing projectile velocity. The linear model predicted a few outliers at lower velocities but provided more 253 stable predictions throughout the entire velocity range. The dashed lines in Figure 5b demonstrate that the 254 average relative error for the linear regression predictions (5.4%) was dramatically lower than those for the 255 PCLGGP model (15.8%). 256 The results of LASSO (RMSE = 0.259 km/s), ridge (RMSE = 0.260 km/s), and elastic-net (RMSE = 257 0.260 km/s) regression were almost identical to the linear regression results. In all cases, the penalty terms 258 were negligibly small. Recall that with a null penalty term the LASSO, ridge, and elastic-net regression 259 predictions will be identical to that for basic linear regression with no reduction in overfitting. Hence, a 260 penalty term near zero implies that, for a comparable linear model, there is little to no benefit in removing 261 any independent variables. The support vector regression (RMSE = 0.263 km/s) predictions approach, 262 but do not improve upon, the linear model results even with significant hyperparameter tuning. Hence, 263 the support vector regression model was the only model considered that performed worse than basic linear 264 regression. A summary of all model results is presented in order of descending RMSE values in Table 2. 265 11 Figure 4: PCLGGP (Charters’ code) results compared to linear regression results: (a) predicted muzzle velocity versus measured muzzle velocity and (b) normalized absolute error versus measured muzzle velocity. A reasonable explanation exists as to why the LASSO, ridge, elastic-net, and support vector methods 266 did not perform better than simple linear regression. While 171 2SLGG launches are significant from a 267 cost and time standpoint, there are still relatively few data points for regression, especially when using 268 10 independent variables. Furthermore, significant random error between nearly identical shots makes it 269 difficult to differentiate trends in the data from random noise. For example, when linear regression was 270 performed with the secondary powder mass as the only independent variable, the model achieved an RMSE 271 of 0.439 km/s and an R-squared value of 0.92 ( i.e., 92% of all the variance in the data can be explained 272 solely by the secondary powder mass). The coupled effects of other variables are likely numerous, complex, 273 and largely overshadowed by random error and the dominant effect of the secondary powder mass. This 274 makes it more difficult to model weaker trends without overfitting the limited data. 275 Gaussian process regression showed a noticeable improvement (RMSE = 0.240 km/s) over linear regres- 276 sion (RMSE = 0.260 km/s). Moreover, it also gives an estimate of the variance in the prediction at any 277 evaluated point— this is useful in determining the model’s confidence in each prediction. The linear, LASSO, 278 Table 2: The root-mean-square error (RMSE) over holdout data for each model over the 2SLGG operational velocity range 1.5- 6.8 km/s (The Quad. RMSE column represents the results when quadratic features were passed through the model). LGGUN results (*) correspond to a different set of experiments performed using multiple 2SLGGs [38, 39]. No. Model RMSE (km/s) Quad. RMSE (km/s) 1 PCLGGP (Charters’ code) 0.597 . . . 2 LGGUN [38, 39] 0.302*. . . 3 Support Vector 0.263 . . . 4 Linear 0.260 1.445 5 Elastic-Net 0.260 0.255 6 Ridge 0.260 0.249 7 LASSO 0.259 0.248 8 Gaussian Process 0.240 . . . 9 Neural Network 0.234 . . . 12 ridge, and elastic-net methods were also expanded to quadratic regression by using a quadratic expansion 279 of the variables. The basic quadratic model (linear regression model with quadratic variables) performed 280 significantly worse (RMSE = 1.445 km/s) than basic linear regression. In contrast, the quadratic LASSO, 281 ridge, and elastic-net models provided a slight improvement in prediction error (RMSE = 0.248-0.255 km/s) 282 relative to basic linear regression (RMSE = 0.260 km/s; cf., Table 2). The difference between the basic 283 quadratic model results and the results of the quadratic LASSO, ridge, and elastic-net models is explained 284 by the lack of weight (coefficient) regularization in basic polynomial regression, where the coefficients of all 285 polynomial terms must be determined, leading to substantial overfitting in higher order regression as the 286 number of terms increases. In contrast, the LASSO, ridge, and elastic-net regression techniques can eliminate 287 excess (non-contributing) variables to avoid overfitting. Moreover, since the RMSE is calculated based on 288 unseen data (instead of the training data), the overfitted basic quadratic model performed much worse than 289 the basic linear model. Note that the RMSE values of the quadratic models were all greater than that for 290 the Gaussian process model (RMSE = 0.240 km/s). 