A pressure precision compensation method based on traditional polynomial fitting superposition PINN

Abstract

The present application belongs to the technical field of automobile electronic sensing chip, and particularly relates to a pressure measurement precision optimization technology of a tire pressure monitoring sensing chip, and particularly discloses a pressure precision compensation method based on a traditional polynomial fitting superposition PINN. The method aims to solve the problems of weak multi-physical coupling modeling capability, easy overfitting of small samples, poor extrapolation stability, insufficient interpretability, high calibration cost and difficult embedded deployment of traditional compensation methods. First, a traditional quadratic polynomial model is used to provide physical prior, and then a PINN network is used to learn a residual correction term, and finally a calibration model with good smoothness and physical consistency in the full range is obtained. The present application effectively improves the measurement precision of the sensor under various working conditions, greatly reduces the data dispersion degree, and comprehensively improves the stability and global consistency.

Description

Technical Field

[0001] This invention relates to the field of automotive electronic sensor chip technology, specifically to a pressure accuracy compensation method based on traditional polynomial fitting superimposed with PINN, belonging to the field of pressure measurement accuracy optimization technology for tire pressure monitoring sensor chips. Background Technology

[0002] Tire Pressure Monitoring System (TPMS) is a critical onboard electronic system for ensuring vehicle safety. It collects acceleration, tire pressure, and temperature information through sensors built into the tires to monitor their operating status in real time. Pressure detection data is the core foundation for TPMS to identify tire conditions and determine abnormalities; its measurement accuracy directly affects the system's monitoring accuracy. Besides automotive TPMS, pressure detection chips are widely used in industrial inspection and environmental monitoring. However, due to process deviations, temperature drift, and hardware nonlinearities, chip detection errors vary significantly with temperature and pressure conditions. Traditional compensation methods often employ polynomial fitting, lookup table correction, and ordinary neural network fitting.

[0003] However, the measures currently being taken have at least the following problems: 1) Difficulty in achieving high-precision global fitting across the entire range with limited calibration data: Traditional polynomial fitting, lookup table correction, and ordinary neural networks all rely heavily on the number and distribution of calibration samples, resulting in poor adaptability. Traditional polynomial fitting relies on only a small number of calibration points for fitting, leaving residual errors within the calibration interval. Its extrapolation fitting ability is even worse in non-calibrated regions such as extreme temperatures, leading to a significant decrease in accuracy. Lookup table correction relies on interpolation calculations using discrete calibration points, which introduces interpolation errors in non-calibrated intervals, making continuous high-precision compensation impossible. Ordinary neural networks are prone to underfitting and poor generalization with limited samples, lacking sufficient adaptability to all operating conditions and making it difficult to achieve high-precision global fitting across the entire range with limited calibration data.

[0004] 2) Pure data fitting is prone to producing non-physical behaviors, deviating from the actual working law of the sensor: Traditional polynomial fitting and ordinary neural networks are both pure data-driven fitting modes, without considering the constraints of the sensor's hardware physical mechanisms. The fitting results are prone to non-physical behaviors that violate the actual working law. In practical applications, simple data fitting is prone to problems such as non-monotonic fluctuations in pressure changes with ADC sampling values, abnormal local oscillations of the curve, and distortion, which violate the inherent physical characteristic of the monotonic output of the pressure sensor and greatly reduce the stability and reliability of TPMS tire pressure detection. The overall fitting model lacks effective monotonicity and smoothness physical constraints, and the output results cannot fully fit the actual physical working law of the sensor. Summary of the Invention

[0005] To address the aforementioned problems, the purpose of this invention is to provide a pressure accuracy compensation method based on traditional polynomial fitting superimposed with PINN. This method utilizes a traditional quadratic polynomial model to provide physical priors, and then learns residual correction terms through a PINN network, ultimately obtaining a calibration model with good smoothness and physical consistency across the entire measurement range.

