Predicting Airplane Cabin Temperature Using a Physics-Informed Neural Network Based on a Priori Monotonicity

Abstract

Airplane cabin temperature is a critical environmental factor governing the safety and reliability of airborne equipment. Compared with measuring temperature, predicting temperature is more cost- and time-saving and can cover an extreme flight envelope. Physics-informed neural networks (PINNs) offer a promising prediction solution whose performance hinges on the availability of precise governing differential equations. However, building governing differential equations between flight parameters and cabin temperature is a great challenge, as it is comprehensively influenced by aerodynamic heat, avionic heat, and internal flow. To solve this, a new PINN framework based on “a priori monotonicity” is proposed. Underlying physical trends (monotonicity) from flight data are extracted to construct the loss function as a data-driven constraint, thus eliminating the need for any governing equations. The new PINN is developed to estimate the seven cabin temperatures of an unmanned aerial vehicle. The model was trained on data from four flight sorties and validated on another four independent sorties. Results demonstrate that the proposed PINN achieves a mean absolute error of 1.9 and a root mean square error of 2.6, outperforming a conventional neural network by approximately 35%. The core value of this work is a new PINN framework that bypasses the development of complex governing equations, which enhances its practicality for engineering applications.

1. Introduction

Cabin temperature data of the full-flight regime is fundamental for designing and testing of airborne equipment. In the design stage of airborne equipment, temperature data is used to define thermal performance parameters. In the testing stage of airborne equipment, the temperature data is used to develop safety indicators and formulate reliability metrics. Measurement [1,2], computational fluid dynamics [3,4], and neural networks [5,6] are conventional methods for obtaining cabin temperature. Among these, measurement is the most accurate method, although it is both costly and time-consuming. In addition, temperature data from extreme weather conditions is hard to measure. Computational fluid dynamics can predict the temperature field at any flight regime, although this method requires larger computing power, especially when modeling a dynamic process. The neural network method [7] introduces artificial intelligence to predict temperature, including feedforward neural networks [8], backpropagation neural networks [9], etc. This method needs less computing power compared with the calculated fluid dynamics. However, large amount of measured data is required to train the neural network, and the prediction accuracy highly relies on the data sample size. The methods above are costly and time-consuming.

Physics-informed neural network (PINN) is an emerging prediction method that introduces physical laws into neural networks [10,11]. Compared with the conventional neural network, the distinct difference that PINN introduces is the loss function of physics inconsistency [12,13]. When prediction results do not follow the physical law, the loss function of physics inconsistency exerts a penalty on the neural network. This penalty will be used to update the neural network structure. Li et al. [14] apply PINN to predict the temperature field in laser metal deposition, and the physics inconsistency loss includes the heat conduction, convection, and radiation. Introducing physics inconsistency loss reduces the training time by 1/3 with 2% maximum relative error. Zhang et al. [15] build a general loss function to simulate the flow field. The general loss function bridges the flow field and temperature field. Results show that using temperature data can effectively estimate flow fields. Xia et al. [16] use PINN to solve the frictional contact temperature. Zerrougui et al. [17] evaluated temperature fluctuations in proton exchange membrane electrolysis by the PINN. They use differential governing equations to develop the loss function of physics inconsistency.

However, this approach encounters a fundamental bottleneck in scenarios where precise governing equations are unavailable or excessively complex to derive. This is particularly true for aircraft cabin thermal systems, where temperature distribution results from the coupled effects of aerodynamic heating, internal avionics heat dissipation, complex airflow, and solar radiation. The difficulty in formulating accurate governing equations for such a system significantly hinders the application of conventional PINNs. A promising strategy to overcome this limitation is to shift from equation-based constraints to data-driven physical trends. Recent research has demonstrated that fundamental physical relationships, such as monotonicity, can serve as effective, low-fidelity physical constraints [18]. Monotonicity is a consistently increasing or decreasing tendency between system inputs and outputs. This approach has been successfully implemented in diverse fields, including the prediction of biomass gasification outcomes [19] and NOx emissions from combustion processes [20], where it improved model generalizability and physical consistency without requiring full governing equations.

