Pin-based integrated performance prediction model construction method and system based on pinn

Abstract

The application relates to the technical field of aero-engine control, and specifically discloses a propeller-engine integrated performance prediction model construction method and system based on a physical information neural network (PINN), which comprises the following steps: a PINN network taking a multilayer perceptron as a core is built, and an input layer, a hidden layer and an output layer structure are configured; a two-stage hierarchical training strategy of pre-training and fine-tuning is adopted to optimize parameters of the PINN network; the parameter optimization comprises the following steps: in the pre-training stage, mechanism simulation data is combined with physical constraints to learn global dynamic characteristics of a propeller-engine system; and in the fine-tuning stage, sea level static bench test data is introduced to correct local parameters of the pre-training network. Results show that the error level of the application scheme meets the engineering application requirements and is better than all the comparison models, the application scheme has good cross-condition generalization capability, can significantly reduce non-physical deviation in a sparse sample area, and has onboard real-time application conditions.

Description

Technical Field

[0001] This invention relates to the field of aero-engine control technology, and particularly to control systems and their electronic digital data processing. Specifically, it relates to a method and system for constructing a propeller-engine integrated performance prediction model based on a physical information neural network (PINN). Background Technology

[0002] The propulsion system, consisting of an aero-engine and a propeller (referred to as "propeller-engine" for convenience in this invention), is the main power source for medium-altitude long-endurance unmanned aerial vehicles (UAVs). Its core performance lies in achieving efficient matching between engine output power and propeller absorbed power under all flight conditions (including takeoff, climb, and cruise). However, there is a strong coupling relationship between the characteristics of the engine and the propeller, and both are highly sensitive to parameters such as flight altitude, speed, and power requirements, resulting in complex nonlinear characteristics of the entire propulsion system.

[0003] The integrated propulsion system of an aircraft piston engine and propeller (hereinafter referred to as the propeller-engine integrated propulsion system) is the core power plant of low-speed general aviation aircraft, unmanned aerial vehicles, and special-purpose aircraft. There is a strong coupling relationship between the engine thermodynamic cycle and the propeller aerodynamic load within the system: when the input conditions such as flight altitude, flight speed, throttle opening, and blade angle change, the engine output power, speed, torque, and propeller propulsion performance do not change proportionally and synchronously; the input variables interfere with and constrain each other, and a change in one parameter will simultaneously induce a linked change in the thermodynamic cycle conditions and propeller aerodynamic loads. There is no simple linear correspondence between input and output, exhibiting significant nonlinear characteristics. To address this coupled nonlinearity, the system must possess online calculation and real-time control capabilities. Therefore, constructing a propeller-engine integrated propulsion system model that combines physical constraints, predictive accuracy, and computational efficiency has become an important foundation for propulsion system design optimization, condition monitoring, and intelligent control.

[0004] Currently, propeller-engine system modeling is mainly divided into two categories: mechanistic modeling and data-driven modeling. Aero-engine piston engines typically use Mean Value Engine Models (MVEMs) to describe the dynamic response of key state variables such as inlet manifold pressure, crankshaft speed, and output torque; the propeller portion is modeled based on blade element momentum theory or experimental characteristic diagrams. This approach offers clear physical interpretability and possesses a certain degree of operational condition extrapolation capability. However, mechanistic models usually rely on extensive experimental data for parameter calibration, and their accuracy is highly dependent on the model structure and parameter identification quality. Prediction biases exist under complex operating conditions, especially in areas where the model is not fully calibrated. Under high load, low speed, and the dynamic effects of fuel, there are significant deviations between model predictions and actual conditions. Furthermore, this type of model suffers from problems such as large parameter identification workload, high calibration costs, and limited real-time computation efficiency, making it difficult to simultaneously meet the requirements of accuracy and real-time performance.

[0005] With the development of machine learning technology, data-driven methods have been widely applied in the field of aerospace propulsion system modeling. Machine learning methods, represented by Back Propagation Neural Network (BPNN), Support Vector Machine (SVM), and Deep Neural Network (DNN), have achieved significant results in engine performance prediction, health status assessment, fault diagnosis, and remaining service life prediction. Compared with mechanistic models, these methods can effectively capture complex nonlinear features, but their modeling effectiveness is highly dependent on the size and coverage of the training samples. For propeller-engine integrated propulsion systems, due to the difficulty in obtaining complete flight envelope test data, the test samples are difficult to cover parameters such as altitude, speed, and blade angle, making traditional data-driven models prone to overfitting and reducing the reliability of predictions under untested conditions.

[0006] Physics-Informed Neural Networks (PINNs) embed physical priors such as conservation relations, governing equations, and boundary conditions into the network training process, achieving synergistic optimization of data constraints and physical constraints. This provides a new research approach for modeling complex dynamic systems under small sample conditions. In recent years, PINNs have demonstrated good modeling performance in fields such as fluid mechanics, heat transfer analysis, and combustion modeling. However, existing research mainly focuses on solving partial differential equations and modeling single physical systems. For complex dynamic systems such as propeller-engine integrated propulsion systems, which have multi-source coupling characteristics of thermodynamics, aerodynamics, and propulsion dynamics, systematic research on physical constraint construction methods, training strategies, and small sample adaptation mechanisms is still lacking. In particular, under conditions of insufficient experimental data, how to utilize mechanistic models to provide prior knowledge and achieve local model correction through experimental data is a pressing problem that needs to be solved in current propeller-engine system modeling. Summary of the Invention

[0007] To address the aforementioned problems, this invention proposes a two-stage (pre-training-fine-tuning) PINN modeling method that integrates mechanism and data-driven information. In the pre-training stage, a mechanism model of the integrated propeller-engine propulsion system is constructed based on the average value model of an aero-piston engine and the aerodynamic model of a propeller, generating simulation samples covering the entire operating range. This pre-trains the PINN network, enabling it to acquire global dynamic feature representation capabilities. In the fine-tuning stage, hierarchical parameter fine-tuning is performed using static bench test data at sea level, achieving the synergistic fusion of mechanism knowledge and measured information, thereby improving the prediction accuracy for key operating conditions. During the training process, propeller-engine matching relationships and physical constraints of the propulsion system are further introduced to ensure that the prediction results meet actual operating laws, thus providing a new technical approach for high-precision modeling and intelligent control of integrated propeller-engine propulsion systems. To achieve the above objectives, this invention specifically adopts the following technical solutions:

[0008] This invention provides a method for constructing a propeller-engine integrated performance prediction model based on PINN, comprising the following steps: constructing a PINN network with a multilayer perceptron as the core and configuring an input layer, a hidden layer, and an output layer structure; the input layer imports input variables consisting of flight state parameters and control command parameters; the hidden layer includes at least two layers, wherein the first hidden layer is configured with a nonlinear activation function to extract the nonlinear coupling features of the input layer parameters, and the second hidden layer is used for feature compression; the output layer outputs the core performance response parameters of the propeller-engine system.

[0009] A two-stage hierarchical training strategy of pre-training and fine-tuning is adopted to optimize the parameters of the PINN network. The parameter optimization includes: in the pre-training stage, learning the global dynamic characteristics of the propeller-generator system based on mechanism simulation data and physical constraints; in the fine-tuning stage, local parameter correction of the pre-trained network is performed by introducing sea-level static test bench data, and a hierarchical freezing strategy is adopted to freeze all network parameters of the first hidden layer, while only the second hidden layer is open; all input and output variables are normalized and preprocessed to map the variables to a fixed numerical range and output the performance prediction value through linear transformation.

[0010] During the construction process, the PINN network is constrained by a composite loss function, which is composed of a weighted combination of a data loss term and a physical loss term. The data loss term is used to measure the fitting error between the predicted value and the sample true value, and the physical loss term is a multi-dimensional physical constraint residual term constructed based on the aerodynamics and dynamic mechanism of the propeller-generator system.

[0011] Furthermore, the input variables include flight altitude, airspeed, blade angle, and throttle opening; the output variables include engine speed, effective engine output torque, propeller thrust, and fuel consumption rate.

[0012] Furthermore, the forward-forward nonlinear mapping relationship of the hidden layer is specifically as follows: In the formula For network input vectors, and Let be the weight matrix and bias vector of the k-th layer network, respectively. is the ReLU activation function, k is the number of hidden layers in the network and k is an integer greater than or equal to 2 (preferably 2 layers, then k takes the values ​​1 and 2).