[0006] This was achieved through the following technical solutions: A pressure accuracy compensation method based on traditional polynomial fitting and superposition of PINN includes the following steps: S1. Data Acquisition: Record the sensor's raw analog-to-digital converter (ADC) count value, actual temperature value, and actual pressure value at each operating point; S2. Calculate the residual error of the traditional model: Based on the physical priors provided by the traditional quadratic polynomial model, calculate the polynomial coefficients and residual error of the traditional model. S3. Construct the PINN correction model: Set training parameters and generate physical constraint points; S4. Train the PINN correction model: Perform iterative training and save the model and parameters; S5, parameter organization and embedded deployment.

[0007] Optionally, in step S2, the traditional quadratic polynomial model uses data from 3 pressure points and 2 temperature points for data fitting, and the model expression is: P code (T,ADC)=(Ma×ΔT+A)×ADC 2 +(Mb×ΔT+B)×ADC+(Mc×ΔT+C); Among them, P code (T, ADC) represents the ADC digital value calculated using traditional polynomial fitting; T represents the measured temperature under current operating conditions; ADC represents the raw ADC count value output by the pressure sensor; ΔT represents the temperature change, satisfying ΔT = TT. ref T ref The reference temperature is used for calibration; Ma, Mb, and Mc are temperature coupling correction coefficients, which are used to compensate for the drift error of the ADC square term, ADC first term, and constant term caused by temperature changes, respectively; A, B, and C are the basic coefficients for the pressure and ADC second-order fitting at the reference temperature, where A corresponds to the basic coefficient of the ADC square term, B corresponds to the basic coefficient of the ADC first term, and C is the zero-point offset constant term.

[0008] Optionally, in step S3, the training parameters include: optimizer, initial learning rate, learning decay rate, batch size, training epochs, early stopping patience value, and weights of each λ in the loss function.

[0009] Optionally, in step S3, the model building process is as follows: S3.1 Construct a fully connected neural network; S3.2 Construct the total loss function L of the neural network, which combines the data fitting loss, second-order smooth regularization loss, physical monotonicity constraint loss, and first-order gradient smoothing loss to construct the total loss function.

[0010] Optionally, in step S3.1, the structure of the fully connected neural network includes: two nodes in the input layer, with the input variables being the measured temperature T under the working condition and the original analog-to-digital conversion value ADC from the sensor, both of which are pre-normalized using min-max; two hidden layers, each containing 64 neurons, with tanh as the activation function to ensure that the network output is smooth and differentiable, meeting the physical constraint differentiation requirements; and one node in the output layer, which outputs the residual error err_pred predicted by the neural network, to achieve an approximation fit to the residual error err_trad calculated by the traditional model.

[0011] Optionally, in step S3.2, the total loss function is constructed, with the expression: L = L_data + λ_smooth·L_smooth + λ_mono·L_mono + λ_first·L_first. Where λ_smooth, λ_mono, and λ_first are the balancing weight coefficients corresponding to the smoothing term, monotonicity constraint term, and first-order gradient term, respectively; L_data, L_smooth, L_mono, and L_first are the data loss, second-order smoothing regularization term, monotonicity constraint term, and first-order gradient regularization term, respectively.

[0012] Optionally, for the calculation of data loss L_data, the mean absolute error (MAE) is used to measure the difference between the network prediction value and the actual residual error, and the formula is: L_data=mean(|err_pred-err_trad|). The formula for calculating the second-order smoothness regularization term L_smooth is as follows: ; For the calculation of the monotonicity constraint L_mono, the ReLU hinge loss is used to penalize the region where the monotonically increasing condition is not met. The formula is as follows: ; The formula for calculating the first-order gradient regularization term L_first is: ; Where err_pred is the residual error obtained from the neural network prediction, and err_trad is the residual error calculated by the traditional model; T_norm and ADC_norm are the normalized temperature and ADC value, respectively.

[0013] Optionally, in step S4, the process of training the PINN correction model is as follows: the parameters of the traditional physical model are fitted and superimposed with the parameters of the PINN network for training. The training data comes from additional calibration experiments in a wide temperature range and gradient data required for physical constraints.

