Predicting Pressure Sensitivity to Luminophore Content and Paint Thickness of Pressure-Sensitive Paint Using Artificial Neural Network

Abstract

An artificial neural network (ANN) was constructed and trained for predicting pressure sensitivity using an experimental dataset consisting of luminophore content and paint thickness as chemical and physical inputs. A data augmentation technique was used to increase the number of data points based on the limited experimental observations. The prediction accuracy of the trained ANN was evaluated by using a metric, mean absolute percentage error. The ANN predicted pressure sensitivity to luminophore content and to paint thickness, within confidence intervals based on experimental errors. The present approach of applying ANN and the data augmentation has the potential to predict pressure-sensitive paint (PSP) characterizations that improve the performance of PSP for global surface pressure measurements.

1. Introduction

Pressure-sensitive paint (PSP) using a luminophore has been widely used as pressure sensor in fluid dynamics studies [1,2,3,4]. A key feature of PSP is the sensitivity to pressure variations [5,6,7,8,9,10,11]. The luminescence of the PSP is converted to pressure using the Stern–Volmer equation [11]. Given the relationship between the luminescence of the PSP and corresponding pressure, a PSP with higher pressure sensitivity is desirable.

The higher pressure sensitivity is obtained by adjusting the paint formulation of the PSP. The luminophore concentration of the PSP influences the pressure sensitivity [11,12,13]. Other components such as the thickness of the PSP applied to a surface also affect the pressure sensitivity [14,15,16,17]. The pressure sensitivity of the PSP is determined by an experimental calibration. Typically, one component of a PSP is varied at a time to extract the correlation between the component and pressure sensitivity. Depending on the range of required pressure sensitivity, an extensive range of the components sensitive to pressure sensitivity must be experimentally investigated. Such investigations require significant time since many experimental coupons must be created and calibrated. Ideally, all components must be investigated, including their mutual interactions, in order to map their influence on the pressure sensitivity of the PSP. Given the complex nature of the correlations between the pressure sensitivity and components, it is challenging to incorporate all of the components into a parametric study due to experimental time constraints. It is highly desirable to create a model linking the pressure sensitivity and components related to the PSP. In this work, we investigate the use of artificial neural networks (ANNs) to create the model.

ANNs are often used to approximate nonlinear functions with great success in various fields including chemical engineering [18,19,20], civil engineering [21,22], electrical engineering [23], computer engineering [24], and interdisciplinary engineering [25,26,27,28,29,30,31]. In this work, the ANN is applied to predictions of the pressure sensitivity of a PSP. The ANN architecture is constructed and trained using the PSP datasets measured. A practical challenge is that the prediction accuracy of the ANN depends on the number of data points [32]. Large datasets reduce overfitting and generalize the ANN predictive capabilities [32,33,34,35] However, due to the extensive time required for experiments, the number of data points measured is typically small, consisting of less than a hundred data points. Data augmentation techniques are often used to increase the number of data points [36].

In the present paper, we study the suitability of the use of an ANN in predicting the pressure sensitivity of the PSP to luminophore content and paint thickness. The ANN is used instead of typical approaches such as phenomenological modeling or statistical modeling approaches. The correlation among the three parameters is hard to obtain by the typical approaches because the phenomenon that determines pressure sensitivity is coupled by chemical and physical factors. As components increase, the application of typical approaches becomes increasingly difficult. The main goal of the present study is to enable the replacement of time-consuming and expensive experiments by an ANN trained to predict pressure sensitivity to given components related to PSP. To the best of our knowledge, the present study is the first attempt to apply an ANN to PSP development. The ANN is trained using an experimental dataset to obtain the correlation between pressure sensitivity with luminophore content and paint thickness. The training dataset includes the following variables: paint thickness (mm), luminophore content (mg), and pressure sensitivity (%/kPa). This paper also investigates the general applicability of the ANN trained using an augmented dataset utilizing experimental errors. The data augmentation technique is used to increase the number of data points from experimental observations.