[0013] Furthermore, the physical loss term in the composite loss function includes one or more of the following: torque balance constraint, thrust consistency constraint, and speed limiting constraint.

[0014] Furthermore, the total loss function expression of the composite loss function is as follows: In the formula and These are the weighting coefficients for data loss and physical loss, respectively. The mean squared error between the predicted value and the sample label. This refers to the physical residual constraint term; within the physical residual constraint term, the torque balance constraint satisfies the steady-state transmission balance mechanism, and the engine output torque and the propeller aerodynamic torque exhibit a nonlinear coupling relationship. The propeller speed Satisfy conversion formula , For engine output torque, For propeller aerodynamic torque, Engine speed, This represents the propeller power coefficient obtained from the graph under the combined operating conditions of the approach ratio J and blade angle β. air density, The system reduction ratio, propeller diameter To maintain fixed design parameters, the thrust consistency constraint requires the thrust component output by the network to conform to the propeller aerodynamic characteristics. ,in The thrust coefficient is represented by the combination of approach ratio J and blade angle β obtained from the graph; the speed limiting constraint Calculate using the following formula: Where mean represents the arithmetic mean of all samples in the current training batch, when the model predicts the engine speed. When ≤5800, ReLU outputs 0, and this term has no loss; when When the value is greater than 5800, the difference is output and the squared value is used to amplify the over-limit penalty.

[0015] Furthermore, the pre-training phase employs a two-dimensional polynomial fitting method to analytically fit the thrust coefficient and power coefficient, and uses a log-hyperbolic cosine function to measure the physical residual; the pre-training phase uses the Adam optimizer for training, with an initial learning rate set to 1×10⁻⁶. -5 -1×10 -4 The maximum number of training rounds is 1000-2000. The quality control parameters for pre-training include: gradient clipping threshold of 1.0, physical loss weight λ1=0 for the first 200 rounds of pre-training, only optimizing data loss, while after 200 rounds, physical loss weight λ1 increases linearly with the number of iteration rounds to 0.1 and data loss weight λ2 remains constant at 1.0.

[0016] Furthermore, the fine-tuning stage employs a layered freezing strategy to freeze all network parameters in the first hidden layer, while only opening the weights and bias parameters of the second hidden layer and the output layer for gradient updates. The reconstruction method for the loss function and aerodynamic coefficients in the fine-tuning stage is as follows: based on the characteristic that the test data only contains the true values ​​of speed and torque, the data loss only calculates the mean square error between the predicted values ​​of speed and torque and the label values ​​and increases the weight of the data loss, cancels the speed limit constraint, replaces the physical residual measurement method from the logarithmic hyperbolic cosine function with the standard mean square error, and configures differentiated fixed weights for torque and thrust constraints. For the static condition of J=0 on the test bench, the two-dimensional polynomial fitting method is replaced with piecewise cubic Hermite conformal interpolation, and the thrust coefficient and power coefficient are calculated based on the discrete nodes of the test data.

[0017] Specifically, the parameters for the fine-tuning phase are configured as follows: data loss weight is set to 10.0, torque constraint weight is set to 3.0, thrust constraint weight is set to 1.0, the Adam optimizer is used, and the training learning rate is adjusted to 1×10. -4 The maximum number of training rounds is maintained at 2000.

[0018] The present invention also provides a propeller-generator integrated performance prediction system constructed by the above-mentioned PINN-based propeller-generator integrated performance prediction model construction method.

[0019] The present invention also provides a method for predicting the performance of a propeller-engine integrated system using the above-mentioned propeller-engine integrated performance prediction system, including inputting real-time flight status parameters and control command parameters, and outputting the predicted results of engine speed, output torque, and propeller thrust to complete the dynamic prediction of the performance of the propeller-engine integrated system.

[0020] The present invention has the following beneficial effects:

[0021] This invention addresses the limitations of limited measured samples in propeller-generator integrated propulsion systems and the insufficient accuracy of traditional mechanistic models in predicting local operating conditions. It proposes a two-stage PINN modeling method that integrates mechanistic and data-driven approaches. The performance of this method was systematically verified through standard interpolation, cross-blade angle interpolation, and ablation experiments. The main beneficial effects are as follows:

[0022] (1) By embedding the propeller torque balance relationship and thrust consistency constraint into the network training process, the two-stage PINN exhibits high prediction accuracy under sparse sample conditions. Under the standard interpolation condition where each blade angle contains only 6 sampling points, its maximum relative error of torque is 7.29%, which is better than the two-stage MLP and multinomial regression model and meets the engineering accuracy requirements;

[0023] (2) Under the cross-condition interpolation condition where the entire set of blade angle samples is missing, the maximum relative error of the torque of the two-stage PINN is still controlled within 9.35%, showing good cross-condition generalization ability. The results show that introducing physical constraints helps to provide trend guidance when training samples are insufficient, thereby improving the model's prediction stability for unobserved conditions;

[0024] (3) Ablation experiments show that mechanistic pre-training and physical constraints have complementary effects in the model. During the pre-training stage, the full envelope data generated by the mechanistic model improves the quality of parameter initialization, thereby enhancing the ability to express global trends; physical constraints further strengthen the model's adherence to the system's operating rules and reduce error accumulation in sparse regions. The synergistic effect of the two is an important factor in improving the model's accuracy and generalization ability;

[0025] (4) The average inference time of the constructed PINN agent model under a single working condition is about 2.13ms, which is more than two orders of magnitude faster than the original mechanism model, while ensuring prediction accuracy and meeting the requirements of airborne real-time applications. Attached Figure Description

[0026] Figure 1 This is a schematic diagram of the basic process of the two-stage PINN model construction scheme of the present invention.

[0027] Figure 2 This is a graph showing the trend of the predicted torque MAPE and MaxErr under different sampling densities of this invention as a function of the K value. Hollow icons represent torque MAPE, and solid icons represent torque MaxErr. The dashed line extending horizontally from position 10 on the vertical axis indicates that the maximum error in torque prediction should be limited to within 10%.

[0028] Figure 3 This is a graph showing the fluctuation trajectory of the absolute percentage error (APE) of torque at a single point on the test set for each model under the standard interpolation task of this invention. The square icons represent data from scheme 1, the circular icons represent data from scheme 2, the triangular icons represent data from scheme 3, and the inverted triangular icons represent data from scheme 4. The dashed line extending horizontally from position 10 on the vertical axis indicates that the maximum error in torque prediction should be limited to within 10%.

[0029] Figure 4 This is a graph showing the fluctuation trajectory of the absolute percentage error (APE) of torque at a single point on the test set for each model under the cross-blade angle interpolation task of this invention. The square icons represent data from scheme 1, the circular icons represent data from scheme 2, the triangular icons represent data from scheme 3, and the inverted triangular icons represent data from scheme 4. The dashed line extending horizontally from the vertical axis position 10 indicates that the maximum error of torque prediction should be limited to within 10%. Detailed Implementation

[0030] The following detailed description provides further details through specific embodiments. However, it should be noted that the embodiments described below are merely for illustrating the content of the invention and do not represent that the invention is limited to the described embodiments. Therefore, non-essential improvements and adjustments made to the implementation schemes by those skilled in the art based on the above-described invention still fall within the protection scope of the invention, and the scope of protection of the appended claims shall prevail.

[0031] This invention is applicable to various aircraft or flying vehicles, especially those requiring long-duration flight, such as unmanned aerial vehicles (UAVs) for extended medium-altitude flight. The aircraft engine is the "heart" that provides propulsion and support for the aircraft / flying vehicle. Major types include aircraft piston engines and aircraft turbine engines. This invention is particularly applicable to aircraft piston engines with propellers, where the propeller is the core component that converts engine power from rotational force into thrust / pull.

[0032] In this invention, "operating condition" broadly refers to the combination of all external environment and internal system states experienced by an aircraft / aircraft during its mission cycle, including atmospheric conditions, flight status, power / energy requirements, onboard system loads, and other multi-dimensional parameters. In this invention, operating condition specifically refers to the combination of the aircraft / aircraft's takeoff, climb, and cruise flight phases and their flight status parameters (altitude H, airspeed v, and power P).

[0033] The linear interpolation method described in this invention is a simple numerical interpolation method. Its core is to construct a straight line using two known data points to estimate the unknown value at any position between the two points. The basic principle is to assume that two known points (x0, y0) and (x1, y1) have a linear relationship (i.e., a constant rate of change). When the unknown point x lies between x0 and x1, its corresponding y value is derived through the equation of the straight line; essentially, it proportionally distributes the difference. This linear interpolation calculation method is well-known to those skilled in the art.