[0014] Optionally, in step S4, the model training process uses the Adam adaptive optimizer, sets a fixed number of iteration training rounds and an early stopping mechanism, and continuously iterates and updates the PINN network parameters until the total loss function converges and the early stopping condition is met. After optimization to meet the model accuracy, the training is terminated and the optimal model parameters are saved.

[0015] Optionally, in step S5, the polynomial coefficients of the traditional model and the normalized parameters of the PINN modified model after training convergence are sorted out and then embeddedly deployed.

[0016] The beneficial effects of this invention compared to the prior art are: By introducing multiple physical constraints, this invention ensures the model's physical interpretability and avoids non-physical interpretation: The invention simultaneously introduces a first-order gradient regularization term, a second-order smoothness regularization term, and a monotonicity constraint loss into the loss function, forcing the network output surface to be continuous and smooth and to satisfy the physical law of monotonic pressure change. This suppresses network overfitting, improves the model's generalization ability outside the calibration interval, and avoids unreasonable measurement results during embedded runtime.

[0017] 2) Decoupled training architecture for enhanced stability: During the training phase, the parameters of the traditional multinomial model are fixed and do not participate in iterative optimization. Only the PINN network is trained for residual error compensation. Compared to the overall joint training scheme, this scheme separates the basic physical model from the error correction network, resulting in a smoother training loss surface, reducing the likelihood of training divergence and parameter drift, and lowering the difficulty of hyperparameter tuning. Attached Figure Description

[0018] Figure 1 This is a flowchart of the pressure accuracy compensation method of the present invention; Figure 2 This is a graph showing the change in sensor pressure error with temperature under a 104 kPa working condition, as tested by the method of this invention. Figure 3 This is a graph showing the change in sensor pressure error with temperature under a 300 kPa working condition, as tested by the method of this invention. Figure 4 The graph shows the change in sensor pressure error as a function of temperature under a 900 kPa operating condition, as tested by the method of this invention. Figure 5 This is the curve of sensor pressure error versus ambient temperature under 900kPa working condition using the traditional polynomial compensation method in this invention. Figure 6This is the curve showing the change of sensor pressure error with ambient temperature under 900kPa operating conditions using the PINN compensation method in this invention. Detailed Implementation

[0019] The technical solutions in the embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0020] This invention discloses a pressure accuracy compensation method based on traditional polynomial fitting and superposition of PINN. For example... Figure 1 As shown in the flowchart of the pressure accuracy compensation method of the present invention, it includes the following steps: S1. Data Acquisition: Data is acquired from multiple temperature and pressure combinations across the entire temperature and pressure range using existing temperature and pressure control data acquisition platforms; the sensor analog-to-digital converter (ADC) sampled value, actual temperature value, and true pressure value (P) are recorded at each operating point. real ).

[0021] S2. Calculate the residual error of the traditional model: Based on the physical prior provided by the traditional quadratic polynomial model, calculate the polynomial coefficients and residual error of the traditional model. The traditional quadratic polynomial model uses data from 3 pressure points and 2 temperature points for data fitting, abbreviated as 3P2T, and the model expression is: P code (T,ADC)=(Ma×ΔT+A)×ADC 2 +(Mb×ΔT+B)×ADC+(Mc×ΔT+C); Among them, P code (T, ADC) represents the ADC digital value calculated using traditional polynomial fitting; T represents the measured temperature under current operating conditions; ADC represents the raw ADC count value output by the pressure sensor; ΔT represents the temperature change, satisfying ΔT = TT. ref T ref To calibrate the reference temperature, temperature T1 is used here (for example, T1 = 25℃ in the current experiment); Ma, Mb, and Mc are temperature coupling correction coefficients, which are used to compensate for the drift error of the ADC square term, ADC first term, and constant term caused by temperature changes, respectively; A, B, and C are the basic coefficients for the pressure and ADC second-order fitting at the reference temperature, where A corresponds to the basic coefficient of the ADC square term, B corresponds to the basic coefficient of the ADC first term, and C is the zero-point offset constant term.