2. Experimental and Augmented Dataset

An experimental dataset was used to train an ANN and evaluate the prediction performance of the trained ANN. The experimental dataset was collected through the characterization of PSP coupons. Luminophore content and paint thickness are mutually independent variables, and they are chemical and physical factors that impact pressure sensitivity, respectively. The luminophore content and paint thickness of PSP coupons were selected to design a PSP that increases the pressure sensitivity as much as possible. The pressure sensitivity was obtained for different luminophore contents and paint thicknesses. The experimental dataset, D_O, consisting of a total of 84 measurements, was split into 90% for the training dataset; D_O,76, composed of 76 data points and 10% for the test dataset; and D_O,8, consisting of 8 data points. Typically, 80% and 20% of data points or 90% and 10% of data points are used for training and testing, respectively. The present study selected the latter ratio to increase the training dataset because the available experimental data is limited to less than 100 points.

Table 1. Range and experimental errors of experimental dataset for pressure-sensitive paint (PSP) coupons.

Component	Minimum	Maximum	Relative Error
Paint Thickness (µm)	4.532	67.635	±9% of paint thickness
Luminophore Content (mg)	3.0 × 10−8	2.0 × 10−5	±7% of luminophore content
Pressure Sensitivity (%/kPa)	0.2873	0.8094	±5% of pressure sensitivity

shows the range and the experimental errors of the experimental dataset for PSP coupons. Details of the collection of experimental datasets are described in Appendix A.

The augmented dataset, D_A, was used to train the ANN. The augmented dataset was produced using the experimental dataset. Figure 1a shows the schematic concept of the data augmentation used in the present study. The data augmentation produced the data points distributed within the confidence intervals based on experimental errors at each observation point. Here, it is assumed that there is no correlation between the error and the variable (i.e., all variables are mutually independent). The randomness of the dataset produced by the data augmentation follows a Gaussian distribution centered at zero. The data augmentation produced data, D_A,n, consisting of 76, 760, 7600, 76,000, and 760,000 entries starting from the training dataset, D_O,76, with 76 data points. The subscript n in D_A,n, indicates the total number of entries obtained through the augmentation procedure. The number of augmented data points was varied to find the one that yields the most accurate ANN model. Figure 1(b) shows an example of the augmented dataset, D_A,7600, with 7600 data points created around each observation point. The data augmentation process is repeated for each experimentally observed data point corresponding to various paint thicknesses and luminophore contents.

Dataset standardization was performed to avoid numerical issues and avoid divergence of the training process during the training caused by different input scales in a dataset with differing physical units and values [37]. Standardization of the dataset was achieved by normalization using the mean and standard deviation.

3. ANN Model Development

3.1. Architecture of ANN Models

Figure 2 shows the schematic description of the architecture of the ANN model used in the present study. The architecture was realized through a fully connected, deep ANN consisting of the input layer with two units, the hidden layers with four units, and the output layer with a single unit. Here, units are also called neurons in terms of a biological brain. Luminophore content and paint thickness are considered to constitute inputs of the ANN models, while the pressure sensitivity constituted an output of the ANN models. The activation function, rectified linear unit (ReLU), was used in the hidden layers [38]. Further details regarding the architecture and studies used to define the final architecture are provided in Appendix B. The selected architecture of the ANN model is summarized in Figure 2.

3.2. Training of ANN Models

ANN models were trained using the augmented datasets, D_A,n. An ANN model was also trained using the experimental training dataset, D_O,76, in order to compare with the ANN models trained by augmented datasets. The ANN models were trained for 40,000 epochs. The ANN model weights were optimized using the Adam optimizer [39]. Because the random selection of the initial ANN model weights in the training influences updated weights, resulting in different predictions, each ANN model was trained 10 times to obtain the median. The learning rate was set at 10⁻³.

Table 2. ANN models. D_O,n represents an experimental dataset and D_A,n represents an augmented dataset. n is the number of data points.