[0034] The polynomial interpolation described in this invention is a more accurate numerical interpolation method. Ordinary Lagrange interpolation constructs a polynomial using only the function values ​​at discrete points; Hermite interpolation additionally requires strict matching of both function values ​​and derivative values ​​(slopes) at the interpolation nodes; while the cubic Hermite interpolation used in this invention employs a cubic polynomial within a single interval. Each subinterval requires four constraints: the function values ​​and derivative values ​​at the left and right endpoints of the interval, which precisely determine the four coefficients of the cubic polynomial. The function construction methods used in these interpolation methods are all common knowledge in the field.

[0035] The PINN obtained in this invention stands for Physics-Informed Neural Networks, similar to the one proposed by Raissi et al. in 2017 (see Raissi M, et al. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 2019, 378: 686-707). It belongs to a modeling framework that integrates deep learning and physical mechanisms. Unlike purely data-driven neural networks that rely solely on sample fitting, PINN transforms the physical laws governing the system into differentiable soft constraints embedded in the network loss function. This achieves dual-constraint training—data fitting plus physical conservation—overcoming the shortcomings of traditional neural networks, such as lack of physical boundaries in extrapolation and poor generalization under small sample conditions.

[0036] This invention introduces the training model into the field of aero-engines. The mechanistic model used is a crucial foundation for constructing a Physical Information Neural Network (PINN), and its modeling accuracy directly affects the effectiveness of physical constraints and the quality of prior knowledge during the pre-training phase. For integrated propeller-engine propulsion systems, the mechanistic model in this invention primarily undertakes two functions: first, it generates simulation data covering the entire operating range of the propeller and engine, providing sample support for network pre-training and enabling the network to establish a preliminary understanding of the system's global operating characteristics; second, it provides propeller-engine matching relationships and aerodynamic constraints for the physical constraint terms in the PINN loss function, thereby ensuring that the model's prediction results conform to the basic operating laws of the propulsion system.

[0037] Example 1: Method for constructing a propeller-generator integrated performance prediction model based on PINN

[0038] This invention first establishes an average value model for an aero-piston engine and an aerodynamic characteristic model for a propeller. Then, based on the power matching and torque balance relationship, it constructs a mechanism model for an integrated propeller-engine propulsion system. Based on the mechanism model, bench test data is introduced to validate the model, analyzing its trend-following capability across the entire operating range and the error distribution characteristics under typical operating conditions. The main error sources under non-design point conditions are identified, providing a basis for the subsequent two-stage PINN model's mechanism priors and physical constraint design.

[0039] 1. Construction of the engine average value model

[0040] Considering the requirements for computational efficiency and numerical stability in real-time simulation of propeller-engine integrated propulsion systems, this invention employs an average value engine model to establish a mechanistic model of a turbocharged aero-piston engine. This model focuses on key components such as the intake manifold and crankshaft system, ignoring high-frequency dynamic characteristics such as pressure fluctuations and combustion details within the cylinder working cycle through state averaging, and emphasizing the average energy transfer and dynamic response characteristics of the engine at the system scale. The overall model mainly consists of an intake manifold gas dynamics sub-model and a crankshaft rotor dynamics sub-model, combined with a multi-dimensional MAP plot to achieve parameter characterization of highly nonlinear processes.

[0041] In this modeling process, the intake manifold is simplified as a uniformly parameterized concentrated cavity, whose internal state changes satisfy the ideal gas law of state and the mass conservation relationship. The dynamic change of manifold pressure is mainly determined by the difference between the mass flow rate of air entering the throttle and the mass flow rate of air drawn into the cylinder. The throttle air flow rate is calculated based on the isentropic one-dimensional nozzle flow relationship, and a nonlinear effective flow area mapping function related to the throttle opening is introduced to characterize the nonlinear characteristics of the throttle flow capacity changing with the opening. The air flow rate entering the cylinder... The Speed-Density method is used to solve this problem, and its expression is as follows:

[0042]

[0043] in: Air flow rate entering the cylinder (unit: kg / s); Engine displacement (unit: m) 3 ), The value represents the engine speed (in r / min); the 120 in the denominator corresponds to the conversion relationship of a four-stroke engine completing one intake cycle every two crankshaft rotations. This is an empirical correction coefficient for intake dynamic effects, used to compensate for the additional dynamic pressure effect under high-speed flow conditions (dimensionless). The volumetric efficiency (dimensionless) is determined by both engine speed and manifold pressure, and is obtained through interpolation using a two-dimensional MAP plot; p man T is the manifold pressure (in Pa); R is the air intake constant (in J / kg⋅K); T man The temperature of the gas in the manifold is in K.

[0044] In addition, a simplified turbocharger subsystem was constructed in the model to solve for the changes in compressor outlet boost pressure and intake air temperature. This subsystem performs dynamic calculations based on the steady-state characteristic diagrams of the compressor and turbine, combined with the relationship between environmental pressure conditions and exhaust energy balance, thereby realizing the description of the coupled influence of the turbocharger on the engine intake state.

[0045] In power output modeling, this invention ignores the dynamics of fuel evaporation film and in-cylinder combustion details, and uses a quasi-steady-state air-fuel ratio mapping relationship to calculate fuel consumption rate. The engine's effective output torque is calculated based on the energy conservation relationship and a comprehensive drag loss model. To improve the model's fitting ability to steady-state test data, this invention introduces empirical correction terms for specific speed and load regions to improve the model's output accuracy under local operating conditions. Based on the crankshaft system rotational dynamics equations, the engine's transient speed response under the combined action of effective torque and external load torque is calculated, and its expression is:

[0046]

[0047] in: Equivalent moment of inertia of engine and transmission system (unit: kg / m) 2 ), The crankshaft angular velocity (unit: rad / s). represent The angular acceleration obtained by differentiating with respect to time t (in seconds) (in rad / s²) 2 ), The effective output torque of the engine (unit: Nm). The torque applied by the propeller to the engine output shaft (in Nm) is the load torque. This equation describes the dynamic evolution of the engine speed under propeller-engine coupling conditions and is an important foundation for subsequent propeller-engine integrated dynamic modeling.

[0048] 2. Construction of the propeller model

[0049] As the core propulsion component of an integrated propeller-engine propulsion system, the propeller's aerodynamic characteristics not only determine the aircraft's thrust output level but also directly affect the engine load characteristics and system power distribution. To balance the computational accuracy and real-time requirements of full-condition envelope simulation, this invention employs a variable-pitch propeller steady-state model based on dimensionless aerodynamic coefficients, calculating propeller thrust and aerodynamic drag torque through multidimensional interpolation of propeller aerodynamic characteristic diagrams. The key dimensionless parameter describing the propeller's operating state is the advance ratio J, defined as:

[0050]

[0051] in: Airspeed (in m / s); denoted by , where is the propeller rotational speed (r / s); and D is the propeller diameter (m). The advance ratio reflects the relative relationship between the axial incoming velocity of the aircraft and the characteristic rotational velocity of the propeller, and is an important dimensionless parameter characterizing the aerodynamic working state of the propeller.

[0052] During the model calculation, based on the combined operating conditions of the current advance ratio J and blade angle β, the corresponding thrust coefficient and power coefficient are obtained from the propeller performance spectrum through interpolation. This performance spectrum, constructed from wind tunnel test data, can accurately reflect the aerodynamic performance variations under different operating conditions.

[0053] Based on the aerodynamic similarity theory, the actual thrust generated by the propeller can be expressed as:

[0054]

[0055] Where: F is the propeller thrust (unit: N); The thrust coefficient (dimensionless) is obtained from the graph under the combined flight conditions of approach ratio J and blade angle β as described above; ρ is the atmospheric density (unit: kg / m³) at the current flight altitude. 3 As can be seen from the above formula, propeller thrust is not only affected by the aerodynamic coefficient, but also closely related to parameters such as air density, rotational speed, and propeller diameter.

[0056] Similarly, the power absorbed by the propeller can be expressed as:

[0057]

[0058] in: Power absorbed by the propeller (in W); This represents the power coefficient (dimensionless) obtained from the graph under the combined operating conditions of the approach ratio J and blade angle β, as described above. This power essentially originates from the mechanical energy provided by the engine output shaft and is ultimately converted into propulsive work and aerodynamic loss work.