[0022] The residual error, calculated using traditional polynomial fitting, is in the same unit as the calibration pressure (kPa); it is calculated from the current temperature T and the original ADC sampling values. The specific calculation process for the stepwise parameter fitting method of the 3P2T model is as follows: S2.1 Calculate the nonlinear coefficients A1, B1, and C1 of ADC and pressure.

[0023] Under the reference calibration temperature T1 (temperature point 1), three sets of standard known pressures P are first selected. n (n=1,2,3), a second-order polynomial was used to fit the relationship between pressure and ADC acquisition values. The original fitting coefficients were obtained by solving three sets of calibration data: A raw1 B raw1 C raw1 The fitting equation is: P ADCn =A raw1 ×P n 2 +B raw1 ×P n +C raw1 , Among them, P ADCn The ADC sampled values ​​output by the sensor under the corresponding calibration conditions; n=1,2,3, A raw1 B raw1 C raw1 P represents the original fitting coefficients of the second-order polynomial obtained through multi-point pressure calibration; n The actual pressure value set for the high-precision pressure controller under the nth working condition.

[0024] Secondly, the interpolated pressure point ADC numerical derivation: using the original fitting coefficients mentioned above, the theoretical ADC count values ​​corresponding to the characteristic pressure points are calculated, resulting in three sets of corresponding pressure characteristic count values: P count1 P count2 and P count3 The formula is: P countn =A raw1 ×P n 2 +B raw1 ×P n +C raw1 , Where n=1,2,3,P count1 P represents the characteristic ADC count value corresponding to the minimum pressure operating condition. count2 P represents the characteristic ADC count value corresponding to the intermediate pressure condition. count2 This represents the characteristic ADC count value corresponding to the maximum pressure condition.

[0025] Finally, the global nonlinear calibration coefficients are solved as follows: Using the three characteristic points of the sensor's ADC—zero, half-range, and full-scale—as constraints, corresponding to the lower limit, midpoint, and upper limit of the analog-to-digital converter (ADC), a second-order polynomial equation system is constructed and fitted to obtain the final global nonlinear calibration coefficients: A1, B1, and C1. The three sets of constraint equations are as follows: P0=A1×P count1 2 +B1×P count1 +C1, P Max / 2 =A1×P count2 2 +B1×P count2 +C1, P Max =A1×P count3 2 +B1×P count3 +C1, Where ADC=0 represents the theoretical value of the ADC sample corresponding to the minimum range; ADC=Max represents the theoretical value of the ADC sample corresponding to the full scale, i.e., the maximum range; ADC=Max / 2 represents the theoretical value of the ADC sample corresponding to the middle range, i.e., half of the range; P0 is the zero-point standard pressure, P... Max / 2 For half-range standard pressure, P Max This is the full-scale standard pressure.

[0026] Furthermore, selecting the ADC zero-point, half-range, and full-scale characteristic points is a balance between engineering accuracy and practicality. This ensures uniform coverage of the ADC range, improves the numerical stability of the solution process, and reduces the sensitivity of the fitting formula to noise. Simultaneously, it guarantees that the input / output range of the polynomial model matches the hardware range, and each point has a relatively clear physical meaning, facilitating the establishment of an intuitive correspondence during verification. In the corresponding tests of this application, the sensor's digital values ​​0, 1023.5, and 2047 correspond to the following three pressure values: the lower limit of the range corresponding to standard atmospheric pressure (101.35 kPa), 500.6 kPa being the geometric midpoint of the range, and 900 kPa being the upper limit of the full-scale range.

[0027] S2.2 Calculate the nonlinear coefficients A2, B2, and C2 of ADC and pressure.