ANN Model	Training Dataset	Test Dataset
MA,76	DA,76	DO,8
MA,760	DA,760	DO,8
MA,7600	DA,7600	DO,8
MA,76000	DA,76000	DO,8
MA,760000	DA,760000	DO,8
MO	DO,76	DO,8

summarizes the ANN models for different training datasets.

4. Evaluation of the Prediction Performance of Trained ANN Models

The performance of the trained ANN models was assessed by the accuracy of the prediction of the ANN models when the test dataset, D_O,8, was used. A metric, mean absolute percentage error (MAPE), was used in quantitatively assessing the accuracy of the predictions [40,41]. The MAPE was calculated using the following equation [40,42]:

$$\mathrm{MAPE}\%=\frac{1}{N}{{\displaystyle \sum }}_{i=1}^N\frac{\left|o_i-p_i\right|}{o_i}\times 100.$$

(1)

Here, o_i is the vector of observed values, p_i is the vector of predicted values, i is the index, and N is the number of data points.

5. Results and Discussion

Figure 3 shows the comparison of the MAPE obtained in testing for different ANN models. All models use the same architecture (i.e., the size of layer and unit is the same for all models). However, each model was trained using a different training dataset, as shown in Table 2, and MAPE was computed in the testing phase. The procedure of obtaining the MAPE (for example, M_A,760, shown in Figure 3) is as follows: (1) the architecture is trained using a training dataset of 760 points; (2) prediction is obtained using the test data of 8 points; (3) MAPE is computed; (4) step (1) to (3) is repeated 10 times. Maximum, minimum, 75% quartile, 25% quartile, and median of MAPE are presented in the boxplot. MAPE is a measure of the mean of the absolute error relative to the test datasets. MAPE scales the overall accuracy of the prediction through the entire domain. The lower the MAPE values, the higher the accuracy of the prediction. M_A,760 showed the lowest median MAPE of 8.9% in ANN models using the augmented datasets. As the number of augmented data points increased, the median MAPE increased where larger augmented data points than M _A,760 were used, as shown in M_A,7600, M_A,76000, and M_A,760000. By comparing all M_A,n models trained by using augmented datasets, it was found that the number of augmented data points influences the MAPE. It was also found that an investigation of the effect of the number of augmented points is required to minimize the MAPE. M_O showed a median MAPE of 9.4%, which was larger than that of the MAPE of 8.9% for M_A,760. By comparing M_O with all M_A,n models, it is found that the use of augmented datasets minimizes the MAPE more than that of M_O. The augmentation achieved greater accuracy. In the present study, the augmented data set consisting of 760 points is considered sufficient to lower the MAPE in the testing.

Figure 4 shows the accuracy of pressure sensitivity predicted by M_O and M_A,760 models when MAPE was the median of the MAPE values for those models. The confidence interval based on the experimental error of pressure sensitivity (i.e., ±5%) was selected as an interval to evaluate how precisely the ANN approximates the pressure sensitivity. By comparing the relative error of M_O with M_A, it was shown that four of eight test points were within 5% of the experimental measurement for both models. This indicates that both models predict the pressure sensitivity within ±5% of experimental measurement, while M_A showed the more accurate and lower MAPE than M_O, as shown in the comparison of the MAPE in the previous section.

6. Conclusions

Reducing time-consuming and expensive experiments is essential to enhancing PSP development. The present study investigated the application of an ANN to predict pressure sensitivity of the PSP to luminophore content and paint thickness with fewer than 10² data points. The ANN model was built and trained on the dataset produced by a data augmentation technique based on the experimental errors. It was concluded that the ANN model can obtain the correlation between pressure sensitivity with luminophore content and paint thickness in PSP development. The ANN model will reduce experimental costs. The ANN models trained by augmented datasets achieved 8.9% of MAPE in predicting pressure sensitivity_. Augmented datasets have the potential to reduce MAPE with a small number of experimental data points.