[0059] Based on the relationship between the power and torque of rotating machinery, the aerodynamic drag torque exerted by the propeller on the engine output shaft can be further obtained:

[0060]

[0061] in: This is the propeller aerodynamic drag torque (unit: Nm), which constitutes the main external load on the engine output shaft.

[0062] From the perspective of propeller-generator coupling, when flight altitude, airspeed, or blade angle changes, the advance ratio J and the corresponding thrust coefficient... This will change accordingly, thus affecting the propeller thrust F and drag torque. The propeller's aerodynamic model changes over time. If an imbalance occurs between the torque absorbed by the propeller and the torque output by the engine, the system speed will dynamically adjust until a new equilibrium is reached. Therefore, the propeller aerodynamic model not only performs thrust calculations but is also an important component in establishing the engine-propeller coupled dynamic relationship. Its output aerodynamic drag torque serves as the external load input to the engine dynamic model, jointly determining the dynamic operating characteristics of the propeller-engine integrated propulsion system.

[0063] 3. Coupled Model Construction and Error Analysis

[0064] After constructing the engine average value model and the propeller aerodynamic model, the two were integrated through the speed-torque coupling relationship to establish a mechanism simulation model of the propeller-engine integrated propulsion system. Under standard sea-level static conditions (H=0 m, v=0 m / s), the predicted results of the mechanism model were compared with the bench test data of the propeller-engine integrated propulsion system. The results show that the model can well describe the changes in engine speed and output torque with throttle opening and blade angle, and accurately reflect the power matching characteristics of the propeller-engine system. However, the model error increases in the high-load region, and the overall predicted output torque value is too high, mainly manifested in the predicted output torque value being higher than the measured value.

[0065] To analyze the sources of model error, the internal state calculation process was traced. The results show that the prediction deviation mainly stems from the extrapolation error of the empirical correlation in the average value model under non-calibration conditions. Key parameters such as charging efficiency and overall thermal efficiency are calibrated using experimental MAP charts, which can effectively characterize the overall engine performance; however, intermediate state quantities such as heat transfer loss, flow loss, and efficiency correction still rely on empirical model calculations, and their parameters are identified from limited experimental data. Under high load conditions, the in-cylinder thermodynamic process, intake flow characteristics, and turbocharger system operating state deviate from the calibration range, leading to a mismatch between empirical parameters and actual operating conditions, and an increase in the calculation error of intermediate state quantities. This error gradually amplifies after state iteration propagation, ultimately resulting in an overestimation of the output torque prediction.

[0066] Although the mechanistic model exhibits some predictive bias under local operating conditions, it is based on engine thermodynamics and propeller aerodynamics, and can reasonably characterize the coupling relationships between variables such as flight altitude, flight speed, throttle opening, and blade angle, accurately reflecting the development trend of the propulsion system's operating characteristics under all operating conditions. Given its good physical consistency and operating condition coverage, this invention uses it as a priori data generation model. A sample set is constructed through full-condition simulation to compensate for insufficient experimental data, providing a physical prior and sample foundation for subsequent data-driven model training.

[0067] To address the prediction bias of the mechanistic model under local operating conditions, this invention further introduces a Physical Information Neural Network (PINN) for modeling. This network fully integrates the global trend information provided by the mechanistic model with the local features of bench test data, achieving a fusion of physical consistency and experimental constraints. A two-stage training strategy based on mechanistic simulation and measured data can simultaneously improve the model's prediction accuracy and generalization ability under all operating conditions.

[0068] Example 2: Method for Constructing a Performance Model of Paddle-Generator Integration Based on PINN

[0069] To address the prediction bias of the mechanistic model under local operating conditions, this invention introduces a Physical Information Neural Network (PINN) to construct a proxy model for the propeller-generator integrated performance. Unlike traditional data-driven neural networks, PINN embeds the system's physical laws into the network optimization process, enhancing physical consistency and generalization ability across all operating conditions while maintaining data fitting accuracy. The fundamental aerodynamic principles of the propeller are embedded in the network loss function as a regularization term, ensuring that the model follows physical laws when fitting observed data.

[0070] In terms of training strategy, a two-stage hybrid framework of "pre-training-fine-tuning" is adopted. The first stage uses simulation data generated by the mechanistic model to pre-train the network, enabling it to learn the distribution characteristics of samples under all operating conditions, inherit prior knowledge of thermodynamic and aerodynamic laws, and acquire global trend representation capabilities, providing reasonable initialization for small-sample fine-tuning. The second stage introduces static bench test data at sea level to perform hierarchical fine-tuning of the pre-trained network, updating only some network parameters to retain global feature representations. This corrects local biases in the mechanistic model, effectively improving prediction accuracy under key operating conditions.

[0071] This hybrid driving strategy integrates pre-training and fine-tuning mechanisms, and the constructed PINN proxy model has the ability to describe trends under all working conditions, while ensuring the local prediction accuracy under actual bench test conditions.

[0072] 1. PINN Network Architecture Design

[0073] The physical information neural network constructed in this invention uses a multilayer perceptron (MLP) as its core mapping architecture for propeller-generator integrated performance prediction. The network computation process includes three stages: forward data inference, physical constraint calculation, and error backpropagation.

[0074] In the input layer design, the network input vector Defined as a combination of flight status parameters and control command parameters (including flight altitude H, airspeed v, propeller angle β, and throttle opening ε):

[0075]

[0076] In the hidden layer design, the propeller-generator integrated system exhibits strong nonlinear coupling characteristics between aerodynamics and thermodynamics. Therefore, the MLP is configured with K=2 hidden layers, with the number of neurons in each layer being 64 and 32 respectively. The first layer is used to extract the nonlinear interaction features between input variables; the second layer is used for feature compression to filter out redundant information. The forward-step nonlinear mapping relationship between the layers can be expressed as:

[0077]

[0078] in: The network input vector; and Let represent the weight matrix and bias vector of the k-th layer, respectively; the activation function σ is the ReLU function, which is used to improve nonlinearity and gradient propagation efficiency.

[0079] In the output layer design, the network output vector represents the key performance response parameters of the propeller-generator system. Due to significant differences in the dimensions and numerical ranges of the input and output variables, to avoid convergence difficulties caused by gradient scale imbalance during training, this invention employs a Min-Max normalization method for all variables, mapping them to the [0,1] interval, thereby enhancing training stability and accelerating parameter convergence. For the normalized continuous physical quantities, the output layer uses a linear transformation to generate predicted values.

[0080] The specific process of model operation and training in this invention is as follows:

[0081] During forward propagation, the normalized input is mapped by an MLP to generate performance predictions. The predicted engine speed, output torque, and propeller thrust, among other state variables, are input into the physical constraint module. These parameters, combined with environmental parameters and the propeller aerodynamic model, are used to calculate physical residuals, forming physical consistency constraints to ensure the network output meets the propeller aerodynamic equilibrium conditions. The data error between the network output and the sample labels, along with the physical residuals, constitutes the total loss function. During backpropagation, the gradient of the loss function with respect to the network parameters is calculated using automatic differentiation, and the weight matrix and bias parameters are iteratively updated using an optimization algorithm to achieve coordinated optimization of data fitting and physical consistency.

[0082] The training employs a two-stage strategy. The first stage utilizes simulated data for pre-training with enhanced physical information, enabling the network to acquire overall trend representation capabilities across the entire operating envelope. The second stage introduces bench test data, correcting local prediction biases through layered freezing and partial fine-tuning. This strategy, combining trend inheritance and local accuracy calibration mechanisms, constructs a PINN surrogate model that possesses both full-condition trend description capabilities and ensures prediction accuracy under key operating conditions.

[0083] 2. Physical Constraints and Loss Function Construction

[0084] This paper constructs a composite loss function system for PINN, clarifying how propeller aerodynamics is embedded in the network training process. Unlike traditional data-driven MLPs that rely solely on sample error optimization, PINN introduces a physical regularization term into the loss function. The total loss function for network training is defined as a weighted combination of data loss and physical loss: where and are the weight coefficients of the corresponding terms. measures the mean squared error between the network's predicted value and the label value. is then constructed based on system dynamics, specifically including the following three physical constraints.

[0085] 1) Torque balance constraint

[0086] Under steady-state operating conditions, the engine output torque should satisfy the transmission balance relationship with the propeller aerodynamic torque. The engine output torque is calculated using the following formula:

[0087]

[0088] in This represents the engine's output torque. Represents the aerodynamic torque of the propeller; The propeller speed is calculated from the network-predicted rotational speed. Where is the engine speed; D=1.92m is the propeller diameter; G=2.43 is the reduction ratio. This constraint mandates that the torque output by the network must satisfy a nonlinear quadratic coupling relationship with the speed.