Building upon this concept, this study introduces a new PINN framework that leverages a priori monotonicity for predicting airplane cabin temperatures. The core contribution is the replacement of unattainable governing equations with empirically derived monotonicity constraints, which are seamlessly integrated into the network loss function. This framework is validated using a dataset from four flight sorties of an unmanned aerial vehicle, encompassing cabin temperatures, altitude, Mach number, and ambient temperature. The monotonicity between variables is acquired from measurement data.

2. Method

A physics-informed neural network is developed in this section. Compared with a conventional neural network, a distinct difference between physics-informed neural network is the physics loss function. The physics loss function ensures the neural network obeys the physics rules. Monotonicity between the variables is selected as physical rules to establish the loss function. Accordingly, experimental measurements are conducted to determine the monotonicity among the different variables and provide a database for network training and testing. The research workflow of this study is illustrated in Figure 1.

2.1. Development of Physics-Informed Neural Network

The new PINN is aimed at predicting the temperatures of seven cabins of an unmanned aerial vehicle (UAV). The shape of the unmanned aerial vehicle and the relative position of each cabin are illustrated in Figure 2. The cabin temperatures are mainly influenced by the flight state of the unmanned aerial vehicle.

Building the conventional PINN relies on the governing equation of the UAV heat transfer process, although this process is complex, as shown in Figure 3. Cabin temperatures are influenced by aerodynamic heating, avionic heat, natural convection in the cabin, and solar radiation [21,22]. Hence, it is difficult to build the governing equation of this process. In addition, the thermal properties of air, skin materials, and avionic materials are necessary to calculate cabin temperature by conventional PINN.

In comparison, using monotonicity to replace the governing equation in PINN eliminates the need for any governing equations, as depicted in Figure 4. Monotonicity is the underlying physical trend between the flight parameters and cabin temperature. This changing tendency also contains physical constraints and can be injected into PINN as physics-informed parts. The advantages of using monotonicity are simplifying physics-informed parts.

Based on monotonicity, a physics-informed neural network is developed to predict cabin temperature, and its structure is shown in Figure 5. It contains two parts: (1) a conventional neural network and (2) a loss function. The loss function imposes a penalty when the prediction result of conventional neural work does not follow the physics rule.

The structure of a conventional neural network includes three inputs: altitude and Mach number of the unmanned aerial vehicle, and ambient temperature. The output of the neural network is cabin temperature. When N training samples ${\{(X^i,Y^i)\}}_{i=1}^N$ are measured, the prediction results of conventional neural networks are shown in Equation (1). $X^i$ and $Y^i$ are the input and output vectors of ith sample.

$$O^i=\beta g(w{X}^i+b)$$

(1)

where Oⁱ is the prediction result of the conventional neural network. β is the bias vector connecting the hidden and output layers. w is the weight matrix that connects the input and hidden layers. g(·) is the activation function. b is the bias vector of the hidden layer.

The prediction error of a conventional network is calculated according to Equation (2).

$$E_{\mathrm{Reconstruction}}={\displaystyle \sum _{i=1}^N{(O^i-Y^i)}^2}$$

(2)

The loss function includes physics loss function E_Physics, reconstruction loss function E_{Reconstruction}, and structural loss function E_Structural, as calculated in Equation (3).

$$L=E_{\mathrm{Reconstruction}}+{\lambda }_{\mathrm{Structural}}{E}_{\mathrm{Structural}}+{\lambda }_{\mathrm{Physics}}{E}_{\mathrm{Physics}}$$

(3)

where ${\lambda }_i$ is the weight coefficient.

Typically, the physics loss function is construed based on the governing equation of the physical process. However, two problem: (1) It relies on a deep understanding of the physics process. If some influencing factors are incidentally omitted, the accuracy would be affected. (2) Input partial differential equations, increasing the modeling difficulty. Hence, this work uses prior monotonicity to represent the governing equation as the prior physics rules. This method can easily build a physics loss function by analyzing measured data.