291 In contrast, the neural network model (RMSE = 0.234 km/s) was the most accurate of all regression 292 models considered and exhibited reliable (albeit nominal) improvement over linear regression (RMSE = 293 0.260 km/s) and Gaussian process regression (RMSE = 0.240 km/s). Figure 5a compares the linear and 294 neural network predictions to measured velocities, and Figure 5b compares the relative error of the linear 295 and neural network predictions at different muzzle velocities. In both Figure 5a and Figure 5b, the modest 296 improvements in predicted launch velocities obtained using the neural network model are readily apparent 297 but are much less pronounced than the differences between PCLGGP and linear regression predictions shown 298 in Figure 4. Moreover, error histograms corresponding to the neural network and PCLGGP results (Figure 6) 299 clearly show that the neural network model’s error distribution has a lower standard deviation and less bias 300 towards over-prediction. As an aside, as the number of experiments is further increased, the accuracy of 301 the neural network and Gaussian process models are expected to increase dramatically relative to the other 302 regression models considered here since they are well suited for approximating the nonlinear relationships 303 between independent variables that a larger data set would reveal. As the number of experiments increases, 304 however, the neural network model will scale better with the size of the data set. 305 Despite the neural network model’s relative increase in accuracy and potential for improved predictions 306 on larger data sets, it has some unavoidable limitations relative to the other empirical models. For example, 307 in traditional polynomial regression, the magnitude and sign of each coefficient in the fitted model defines the 308 influence of each independent variable (projectile package mass, working gas pressure, etc.) on the predicted 309 response ( i.e.,projectile velocity), as well as how coupled interactions between independent variables affect 310 the model outputs. The neural network model, however, lacks such intuition. Relationships between inputs 311 and outputs can be estimated by evaluating the neural network model at different points, but not directly 312 interpreted from the trained weights and biases. Similarly, while the Gaussian process model was slightly 313 less accurate than the neural network model, it provides a prediction of uncertainty that the neural network 314 13 Figure 5: Neural network model results compared to linear results: (a) predicted muzzle velocity versus measured muzzle velocity and (b) velocity normalized absolute error versus measured muzzle velocity. Figure 6: Histogram of errors for (a) the neural network model and (b) the PCLGGP predictions. Over-predictions by the model are represented as negative to match y=by+ε. model cannot. The slight loss of accuracy associated with use of the Gaussian process model may be offset 315 by the ability to estimate prediction uncertainty. Overall, most of the regression methods considered in this 316 study had an RMSE value less than one-half that for the PCLGGP model. This demonstrates that even 317 simple empirical models are well-suited for predicting 2SLGG muzzle velocity if sufficient experimental data 318 exists. 319 3.2. Projectile Velocity Absolute Error Percentiles and Comparison to LGGUN 320 Using data from the 171 TAMU HVIL 2SLGG experiments with muzzle velocities in the range 1.5- 321 6.8 km/s, projectile velocity error estimates obtained using the PCLGGP model and 11 regression models 322 14 were expressed in terms of absolute error percentiles. For example, the 50thpercentile (2ndquartile, Q2) 323 would define the median absolute error associated with each model. Similarly, the absolute error associated 324 with the 25thpercentile (1stquartile, Q1) means that 25% of the predicted values would have absolute errors 325 less than or equal to the specified value. Hence, absolute error percentiles can be used to assess how well a 326 given model approximates the actual velocity over the entire range of experimental observations and estimate 327 the likelihood of different magnitudes of error in future predictions. The absolute error estimates generated 328 using the 171 TAMU 2SLGG experiments in this study were compared to published values predicted using 329 LGGUN [8] for 52 experiments performed using five different 2SLGGs over a velocity range of 3-11 km/s [38, 330 39]. As mentioned previously, LGGUN is a leading