[0089] 2) Thrust Consistency Constraints

[0090] As a constraint in conjunction with torque balance, thrust consistency requires that the thrust component output by the network conforms to the propeller aerodynamic characteristics, as shown in the following formula:

[0091]

[0092] 3) Speed ​​limiting constraint

[0093] To prevent the network from predicting rotational speeds exceeding physical limits under sparse and unexplored conditions, soft boundary constraints are introduced. Calculate using the following formula:

[0094]

[0095] Where: mean represents the arithmetic mean of all samples in the current training batch; when the model predicts the engine speed. When ≤5800, ReLU outputs 0, and this term has no loss; when When the value exceeds 5800, the output difference is squared and the over-limit penalty is amplified; thus, the gradient constraint is strengthened in backpropagation, guiding the network output back to the physically feasible range. When the predicted speed exceeds the engine's upper limit (e.g., 5800 r / min), the over-limit penalty term increases rapidly, thereby strengthening the gradient constraint in backpropagation and guiding the network output back to the physically feasible range.

[0096] The aforementioned loss functions are configured differently during the pre-training and fine-tuning phases. Physical loss can be measured using mean squared error or a log-hyperbolic cosine function, with the weight range of each loss term adjusted according to the phase objectives. Key parameters such as aerodynamic coefficients are obtained through experimental calibration or mechanistic model identification, used to adapt to the data characteristics and calibration requirements of different phases.

[0097] 3. Pre-training phase

[0098] The pre-training phase aims to use simulation data generated by the mechanistic model to establish initial weights for the network that reflect the global dynamic trends of the propeller-generator system. During this phase, no network parameters are frozen, and all parameters of the hidden and output layers participate in gradient updates.

[0099] In the composite loss function, the data loss consists of the mean square error between the predicted value and the simulation label. When designing the physical loss, the differentiability of the aerodynamic coefficients must be considered: since discrete propeller aerodynamic data tables cannot directly provide gradients, this stage uses a two-dimensional polynomial to analytically fit the thrust coefficient and power coefficient to ensure the continuous differentiability of the physical calculation process.

[0100] Because the simulation data exhibits systematic biases in local operating conditions, the physical residuals in this stage are measured using the log-hyperbolic cosine function (Log-Cosh). This function approximates a quadratic loss when the residuals are small, and the gradient tends to be constant when the residuals are large, thus reducing the impact of outliers on training and making it more suitable for handling large-scale simulation data with local biases. The physical residual terms for torque constraints and thrust constraints are defined as follows:

[0101]

[0102]

[0103] in: The physical residual term representing the torque constraint, The superscript ^ represents the neural network's prediction of the engine's effective output torque. The effective output torque of the engine, calculated using physical formulas (superscript) phys (Represents the actual physical value) The physical residuals representing the thrust constraint, The superscript ^ represents the propeller thrust predicted by the neural network model. The propeller thrust calculated using physical formulas (superscript) phys (Represents the actual physical value). In the operational parameters, `mean` represents the arithmetic mean of all samples in the current training batch, ensuring the consistency of the loss gradient's dimensions across different batch sizes; `logcosh` represents the log-hyperbolic cosine function used to calculate the loss, approximating the value when the loss variable `x` is very small. Consistent with the mean squared error (MSE), when the loss variable x is very large, it approximates... It degenerates into Mean Absolute Loss (MAE), which does not excessively amplify the penalty for outlier errors like MSE. The denominator of torque is 100 and the denominator of thrust is 1000. Essentially, this scales the residuals of physical quantities to a similar numerical range. Torque is typically in the tens or hundreds of Nm range, divided by 100, and thrust is typically in the thousands of Nm range, divided by 1000. This makes the numerical ranges of the two types of losses close, so that during training, one loss will dominate the gradient while the other is ignored, which is convenient for multi-objective joint training.

[0104] Incorporating the soft constraint on rotational speed, the total physical loss during the pre-training phase consists of a weighted combination of torque residual, thrust residual, and rotational speed constraint. The model is trained using the Adam optimizer with a learning rate of 5×10⁻⁶. -5 The maximum number of training epochs is 1500, and the gradient clipping threshold is 1.0. To mitigate the impact of excessively strong physical regularization on fitting in the early stages of training, a gradual physical constraint strategy is introduced during the pre-training phase: the data loss weight is kept constant at 1.0; the physical loss weight coefficient is set to 0 in the first 200 epochs, and then linearly increases to 0.1 with each iteration, thus achieving a smooth introduction of physical constraints.

[0105] After the pre-training phase, the network has captured the nonlinear mapping relationship of the mechanistic model within the full envelope, providing initial conditions for introducing bench test data in the fine-tuning phase.

[0106] 4. Fine-tuning stage

[0107] After pre-training, the network has captured the nonlinear mapping relationship of the propeller-generator integrated system across the entire system envelope. However, systematic deviations still exist between the mechanism simulation and the real system, mainly due to unmodeled factors such as transmission chain mechanical losses and bearing friction. To reduce the impact of these deviations, static bench test data at sea level is introduced during the fine-tuning stage to locally correct the model. Considering the limited size and high reliability of the measured samples, a layered freezing strategy is adopted during the fine-tuning stage. Only the parameters of the second hidden layer and the output layer are updated, while the parameters of the first layer are frozen to retain the global feature representation learned during the pre-training stage, thereby suppressing the risk of overfitting under small sample conditions.

[0108] In terms of loss function design, the fine-tuning stage involves reconstruction based on the characteristics of the bench data. Since the measured data only contains speed and torque information, the data loss consists only of the mean square error of the above two types of variables, and the weighting coefficient is increased to 10.0 to enhance the fitting ability to high-confidence data.

[0109] The loss function in the fine-tuning phase was readjusted based on the characteristics of the bench test data. Since the bench test samples are sparse but their confidence level is higher than the simulation data, the data loss consists only of the mean square error of the two types of variables mentioned above, and the weighting coefficient for the data loss is increased to 10.0. Furthermore, the bench test data only provides the true values ​​of speed and torque; therefore, the data loss in this phase only calculates the mean square error between these two network predictions and the label values. The adjustment of the physical loss term is mainly reflected in the following two points:

[0110] 1) Change in the method of extracting aerodynamic coefficients

[0111] In the fine-tuning phase, since the bench tests are all conducted under static conditions with a pitch ratio of J=0, the propeller aerodynamic characteristics can be simplified to a single-variable function that varies only with the blade angle. The two-dimensional polynomial fitting used in the pre-training phase may introduce truncation errors at the sample boundaries. Therefore, this phase replaces it with discrete thrust and power coefficients based on experimental data, calculating the aerodynamic coefficients at the corresponding blade angles using piecewise cubic Hermite interpolating polynomial (PCHIP). This method strictly passes through all known nodes and maintains the inherent monotonicity of the data, effectively reducing endpoint fitting errors.

[0112] 2) Adjustment of the physical error measurement function.

[0113] During the pre-training phase, to reduce the impact of local biases in the simulation data on physical constraints, the physical residuals were measured using the Log-Cosh function. In the fine-tuning phase, due to the high reliability and sparse sample size of the bench data, the residual metric was changed to Standard Mean Square Error (MSE) to enhance the network's sensitivity to local errors; simultaneously, since the bench operating speed was far below the engine's structural upper limit, the speed limit constraint was removed. The adjusted torque and thrust physical residuals are as follows:

[0114]

[0115]

[0116] in and These represent the physical residuals of the torque and inference constraints during the fine-tuning phase, respectively, with the parameters defined as before. The physical loss weights for the fine-tuning phase are set to: torque 3.0, thrust 1.0. The network training still uses the Adam optimizer, with the learning rate adjusted to 1×10⁻⁶. -4 To enhance local convergence accuracy, the maximum number of training rounds is 2000.

[0117] By employing a two-stage training strategy of pre-training and fine-tuning, the model achieves local accuracy correction for actual test conditions on the test bench while retaining the global mapping capability of the full envelope of the mechanistic model.