[0028] Under the reference calibration temperature T2 (temperature point 2), three sets of standard known pressures P are first selected. n (n=1,2,3), a second-order polynomial was used to fit the relationship between pressure and ADC acquisition values. The original fitting coefficients were obtained by solving three sets of calibration data: A raw2 B raw2 C raw2 The fitting equation is: P ADCn =A raw2 ×P n 2 +B raw2 ×P n +C raw2 , Among them, P ADCn The ADC sampled values ​​output by the sensor under the corresponding calibration conditions; n=1,2,3, A raw1 B raw1 C raw1 P represents the original fitting coefficients of the second-order polynomial obtained through multi-point pressure calibration; n The actual pressure value set for the high-precision pressure controller under the nth working condition.

[0029] Secondly, the interpolated pressure point ADC numerical derivation: using the original fitting coefficients mentioned above, the theoretical ADC count values ​​corresponding to the characteristic pressure points are calculated, resulting in three sets of corresponding pressure characteristic count values: P count1 P count2 and P count3 The formula is: P countn =A raw2 ×P n 2 +B raw2 ×P n +C raw2 , Where n=1,2,3,P count1 P represents the characteristic ADC count value corresponding to the minimum pressure operating condition. count2 P represents the characteristic ADC count value corresponding to the intermediate pressure condition. count2 This represents the characteristic ADC count value corresponding to the maximum pressure condition.

[0030] Finally, the global nonlinear calibration coefficients are solved as follows: using the three characteristic points of the sensor ADC—zero, half-range, and full-range—as constraints, a second-order polynomial equation system is constructed and fitted to obtain the final global nonlinear calibration coefficients, namely A1, B1, and C1. The three sets of constraint equations are as follows: P0 = A2 × P count1 2 +B2×P count1 +C2, P Max / 2 =A2×P count2 2 +B2×P count2 +C2, P Max =A2×P count3 2 +B2×P count3 +C2, Where ADC=0 represents the theoretical value of the ADC sample corresponding to the minimum range; ADC=Max represents the theoretical value of the ADC sample corresponding to the full scale, i.e., the maximum range; ADC=Max / 2 represents the theoretical value of the ADC sample corresponding to the middle range, i.e., half of the range; P0 is the zero-point standard pressure, P... Max / 2 For half-range standard pressure, P Max This is the full-scale standard pressure.

[0031] S2.3 Calculate the linear slope coefficient of temperature. Two sets of calibration temperatures are selected, and the formulas for calculating the slope of each temperature component are as follows: Quadratic temperature slope: Ma = (A2 - A1) / (T2 - T1); Temperature slope of the first term: Mb=(B2-B1) / (T2-T1); The constant term temperature slope: Mc = (C2 - C1) / (T2 - T1); Among them, the pressure-ADC basic coefficients at the reference temperature are A, B, and C, with reference values ​​of A=A1, B=B1, and C=C1; A1, B1, and C1 are the basic coefficients of the second-order polynomial obtained by fitting the pressure and ADC at the reference temperature; A2, B2, and C2 are the temperature nonlinear fitting coefficients obtained by solving at another set of calibration temperatures; T1 and T2 are the test temperatures corresponding to the two sets of calibration points, respectively; Ma, Mb, and Mc represent the temperature coupling slopes of the ADC square term, the ADC first-order term, and the constant term, respectively, used to quantify the linear influence of temperature changes on the coefficients of each pressure fitting component.

[0032] After selecting multiple batches of chips for verification and fitting to obtain their respective 3P2T parameters, the test conditions of the selected chips are input into the corresponding 3P2T model to calculate the corresponding model-predicted ADC digital quantity P. code (T, ADC); then, the residual error err_trad of the traditional model is calculated based on the digital quantity. The effective output range of the ADC digital quantity of the sensor used in this test is 0 to 2047, corresponding to a lower limit of pressure measurement of the current standard atmospheric pressure (101.35 kPa by default in this test) and an upper limit of 900 kPa (i.e., the pressure value corresponding to full scale). According to the definition of resolution (the difference between the upper and lower limits of the range divided by the ADC digital quantity range), the resolution of this system is: Res = (900 - 101.35) / 2047 (kPa / LSB). Based on this, the pressure value calculated by the traditional model is P. trad =P code (T,ADC)*Res+101.35, the residual error of the traditional model err_trad=P trad -P real (where P) realThe actual pressure value under this working condition is used, and then the residual error err_trad of the traditional model is used as the fitting target of the PINN network.