edge 2SLGG performance prediction code. 331 Excluding the basic quadratic model ( i.e., linear regression model with quadratic variables), all of the 332 regression models’ root mean square error (RMSE) values (0.234-0.263 km/s) were lower than that for the 333 LGGUN model (0.302 km/s; cf., Table 2). Similarly, Table 3 includes a summary of predicted velocity 334 absolute error estimates corresponding to the 25th, 50th, 75th, and 100thpercentiles ( i.e., quartiles Q1, Q2, 335 Q3, Q4, respectively) for the PCLGGP and 11 regression models, as well as generic LGGUN values from 336 data provided by D. W. Bogdanoff and shown in the literature [38, 39]. Figure 7 contains a graphical 337 representation of these same data using “box and whisker” plots [40]. For a given model, the “box” provides 338 the Q1, Q2 (median), and Q3 velocity absolute error estimates, as suggested in the figure. The “whiskers” 339 associated with a given model defines the minimum (Q0) and maximum (Q4) values including outliers. Not 340 surprisingly, the absolute errors associated with LGGUN predictions were over 40% lower than those for 341 PCLGGP for all quartiles. Similarly, all of the regression models significantly outperformed the PCLGGP 342 model. For example, the median absolute error (50thpercentile; Q2) for each of the regression models was 343 at least 62% lower than the PCLGGP value. With the exception of the 100thpercentile (Q4) absolute 344 error for the basic quadratic linear model, all of the regression models outperformed Charters’ code over 345 the entire range of experiments. The regression model Q1-Q3 absolute errors from the TAMU 2SLGG 346 experiments also compared favorably to general absolute error estimates obtained using LGGUN. Excluding 347 the basic quadratic linear model, the 25th(Q1), 50th(Q2, median), and 75thpercentile (Q3) absolute errors 348 for the regression models were at least 10%, 25%, and 30% lower than the corresponding LGGUN values, 349 respectively. For comparison purposes, the neural network model’s predicted 25th, 50th, and 75thpercentile 350 absolute errors were 18%, 39%, and 38% lower than the generic LGGUN values, respectively. All of the 351 regression model absolute errors, however, exceeded the LGGUN 100thpercentile (Q4) value. This latter 352 difference may be attributable to a number of factors including: 1) the increased likelihood of encountering 353 outliers within the larger TAMU dataset (171 experiments) compared to the dataset used to validate LGGUN 354 (52 experiments); 2) the reliance of regression models on data points in the training set with similar or 355 “nearby” inputs ( i.e., experiments conducted at the extreme range of independent variables will be more 356 difficult to predict), and 3) better fidelity over the entire range of independent variables associated with 357 the physics-based LGGUN numerical model. As an aside, if LGGUN was used to predict muzzle velocities 358 15 Table 3: Absolute error percentiles (in km/s) for all considered models. LGGUN results (*) correspond to a different set of experiments performed using multiple 2SLGGs [38, 39]. Absolute error Percentile (Quartile) No. Model 25th(Q1) 50th(Q2) 75th(Q3) 100th(Q4) (km/s) (km/s) (km/s) (km/s) Numerical Models 1 PCLGGP 0.185 0.501 0.677 1.381 2 LGGUN [38, 39] 0.083*0.226*0.395*0.655* Regression Models 3 Linear 0.065 0.168 0.250 0.844 4 Quad. Linear 0.083 0.192 0.377 11.686 5 Ridge 0.065 0.168 0.247 0.839 6 Quad. Ridge 0.058 0.139 0.269 1.162 7 LASSO 0.068 0.155 0.261 0.826 8 Quad. LASSO 0.065 0.142 0.246 1.168 9 Elastic-Net 0.065 0.163 0.256 0.833 10 Quad. Elastic-Net 0.073 0.151 0.257 1.062 11 Support Vector 0.074 0.156 0.264 0.903 12 Gaussian Process 0.069 0.145 0.255 0.992 13 Neural Network 0.068 0.138 0.244 0.865 Figure 7: Box and whisker plots of the prediction absolute error for PCLGGP, LGGUN, and all considered regression models. Note that the Q4 whisker plot for the Quad. Linear model was cutoff since the Q4 value was much larger than that for any other model. LGGUN results correspond to a different set of experiments performed using multiple 2SLGGs [38, 39]. 