[0118] Example 3 Result Verification

[0119] To verify the PINN model and the two-stage "pre-training-fine-tuning" strategy proposed in Examples 1 and 2, this invention further conducts system performance evaluation under standard interpolation and cross-blade angle interpolation tasks. The verification includes: methods for acquiring and preprocessing multi-source data; analysis of the impact of sampling density and blade angle selection on the prediction accuracy of PINN; and a comparison of the proposed method with pure data-driven models and multinomial regression. Furthermore, ablation experiments are used to quantify the independent contributions of physical constraints and pre-training.

[0120] The dataset of this invention consists of two parts: mechanistic model simulation data and bench test data. The former is used in the pre-training stage to learn the global mapping relationship of the system, and the latter is used in the fine-tuning stage to achieve local accuracy correction.

[0121] The pre-training data was generated from the propeller-generator integrated mechanism simulation model constructed in Example 1, covering multi-dimensional parameter combinations such as flight altitude, airspeed, blade angle, and throttle opening. After data cleaning, outliers and samples exceeding physical constraints were removed, ultimately yielding approximately 3000 sets of steady-state valid data. The input feature vector is... These represent flight altitude, airspeed, propeller blade angle, and throttle opening, respectively; the output labels are... , representing the engine speed, effective engine output torque, propeller thrust, and fuel consumption rate predicted by the network model, respectively. All data were normalized to the [0,1] interval, and the normalized coefficient matrix was retained for later use.

[0122] The fine-tuning data comes from a static bench test at sea level of a certain type of aero-engine and propeller coupling system, forming 8 sets of blade angle operating condition sample sets, totaling 72 steady-state operating points. Full throttle scans were performed at each blade angle. Six blade angle sets contained 10 throttle opening points, one small blade angle set contained only 3 operating points due to throttle opening limitations, and another set contained 9 operating points. The input is... The output variables are engine speed and torque. The bench data is scaled for consistency before being input into the network using normalized parameters saved during the pre-training stage to ensure consistency in feature distribution between the simulation data and the measured data.

[0123] To evaluate the model's generalization and interpolation capabilities under different sparse sampling conditions, two partitioning schemes were designed for the bench data:

[0124] Option 1: Standard interpolation scheme. All data for the two extreme blade angles (10.80° and 30.92°) are included in the training set as physical boundary anchor points; K opening points are uniformly sampled from each of the six middle blade angle groups and added to the training set, with the remaining opening points used as the blind test validation set. Experiments use K values ​​of 4, 5, 6, and 7 to examine the impact of sampling density on prediction accuracy.

[0125] Option 2: Cross-blade angle interpolation. Data from the extreme blade angles at both ends, along with four sets of data selected from the six intermediate blade angle sets, are included in the training set. The remaining two sets of blade angle data serve as the test set. Three representative combinations are selected using three logics: uniform interval, concentrated testing, and edge interpolation. This paper focuses on showcasing the results of the uniform interval combination (training set: 10.80°, 30.92°, 14.25°, 20.57°, 23.32°, 28.78°; test set: 17.03°, 26.13°).

[0126] To quantitatively evaluate the predictive performance of the model under sparse sampling conditions, this invention selects the mean absolute percentage error (MAPE) and maximum absolute percentage error (MaxErr) as core indicators, reflecting the model's global average accuracy and local maximum error, respectively. The coefficient of determination R0 is also used. 2 The evaluation model is used to assess how well it fits the observed trends. The definitions of each indicator are as follows:

[0127]

[0128]

[0129]

[0130] Where: N is the number of samples in the test set; This represents the network model prediction value for the i-th sample; To measure the true value on the corresponding test bench, For all N samples The arithmetic mean of the values.

[0131] MAPE provides an average measure of prediction accuracy across the entire test set. R² reflects the model's ability to capture data trends; a value closer to 1 indicates a better fit to the overall trend. MaxErr focuses on the local maximum error under the worst operating conditions. Related studies have shown that the error between pure simulation models and experiments can reach 10%, which is insufficient to meet the accuracy requirements of modern engineering simulations. In piston engine simulations, if the torque prediction error exceeds 10%, it severely weakens the reliability of thrust calculations and structural load analysis. Therefore, this limit is used as a core criterion for verifying the engineering applicability of the model.

[0132] The performance of the PINN model was validated from two dimensions: first, the impact of throttle opening sampling density on prediction accuracy under the same blade angle was analyzed; second, the model's cross-blade angle interpolation capability was evaluated when some blade angle data were missing.

[0133] To verify the impact of different throttle opening sampling densities on the interpolation prediction accuracy of the PINN model during the fine-tuning stage, this invention employs a controlled variable experimental design. The extreme blade angles at both ends of the fixed envelope (10.80°, 30.92°) and the six intermediate blade angles are used as the training domain. For the six intermediate blade angles, K operating points (K∈{4,5,6,7}) are uniformly sampled along the throttle opening for fine-tuning training. The remaining unsampled opening points serve as the test set. The prediction error indices under different sampling densities are shown in Table 1. To directly demonstrate the impact of sampling density on model performance, Figure 1 The trend of torque MAPE and MaxErr with the value of K is shown, where the dashed line represents the maximum torque prediction error of 10%.

[0134] Table 1. Prediction error index under different sampling densities

[0135]

[0136] The predictive performance of the PINN model exhibits a non-monotonic variation with sampling density. When K=4, the training data is too sparse, making it difficult for the model to acquire sufficient local information to correct biases from the pre-training stage during fine-tuning. While physical constraints can maintain overall predictive consistency, their ability to suppress local errors is limited, resulting in a torque MaxErr of 16.35%. As K increases to 5 and 6, the model performance significantly improves. With K=5, the torque MaxErr decreases to 7.44%, and with K=6, it further increases to 7.29%, while the torque R... 2 The MAPE reached 0.9815, stabilizing below 3%. When K=7, the sampling density continued to increase, strengthening the dominant role of the data loss term in the optimization process and increasing the risk of model overfitting. Inherent deviations exist between bench test data and the mechanistic model; local overfitting weakens the regularization effect of physical constraints, causing a decrease in the prediction accuracy of conditions not involved in training.

[0137] It is evident that when the sampling density increases to K=6, the model achieves a better balance between data fitting and physical regularization. At this point, the torque MaxErr decreases to 7.29%, and the torque R² is 0.9815, meeting the engineering accuracy requirements. Therefore, K=6 is chosen for subsequent sampling densities.

[0138] To verify the interpolation capability of the PINN model across blade angles under the condition of limited training blade angles, this invention designs a sparse blade angle sampling experiment. In the experiment, the training blade angles retain their full opening data to test the interpolation stability of the model when some blade angles are missing. According to the data partitioning scheme, three typical combinations were selected for verification: "uniform interval (combination one)," "centralized (combination two)," and "edge interpolation (combination three)." The prediction performance indicators of the model under each combination are summarized in Table 2.

[0139] Table 2. Predictive performance indicators of the model under different combinations of blade angle division schemes.

[0140]

[0141] In terms of average accuracy, the MAPE for both speed and torque was controlled within 5% for all three combinations, indicating that PINN can effectively characterize the overall trend of change across blade angles under physical constraints. However, significant differences were observed among the different combinations in the MaxErr index, which reflects local extreme deviations. For combination one (uniformly spaced), the MaxErr for torque was only 9.35% between adjacent training blade angles (17.03° and 26.13°), meeting the 10% engineering limit. In contrast, the MaxErr for torque increased to 25.25% for combination two and reached 22.16% for combination three, both far exceeding engineering safety requirements. The differences in MaxErr for speed among the three combinations were small (5.59%-6.45%), but the MaxErr for torque showed significant divergence.

[0142] The MAPE (Motor Maximum Effect) for both speed and torque was controlled within 5% for all three combinations, indicating that PINN can effectively characterize the overall trend of change across blade angles under physical constraints. However, significant differences were observed among the different combinations in the MaxErr index, which reflects local extreme deviations. For combination one (uniformly spaced) with test blade angles (17.03° and 26.13°), the torque MaxErr was only 9.35% between adjacent training blade angles, meeting the 10% engineering limit. In contrast, the torque MaxErr for combination two rose to 25.25%, and for combination three it reached 22.16%, both far exceeding engineering safety requirements. The differences in speed MaxErr among the three combinations were small (5.59%-6.45%), but the torque MaxErr showed significant divergence. Combination 2 tested blade angles of 20.57° and 23.32°, which are located in the blank interval between 14.25° and 28.78°. There are no training samples in this segment, and the network relies solely on long-distance interpolation of far-end samples. Combination 3 tested blade angles of 14.25° and 28.78°, which are close to the limits of the large and small angles (10.80° and 30.92°). The nonlinear characteristics of the torque under extreme conditions differ greatly from those of the intermediate angles, making it difficult to provide effective local gradient constraints.