The physics loss function E_Physics and prior monotonicity are calculated in Equations (4) and (5), respectively. If the monotonicity between the input and output parameters is positive, the monotonicity value $m_j^{(i)}$ is set to 1. Otherwise, the value $m_j^{(i)}$ is set to −1. During neural network training, the physics loss function E_Physics serves as a penalty when the calculated monotonicity $\mathrm{sign}[\partial O_j/\partial X_j]$ differs from the prior monotonicity $m_j^{(i)}$.

$$E_{\mathrm{Physics}}={\displaystyle \sum _{j=1}^n{\displaystyle \sum _{i=1}^{T_j}\frac{1-m_j^{(i)}\times \mathrm{sign}[\partial O_j/\partial X_j]}{2}}}$$

(4)

$$m_j^{(i)}=\left\{\begin{array}{c}1, {\displaystyle \frac{\partial O_j}{\partial X_j}}>0\\ -1, {\displaystyle \frac{\partial O_j}{\partial X_j}}<0\end{array}\right.$$

(5)

where T is the number of prior monotonicity. Sign is $\mathrm{sign}(\theta )=\left\{\begin{array}{c}-1,\theta <0\\ 1,\theta \ge 0\end{array}\right.$, and i is ith sample.

The structural loss function E_Structural is used to avoid overfitting and is calculated using Equation (6). When a small quantity of samples is collected to train a neural network, the neural network usually increases the weight on those data points. As a result, overfitting occurs, and the prediction accuracy of other data points is undesirable. The structural loss function calculates the L2-norm of the neural network weight, and imposes a penalty when some weights are so large. This method aims to eliminate overfitting influences.

$$E_{Structural}={\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^{\tilde{N}}{(w_{jk})}^2}}+{\displaystyle \sum _{k=1}^{\tilde{N}}{({\mathit{\beta }}_k)}^2}$$

(6)

2.2. Experimental Measurement Method

The experimental measurement work is conducted to acquire prior monotonicity and a database for neural network training. The measurement is conducted in Jiangxi, China. The measurement system is illustrated in Figure 6, and the measurement instrument parameters are listed in

Table 1. Measurement instrument parameters.

Instruments	Models	Parameters
Data acquisition instrument	KAM-500, ACRA CONTROL	Maximum sample rate: 500 K/s
Battery	DH15206, DONGHUA Co., Ltd.	14.8 V, 6.6 Ah
Thermocouple	K-type, DONGHUA Co., Ltd.	Measurement range: −200~200 °C; Accuracy: 0.1 K

. On the ground, the system components, including the K-type thermocouples, battery, and computer, were integrated with the data acquisition instrument. Key parameters of the data acquisition instrument, such as sample rate and active channels, were then configured to align with the objectives of the flight test. During flight operations, the system transitioned to autonomous data logging, with the data acquisition instrument sampling temperature data throughout the flight sortie. All K-type thermocouples were mounted at the geometric center of each cabin, away from any corner or structural member to avoid localized thermal boundary effects.

Flight parameters of 8 flight sorties are measured, including the altitude and Mach number of the unmanned aerial vehicle, ambient temperatures, and temperatures of seven cabins. A flight sortie contains the process from take-off to landing.

The measurement procedures are listed as follows.

(1)

On the ground, configure the data acquisition instrument and check the temperatures of the K-type thermocouple.

(2)

Stick the K-type thermocouple on the airplane and fix it with aluminum foil tape.

(3)

Configure the time of flight control computer and data acquisition instrument in consensus.

(4)

Data sampling during a flight sortie.

(5)

Collect flight parameters from the flight control computer, including ambient temperature, flight altitude, and Mach number.

(6)

Build a database consisting of flight parameters and cabin temperature.

To ensure the accuracy and reliability of the flight test data, an assurance procedure was implemented throughout the measurement process:

(1)

Instrument Calibration and Verification: All K-type thermocouples were calibrated prior to installation.

(2)

System Integration Testing: On the ground, the complete measurement system, including thermocouples, battery, and data acquisition instrument, was fully integrated and tested. This pre-flight check verified the proper functionality and interconnection of all components.

(3)

Automated Data Logging: During flight, the data acquisition instrument operated autonomously, disconnected from the computer, to sample temperature data continuously. This approach eliminated potential disruptions from ground equipment and ensured uninterrupted data capture.