16 for the 171 experiments using TAMU 2SLGG launch parameters, the predicted absolute error values would 359 likely improve upon the LGGUN values reported here. Nonetheless, these results suggest that easy-to- 360 implement, maintain, and scalable regression models may provide an attractive alternative to more complex 361 computational models such as PCLGGP and LGGUN for 2SLGG launch velocity predictions. While LGGUN 362 can provide crucial information about shock formation, bore erosion, pump tube pressure-time histories, and 363 other key aspects of specific 2SLGG performance, model specification and interpretation of results requires 364 considerable expertise. Clearly, regression models that are amenable to scaling across different 2SLGG 365 platforms can augment physics-based numerical models, particularly as the volume of available experimental 366 data increases. The regression models, such as Gaussian process and neural network, have the potential to 367 markedly improve predictive capabilities, identify complex coupling between experimental parameters, and 368 reduce uncertainty. In the future, it may be possible to significantly reduce the maximum (Q4) absolute 369 errors using physics-informed neural network models or Gaussian process regression models [41]. 370 4. Conclusions and Future Work 371 In this study, linear, ridge, LASSO, elastic-net, support vector, Gaussian process, and neural network 372 regression models were developed to predict two-stage light gas gun (2SLGG) projectile velocities as a 373 function of 10 independent variables (launch parameters): (1) projectile package mass, (2) primary powder 374 mass, (3) secondary powder mass, (4) pump tube fill (working gas) pressure, (5) tankage (backfill) pressure, 375 (6) piston mass, (7) burst disk score depth, (8) piston fit tightness, (9) number of volume reducers in the 376 powder breech, and (10) gunpowder type. The models were fitted to and validated against performance data 377 from 171 experiments (projectile velocities, 1.5-6.8 km/s) conducted using a powder-driven 12.7 mm bore 378 2SLGG at the Texas A&M University Hypervelocity Impact Lab (TAMU HVIL). Regression model muzzle 379 velocity estimates for all 171 shots were compared to numerical predictions obtained using the physics-based 380 Piston Compression Light Gas Gun Performance (PCLGGP) software package ( i.e., “Charters’ code”). In 381 addition, the prediction absolute error distributions from the regression and PCLGGP models were compared 382 to independent values determined using the cutting-edge, physics-based LGGUN code for a different set of 383 experiments involving multiple 2SLGGs. 384 Except for the basic quadratic model ( i.e., linear regression model with quadratic variables), all of the 385 regression models significantly outperformed Charters’ code over the entire range of experiments. Their root 386 mean square error (RMSE) values (0.234-0.263 km/s) were significantly lower than that for the PCLGGP 387 model (0.597 km/s), and their predicted absolute error distributions were clearly superior to that for Charter’s 388 code. The median absolute error (50thpercentile) for each of the regression models was at least 62% lower 389 than the PCLGGP value. In general, there was not a significant difference between simple linear regression 390 (RMSE = 0.260 km/s) and the other regression models. Gaussian process regression (RMSE = 0.240 km/s), 391 however, includes an estimate of prediction confidence. The neural network model (RMSE = 0.234 km/s) 392 was the most accurate regression technique considered and is particularly well-suited to scale with large data 393 17 sets. 394 Not surprisingly, the absolute errors associated with independent LGGUN predictions were at least 395 40% lower than those for PCLGGP for all percentiles. Interestingly, the regression model results from the 396 TAMU 2SLGG experiments compared favorably to LGGUN results. Excluding the basic quadratic model, 397 the regression models’ root mean square error (RMSE) values (0.234-0.263 km/s) were lower than that for 398 the LGGUN model (0.302 km/s). Similarly, the corresponding 25th-75thpercentile absolute errors for the 399 regression models were significantly lower than corresponding LGGUN values. LGGUN, however, predicted 400 a lower maximum (100thpercentile) absolute error than any regression model. In general, regression models 401 may not perform as well as physics-based computational models in predicting individual experimental results 402 involving extreme values of independent variables. In the future, this limitation can be remedied, in part, 403 using physics-informed neural network or Gaussian process models [41] that incorporate both prior data and 404 general physics-based knowledge over the domain. 405 This study demonstrates that straightforward regression models may provide an attractive alternative 406 to more complex deterministic models for 2SLGG launch velocity predictions. Regression models that 407 are amenable to scaling across different 2SLGG platforms can augment physics-based numerical models 408 (particularly as the volume of available experimental data increases) and have the potential to markedly 409 improve predictive capabilities, identify complex coupling between experimental parameters, and reduce 410 uncertainty. 411