[0143] To further verify the effectiveness of the proposed two-stage strategy of "pre-training-fine-tuning" and physical constraint embedding, this invention compares the performance of the constructed PINN model with various benchmark methods under unified experimental conditions, and evaluates the independent contributions of physical constraints and pre-training through ablation experiments.

[0144] This invention selects four modeling schemes for performance evaluation:

[0145] Scheme 1: Two-stage PINN model (the method of this invention). A [64,32] multilayer perceptron structure is adopted, and a two-stage strategy of "simulation data pre-training and bench data fine-tuning" is implemented, with physical constraints of propeller torque balance and thrust consistency embedded throughout the process.

[0146] Option 2: PINN model without pre-training (ablation term). Physical constraints are retained, but the pre-training stage on simulation data is skipped, and the model is trained directly on bench test data to verify the necessity of prior knowledge of the full envelope for establishing the initial physical mapping trend.

[0147] Option 3: Two-stage MLP model (ablation term). The network architecture and two-stage training process are consistent with the two-stage PINN, but the loss function does not contain physical constraints and is used to evaluate the effect of physical priors on suppressing local maximum errors.

[0148] Option 4: Multinomial Regression Model (Traditional Benchmark). Using blade angle and throttle opening as independent variables, third-order response surfaces are fitted to speed and torque respectively. This is a commonly used empirical modeling method in engineering, used to quantify the performance improvement of neural network methods in handling complex nonlinear coupling relationships.

[0149] To comprehensively evaluate the performance of the four models in different application scenarios, the following two test tasks are designed:

[0150] Task A: Standard interpolation task. Based on Scheme 1, all data from the extreme blade angles at both ends and K=6 opening points from each of the six intermediate blade angle groups are uniformly extracted as the training set, while the remaining opening points of the intermediate blade angles are used as the test set. According to experimental results, K=6 is the optimal balance point between data fitting and physical regularization within this operating range. All neural network models are trained with the same initial learning rate, iteration count, batch size, and gradient clipping threshold to ensure that comparative evaluation is only affected by differences in model architecture and training strategy.

[0151] Task B: Cross-blade angle interpolation task. The "uniform interval" combination (Combination 1) of Scheme 2 is adopted. The training set includes all data for the two extreme blade angles and the four middle blade angle sets (14.25°, 20.57°, 23.32°, 28.78°), while the test set includes all data for the remaining two blade angle sets (17.03°, 26.13°). In the experiment, all neural network models had uniform training parameters during the fine-tuning phase: an initial learning rate of 1×10⁻⁶. -4 The training run consisted of 2000 epochs, a gradient pruning threshold of 1.0, and the optimizer was Adam. The two-stage PINN and the two-stage MLP used the same layer-by-layer freezing strategy (freezing the hidden layer fc1), while the unpre-trained PINN, lacking pre-trained weights, used all parameters in the training.

[0152] Table 3 shows the prediction errors of the four models on the test set under the standard interpolation task. Scheme 1 (two-stage PINN) exhibits the best overall prediction performance under the standard interpolation task. Compared to the unpre-trained PINN (Scheme 2), Scheme 1 reduces the MAPE of speed and torque by 2.63% and 5.17%, respectively, and the MaxErr by 7.26% and 8.46%, respectively. This indicates that the pre-training stage can effectively extract global operational features from the mechanistic model and improve the model's generalization ability under small sample conditions. Compared to the two-stage MLP (Scheme 3), Scheme 1 reduces the maximum error of speed from 7.77% to 4.66% and the maximum error of torque from 13.05% to 7.29%, representing reductions of 3.11% and 5.76%, respectively. This demonstrates that physical constraints can effectively suppress the non-physical interpretation of data-driven models. Compared to polynomial regression (Scheme 4), Scheme 1 reduced torque MAPE and MaxErr by 1.78% and 6.62%, respectively, and increased R² from 0.9701 to 0.9815, further validating PINN's advantage in fitting complex nonlinear engine-propeller coupling characteristics. Overall, Scheme 1 achieved the lowest error and highest fitting accuracy in both speed and torque prediction.

[0153] Table 3. Prediction errors of each model under the standard interpolation task.

[0154]

[0155] To further examine the stability of the error distribution in the test sequence, Figure 2 The fluctuation trajectory of the absolute percentage error (APE) of torque at a single point on the test set for each model is shown. The vertical axis APE% in the figure represents the absolute percentage error of a single test sample, i.e.:

[0156]

[0157] MAPE is the arithmetic mean of APE for all test samples. Both the two-stage PINN and the two-stage MLP adopt a two-stage training process of "pre-training-fine-tuning" and the training data comes from the same source. The only difference is whether physical constraints are embedded throughout the process.

[0158] Experimental results show that, under the standard interpolation task, Scheme 1 (Case 1, two-stage PINN) has the lowest prediction error and the smallest error fluctuation range. The absolute percentage error (APE) of all test samples is controlled within the 10% engineering error limit, while other schemes (Case 2–Case 4) have some samples exceeding this limit. Scheme 2 (Case 2, PINN without pre-training), although retaining physical constraints, lacks simulation pre-training priors, resulting in APE exceeding 10% for some test samples, with the maximum error reaching 15.75%, and a significant decrease in error distribution stability. Compared with Scheme 1, this shows that introducing simulation data pre-training can effectively reduce local peak errors and improve the overall prediction stability of the model. Scheme 3 (Case 3, two-stage MLP) uses the same training process and network structure as Scheme 1, only removing the physical constraint term. Its error for some test samples approaches or exceeds the 10% engineering limit, and the maximum local error further increases, indicating that physical constraints can make the prediction results conform to the coupling law and reduce local abnormal errors. Scheme 4 (Case 4, multinomial regression) has low predictive performance and a large number of high-error samples, indicating that traditional regression methods are difficult to effectively characterize the complex nonlinear coupling characteristics between the engine and the propeller.

[0159] It is evident that the two-stage PINN improves the prediction accuracy and stability of the model under conditions of scarce experimental samples by combining the prior information of the system obtained from simulation pre-training with the physical constraints of the propulsion system. The ablation experiment further demonstrates that the two-stage "pre-training-fine-tuning" strategy and the physical constraint fusion method can balance the model's generalization ability and local prediction accuracy.

[0160] Table 4 shows the prediction errors of each model in the cross-blade angle interpolation task. Among them, Scheme 1 (two-stage PINN) still shows the best overall prediction ability, especially in torque prediction. Compared with the untrained PINN (Scheme 2), the torque MAPE of Scheme 1 decreased from 13.65% to 3.58%, and the MaxErr decreased from 70.87% to 9.35%, a reduction of 10.07% and 61.52% respectively, indicating that the pre-training process can effectively reduce local peak error. Compared with the two-stage MLP (Scheme 3), the torque MAPE of Scheme 1 decreased by 0.97%, and the MaxErr decreased by 0.07%, indicating that physical constraints can effectively suppress non-physical oscillations in cross-blade angle prediction and improve model stability. Compared with multinomial regression (Scheme 4), the torque MAPE of Scheme 1 decreased by 1.08%, the MaxErr decreased by 2.11%, and the R² increased from 0.9731 to 0.9759.

[0161] Table 4. Prediction errors of each model under the cross-blade angle interpolation task.

[0162]

[0163] Figure 3 The variation trajectory of torque APE under test conditions was further demonstrated. Scheme 1 (Case 1, two-stage PINN) maintained a generally low prediction error, with the APE of the vast majority of test samples below the 10% engineering error limit, and the error fluctuation was significantly smaller than other methods. In contrast, Scheme 2 (Case 2, PINN without pre-training) showed large error peaks at some test points, indicating that the model's extrapolation ability for unknown blade angle conditions was insufficient without a pre-training stage. While Schemes 3 (Case 3, two-stage MLP) and 4 (Case 4, multinomial regression) had relatively small overall errors, some samples still exceeded 10%. Scheme 1 learns the continuous variation law of propeller aerodynamic characteristics under different blade angles through the pre-training stage and uses physical constraints to limit the prediction results to satisfy the engine-propeller coupling relationship, enabling the model to maintain stable prediction ability in the blade angle region not involved in training.