(4)

Data Consistency Checking: Post-flight, a consistency check was performed by comparing data from multiple sources and across different flight sorties. The consistent physical patterns observed across eight separate sorties confirm the reliability and repeatability of the measured data.

The Pearson correlation coefficient is selected to acquire prior monotonicity. The Pearson correlation coefficient is in the range of −1 to +1. The value of 0 means that two variables have no relationship and −1 or 1 means that two variables have a strong relationship [23]. In this work, if the Pearson correlation coefficient is negative, the prior monotonicity is set as negative. Otherwise, the prior monotonicity is set as positive. The Pearson correlation coefficient of the variables a and b is calculated in Equation (7) [24].

$$rho(a,b)={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^N(a_i-\overline{a})(b_i-\overline{b})}}{\sqrt{{\displaystyle {\sum }_{i=1}^N{(a_i-\overline{a})}^2}}\sqrt{{\displaystyle {\sum }_{i=1}^N{(b_i-\overline{b})}^2}}}}$$

(7)

2.3. Training of Physics-Informed Neural Network

Eight flight parameters are measured, and four groups are used as a training set. Four groups are divided into a testing set. Before neural work training, the measured data are normalized according to Equation (8). Conducting normalization can eliminate dimension differences and avoid weight imbalance.

$$x_{\mathrm{norm}}={\displaystyle \frac{x-x_{\min }}{x_{\max }-x_{\min }}}$$

(8)

The parameter setting during neural network training is listed in

Table 2. Parameter setting during neural network training.

Parameters	Values
Maximum number of iterations	50,000
Sortie number of training set	4
Sortie number of test set	4
Number of hidden layers	2
Number of nodes per layer	30
Activation function	Tanh
Optimizer	Adam
Learning rate	0.01
Decay rate	0.005
Mini-batch size	256

. Training physics-informed neural networks commonly uses adaptive moment estimation (Adam) [25], batch gradient descent [26], and stochastic gradient descent [27] as optimization algorithms. Given that Adam shows advantages in being insensitive to initial learning rate and fast convergence [25,28], this study applies Adam as an optimization algorithm.

Two indicators are introduced to comprehensively evaluate the accuracy of physics-informed neural networks: root mean square error (RMSE) and mean absolute error (MAE). The RMSE is more sensitive to outliers, while the MAE reflects prediction results more directly. Calculation equations of RMSE and MAE are shown in Equations (9) and (10), respectively.

$$RMSE=\sqrt{{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{(x_i-{\widehat{x}}_i)}^2}}{n}}}$$

(9)

$$MAE={\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n\left|(x_i-{\widehat{x}}_i)\right|}$$

(10)

3. Result and Discussion

3.1. Analysis of Measurement Result

The flight altitudes and Mach numbers of eight sorties are shown in Figure 7 and Figure 8, respectively. Due to the unpublished natures of flight parameters, the actual y-axis values are replaced with relative values. In the measurement range, the flight parameter shows great change between different sorties and flight times. This wide range of data can be used to determine the generalizability of the PINN in this study.

The temperature changing tendency of the measured point (taking flight sortie #1 as an example) is depicted in Figure 9. The general temperature changing tendency of each measured point is that the temperature decreases in the take-off stage. Then, as the flight altitude increases, the ambient temperature declines, leading to the cabin temperature decrease. In contrast, during the descent and landing stage, the ambient temperature rises with the decrease in flight altitude, and the cabin temperature increases. Although the temperature-changing tendency is the same, the degrees of change differ. The center cabin is surrounded by other cabins; thus, the temperature change is small. The cabin 6 is close to the engine air intake, and the air flow speed is high, enhancing heat exchange between indoor and outdoor air. Hence, the cabin 6 shows the sharpest temperature change.

According to the measured data, the neural network should have great prediction accuracy in a wide temperature range, which is the main difficulty in neural network development. The maximum and minimum values of each temperature measurement point are illustrated in

Table 3. The maximum and minimum temperatures of ambient and seven cabins (sorties 1).