[0164] Based on the combined results of the two types of tasks, the two-stage PINN of this invention achieves optimal accuracy and stability in both standard interpolation and cross-blade angle interpolation tasks through a collaborative mechanism of "simulation pre-training to construct global trends and physical constraints to guide fine-tuning," and MaxErr always meets the 10% engineering constraint. Although the two-stage MLP has a certain generalization ability, its error fluctuation is more obvious; the PINN without pre-training shows significant performance degradation in both types of tasks, verifying the necessity of the collaborative design of pre-training and physical constraint fine-tuning.

[0165] Based on the evaluation of prediction accuracy, computational timeliness is another key indicator for determining whether a model can be embedded in an airborne system for real-time state estimation. This invention quantitatively analyzes the single-condition inference time of each model under unified testing conditions. The test platform consisted of an Intel Core i5-11400H@2.70 GHz, 16 GB of memory, and an NVIDIA GeForce RTX 3050 Laptop GPU. The software environment was Windows 11 and MATLAB R2024b / Simulink. All inference was executed in single-threaded CPU mode, and the average value of multiple forward propagations was taken to reduce the impact of random fluctuations. Specific results are shown in Table 5 below.

[0166] Table 5 Comparison of computational efficiency for single operating conditions under different model schemes

[0167]

[0168] As shown in Table 5, the original Simulink mechanistic model, due to its involvement of differential equation step integration and system coupling iteration, has a single-point CPU time of 1366.40 ms, which cannot meet the millisecond-level real-time requirements of the flight controller. In contrast, multinomial regression only performs algebraic evaluation, with the shortest time (0.04 ms), but its prediction accuracy is insufficient under extreme conditions (torque MaxErr reaches 13.91%). Both the two-stage MLP and the two-stage PINN online inference logic use matrix multiplication and addition forward propagation, with times of 1.92 ms and 2.13 ms respectively. The difference between the two is mainly due to fluctuations in the underlying computing power, which is negligible. Compared with the mechanistic model, the computational efficiency is improved by more than two orders of magnitude, and it fully possesses real-time state estimation capabilities.

[0169] In summary, the two-stage PINN of this invention maintains millisecond-level online inference speed while embedding physical constraints in offline training, effectively improving local accuracy under sparse sample conditions, thus achieving a balance between high accuracy and high timeliness.

[0170] The above descriptions are merely embodiments of the present invention. Commonly known structures and characteristics of the solutions are not described in detail here. Those skilled in the art are aware of all common technical knowledge in the field prior to the application date or priority date, are aware of all existing technologies in that field, and have the ability to apply conventional experimental methods prior to that date. Those skilled in the art can, under the guidance of this application, improve and implement this solution in combination with their own capabilities. Some typical known structures or methods should not be obstacles for those skilled in the art to implement this application. It should be noted that those skilled in the art can make several modifications and improvements without departing from the structure of the present invention. These should also be considered within the scope of protection of the present invention, and will not affect the effectiveness of the implementation of the present invention or the practicality of the patent. The scope of protection claimed in this application should be determined by the content of its claims, and the specific embodiments described in the specification can be used to interpret the content of the claims.

Claims

1. A method for constructing a propeller-generator integrated performance prediction model based on PINN, characterized in that, Includes the following steps: A PINN network with a multilayer perceptron as its core is constructed and configured with an input layer, a hidden layer, and an output layer. The input layer imports input variables consisting of flight state parameters and control command parameters. The hidden layer includes at least two layers, wherein the first hidden layer is configured with a nonlinear activation function to extract the nonlinear coupling features of the input layer parameters, and the second hidden layer is used for feature compression. The output layer outputs the core performance response parameters of the propeller-generator system. A two-stage hierarchical training strategy of pre-training and fine-tuning is adopted to optimize the parameters of the PINN network. The parameter optimization includes: in the pre-training stage, the global dynamic characteristics of the propeller-generator system are learned by relying on mechanism simulation data and physical constraints. The thrust coefficient and power coefficient are analytically fitted by a two-dimensional polynomial fitting method, and the physical residual is measured by a log-hyperbolic cosine function. In the fine-tuning stage, the local parameters of the pre-trained network are corrected by introducing static test bench data at sea level. A hierarchical freezing strategy is adopted to freeze all network parameters of the first hidden layer, and only the second hidden layer is open. All input and output variables are normalized and preprocessed to map the variables to a fixed numerical range and output the performance prediction value through linear transformation. The fine-tuning stage reconstructs the loss function and aerodynamic coefficient as follows: Based on the characteristic that the test data only contains the true values ​​of speed and torque, the data loss only calculates the mean square error between the predicted values ​​of speed and torque and the label values ​​and increases the weight of data loss. The speed limit constraint is canceled, the physical residual measurement method is replaced by the logarithmic hyperbolic cosine function and the standard mean square error is replaced, and differentiated fixed weights are configured for torque and thrust constraints. For the static condition of J=0 on the test bench, the two-dimensional polynomial fitting method is replaced by piecewise cubic Hermite conformal interpolation. The thrust coefficient and power coefficient are calculated based on the discrete nodes of the measured data. During the construction process, the PINN network is constrained by a composite loss function. The composite loss function is composed of a weighted combination of a data loss term and a physical loss term. The data loss term is used to measure the fitting error between the predicted value and the sample true value. The physical loss term is a multi-dimensional physical constraint residual term constructed based on the aerodynamics and dynamic mechanism of the propeller-generator system. The physical loss term in the composite loss function includes one or more of the following: torque balance constraint, thrust consistency constraint, and speed limiting constraint.

2. The construction method according to claim 1, characterized in that, The input variables include flight altitude, airspeed, blade angle, and throttle opening; the output variables include engine speed, effective engine output torque, propeller thrust, and fuel consumption rate.

3. The construction method according to claim 1, characterized in that, The forward-step nonlinear mapping relationship of the hidden layer is specifically as follows: In the formula For network input vectors, and Let be the weight matrix and bias vector of the k-th layer network, respectively. Here, k is the ReLU activation function, k is the number of hidden layers in the network, and k is an integer greater than or equal to 2.

4. The construction method according to claim 3, characterized in that, The expression for the total loss function of the composite loss function is as follows: In the formula and These are the weighting coefficients for data loss and physical loss, respectively. The mean squared error between the predicted value and the sample label. For physical residual constraints; In the physical residual constraint, the torque balance constraint satisfies the steady-state transmission balance mechanism, and the engine output torque... With propeller aerodynamic torque Nonlinear coupling relationship The propeller speed Satisfy conversion formula , For engine output torque, For propeller aerodynamic torque, Engine speed, This represents the propeller power coefficient obtained from the graph under the combined operating conditions of the approach ratio J and blade angle β. air density, The system reduction ratio, propeller diameter To fix the design parameters; The thrust consistency constraint requires that the thrust components output by the network conform to the propeller aerodynamic characteristics. ,in The thrust coefficient is obtained from the graph under the combined operating conditions of advance ratio J and blade angle β. The speed limiting constraint Calculate using the following formula: Where mean represents the arithmetic mean of all samples in the current training batch, when the model predicts the engine speed. When the value is less than or equal to 5800, the ReLU output is 0, and this term has no loss. When the value is greater than 5800, the difference is output and the squared value is used to amplify the over-limit penalty.

5. The construction method according to claim 4, characterized in that, The pre-training phase uses the Adam optimizer for training, with an initial learning rate set to 1×10⁻⁶. -5 -1×10 -4 The maximum number of training rounds is 1000-2000. The quality control parameters for pre-training include: a gradient clipping threshold of 1.0, a physical loss weight λ1 of 0 for the first 200 rounds of pre-training (only optimizing data loss), and a physical loss weight λ1 that linearly increases to 0.1 with each iteration after 200 rounds, while the data loss weight λ2 remains constant at 1.

0.

6. The construction method according to claim 1, characterized in that, The parameters for the fine-tuning phase are configured as follows: data loss weight is set to 10.0, torque constraint weight is set to 3.0, thrust constraint weight is set to 1.0, the Adam optimizer is used, and the training learning rate is adjusted to 1×10. -4 The maximum number of training rounds is maintained at 2000.

7. A propeller-generator integrated performance prediction system based on the construction method described in any one of claims 1-6.

8. A method for predicting the performance of a propeller-generator integrated system, characterized in that, The propeller-engine integrated performance prediction system described in claim 7 is used to input real-time flight status parameters and control command parameters, and output prediction results of engine speed, output torque, and propeller thrust to complete the dynamic prediction of the propeller-engine integrated system performance.

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