	Area	Maximum Temperature/°C	Minimum Temperature/°C
1	Ambient	35	−47.1
2	Center cabin	48.6	32.9
3	1# cabin	38.2	13.3
4	2# cabin	39.8	4.2
5	3# cabin	35.1	−10.1
6	4# cabin	38	13.3
7	5# cabin	36.2	−12
8	6# cabin	36.1	−12.8

, which shows that every cabin endures a great temperature change. For example, the maximum temperature difference in cabin 6 during the entire flying process is 48.9 K.

The prior monotonicity is obtained by calculating the Pearson correlation coefficient, as depicted in Figure 10. Results indicated that the cabin temperature strongly correlates with ambient temperature, since all Pearson correlation coefficients are close to 0.7. In addition, the monotonicity between ambient temperature and cabin temperature is positive. Regarding flight altitude and Mach number, although the Pearson correlation coefficient shows a great difference in different cabins, the coefficient is below 0. Therefore, the monotonicity between flight altitude and cabin temperatures, and between Mach number and cabin temperatures is negative.

3.2. Prediction Result Analysis of Physics-Informed Neural Network

Comprehensive evaluation of the proposed PINN framework was conducted through multi-sortie validation (sorties 5–8) and detailed error analysis. This section presents the prediction performance across diverse flight conditions and provides statistical characterization of model accuracy. In addition, the influence of the iteration count is analyzed.

The influences of iteration count are shown in Figure 11. In the beginning, the loss function values dramatically decrease with the increase in iteration count. Then, the loss function values approach 3 × 10⁻⁶ and show no change when the iteration count is over 46,000. With the iteration count further increased, the loss function values remain stable. This changing tendency indicates that the training has converged around 46,000 iterations.

The generalization capability of the PINN framework was assessed using four independent test sorties (sorties 5–8), representing diverse flight conditions and operational scenarios. The overall prediction results demonstrate satisfactory accuracy across all test scenarios. As shown in

Table 4. The sortie 5–8 MAE of prediction results.

Sortie	Average MAE	Std Dev	Min MAE	Max MAE
5	1.60	0.68	0.99	3.00
6	1.72	0.65	0.78	2.57
7	1.86	0.68	0.82	2.80
8	2.79	0.93	1.27	3.74

, MAE values range from 0.78 to 3.74 across the four test sorties. The maximum MAE of 3.74, observed in sortie 8, remains within acceptable limits for engineering applications. This comprehensive evaluation confirms the model capability to generalize across unseen flight conditions.

Figure 12 presents the detailed MAE and RMSE comparison across different cabins and sorties. The results reveal a clear performance pattern: cabin 3 consistently achieves the best accuracy with minimum MAE of 0.78, while cabins 5 and 7 exhibit higher errors at 2.57 and 1.76, respectively, across all test sorties.

This performance distribution can be directly attributed to the physical locations and thermal environments of the cabins. Cabin 3 is situated at the nose of the aircraft. The thermal transfer is primarily influenced by flight speed, which serves as an input parameter to the PINN network. In contrast, cabins 5 and 6 are located near engine compartments and air intake systems, where complex heat transfer phenomena occur. These peripheral cabins are subject to strong influences from propulsion system heat dissipation, variable air flow patterns, and rapid temperature fluctuations during flight maneuvers.

The RMSE values follow similar patterns but are generally higher than corresponding MAE values, indicating the presence of some larger prediction errors that are appropriately penalized by the square term in RMSE calculation. This pattern is particularly noticeable in cabins 5 and 6, where complex thermal environments occasionally produce larger prediction errors.

Figure 13 displays the MAE distribution through boxplot analysis, providing insights into prediction consistency across different cabins and sorties. The boxplots reveal that while central tendency values remain within acceptable ranges, the variation patterns differ significantly between cabins.

Cabin 3 shows the most compact distribution with the smallest interquartile range, indicating highly stable prediction performance. Conversely, cabin 6 exhibits wider distributions with higher variability, reflecting the dynamic and complex thermal environments near propulsion systems.

The variation across sorties also follows understandable patterns. Sorties 5–7 show similar distribution characteristics, while sortie 8 exhibits elevated errors across most cabins. This can be attributed to the more challenging flight conditions encountered during sortie 8, including extended high-altitude operations and more complex flight maneuvers that create more dynamic thermal environments.

The temporal analysis of prediction errors confirms consistent performance across all flight phases, including take-off, climb, cruise, descent, and landing. No systematic error accumulation or phase-dependent performance degradation was observed, demonstrating the model robustness under various operating conditions.

The physical validation of error patterns provides strong evidence for the effectiveness of the monotonicity-based approach. The spatial distribution of prediction errors correlates strongly with known thermal characteristics of different cabin locations. This alignment between mathematical performance and physical reality confirms that the prior monotonicity constraints successfully embed essential physical relationships into the data-driven approach.

3.3. Comparison with Convention Neural Network

This section presents a comparative evaluation between the proposed physics-informed neural network and a conventional neural network to quantify the performance improvement achieved through the integration of physical constraints. The analysis focuses on the effectiveness of a priori monotonicity in enhancing prediction accuracy and reliability.

The comparative analysis employs identical network architectures and training procedures for both modeling approaches, with the fundamental distinction residing in the loss function formulation, as depicted in Figure 14. The conventional neural network utilizes a purely data-driven approach, optimizing solely based on prediction accuracy. In contrast, the PINN incorporates additional physics-based constraints through a priori monotonicity relationships derived from flight data analysis. Both models were evaluated using the same four test sorties (sorties 5–8) across seven cabin locations. This controlled experimental design ensures that performance differences directly reflect the impact of physical constraint integration.

The evaluation demonstrates substantial performance advantages for the physics-informed approach. As illustrated in Figure 15, the PINN achieves average MAE and RMSE values of 1.99 and 2.66, respectively. These values represent significant improvements over the conventional neural network, which yields corresponding values of 3.09 and 4.14. The performance differential corresponds to a 35% reduction in MAE and a 35% reduction in RMSE, clearly demonstrating the benefit of incorporating physical knowledge through monotonicity constraints.

The superior performance of the PINN framework can be attributed to the effective integration of domain knowledge through monotonicity constraints. These constraints guide the learning process toward physically meaningful solutions, particularly in regions with limited training data or during extrapolation to unseen flight conditions.

The conventional neural network, lacking such physical guidance, demonstrates higher susceptibility to learning spurious correlations and producing predictions that may violate fundamental physical principles. This limitation becomes particularly evident in complex thermal environments where multiple physical phenomena interact nonlinearly.

Comparative results provide compelling evidence for the value of incorporating physical knowledge into data-driven modeling approaches for complex thermal systems. The a priori monotonicity constraints successfully bridge the gap between purely data-driven methods and physics-based modeling, delivering enhanced accuracy while maintaining physical consistency.

4. Conclusions

This study has developed and validated a new physics-informed neural network (PINN) framework that leverages a priori monotonicity for predicting the cabin temperatures of an unmanned aerial vehicle (UAV). The core innovation of this work is the replacement of complex governing differential equations with data-driven monotonicity constraints, which are derived from flight data and integrated into the neural network loss function. The key findings and contributions are summarized as follows:

(1)

Reliable Prediction Accuracy: The proposed model was trained on data from four flight sorties and achieved satisfactory prediction performance when tested on another four separate sorties (sorties 5–8), across seven cabins. The mean absolute error (MAE) across all test cases is 1.6–2.79, demonstrating the model robustness across diverse flight conditions.

(2)

Performance Enhancement over Conventional Methods: A comparative analysis with a conventional neural network revealed the substantial benefit of incorporating physical constraints. The PINN framework achieved a 35% reduction in both MAE and RMSE, underscoring the value of the priori monotonicity in guiding the learning process toward physically consistent and accurate solutions.

(3)

Practical and Simplified Modeling Method: This approach eliminates the need to derive and solve intricate governing equations for cabin heat transfer, thereby reducing model development complexity and making PINNs more accessible for practical engineering applications in complex systems.

Despite these promising results, the current study has limitations. The framework performance is contingent on the quality and representativeness of the flight data used for monotonicity extraction. Furthermore, the assumption of global monotonicity may not fully capture all nonlinearities in extremely dynamic thermal environments, and the model is static, not explicitly accounting for transient heat accumulation effects.