[0033] S3. Construct the PINN correction model: Set training parameters and generate physical constraint points.

[0034] This model employs the core idea of ​​PINN data-driven training combined with physical priors. It relies on automatic differentiation to obtain the first and second derivatives of the network output, and combines the inherent characteristics of the sensor's temperature drift (monotonic and smooth) to build a lightweight physical loss, belonging to an engineering-simplified physical information neural network. After training, the differential operation and physical constraint modules are removed during the inference stage, retaining only the basic fully connected network for forward computation, facilitating embedded deployment.

[0035] The PINN correction model construction process is as follows: Configure a complete set of training hyperparameters, including optimizer type, initial learning rate, learning rate decay coefficient, batch size, maximum number of training epochs, early stopping patience value, weight coefficients λ of each term in the loss function, etc., start iterative training, and continuously update the network weights by relying on backpropagation driven by the total loss; after the convergence condition is met, save the optimal neural network weight parameters.

[0036] In this step, the parameters are set to 2 hidden layers, 64 neurons per layer, and Tanh activation function; the Adam optimizer has an initial learning rate of 1e-3 and is paired with a StepLR decay strategy; the batch size is 64, the maximum number of iterations is 1200, and early stopping is triggered after 100 iterations without optimization. Choosing a suitable network structure avoids overfitting due to excessive depth, and the number of iterations balances convergence sufficiency and training efficiency. In testing and validation, logs showed that a 2-layer, 64-dimensional network was sufficient for fitting, achieving convergence in about 1000 iterations. Therefore, a relatively suitable value was selected. These parameters, including the values ​​of the various λ weights in the loss function, were gradually adjusted based on observations of the output logs and data fitting effects during the debugging process. The values ​​of each λ weight were adjusted primarily based on the fitting effect of the specific experimental data. The error distribution in the test results was examined, including whether the error distribution trend line was flat; the degree of back-and-forth curves within the same temperature range under fixed temperature conditions; and whether there were sharp divergences at the high and low temperature ends, which were used to adjust the corresponding weights. Then, observe the effect of the adjustment and iterate step by step until the effect of each λ weight setting meets the final compensation accuracy expectation.

[0037] S3.1 Constructing a Fully Connected Neural Network: A fully connected neural network is constructed to fit the residual error of the traditional multinomial model. The structure includes: an input layer with two nodes, the input variables being the measured temperature T under the working condition and the original analog-to-digital conversion value (ADC) from the sensor, both of which are pre-normalized using min-max; two hidden layers, each containing 64 neurons, with tanh as the activation function to ensure that the network output is smooth and differentiable, meeting the physical constraint differentiation requirements; and an output layer with one node, outputting the residual error err_pred predicted by the neural network, thus approximating the residual error err_trad calculated by the traditional model.

[0038] First, the original temperature and ADC values ​​are linearly mapped to the [-1, 1] interval, as shown in the formula: T_norm=2*(T-T_min) / (T_max-T_min)-1, ADC_norm=2*(ADC-ADC_min) / (ADC_max-ADC_min)–1, Where T_norm and ADC_norm are the normalized temperature and ADC value, respectively; T_min and T_max are the upper and lower limits of the dataset temperature; and ADC_min and ADC_max are the upper and lower limits of the original ADC values ​​in the dataset.

[0039] Secondly, the normalized error err_norm is output and then restored to the magnitude of the true residual error through inverse normalization, as shown in the formula: err_norm=2*(err_trad-err_min) / (err_max-err_min)-1, err_pred=err_norm*(err_max-err_min) / 2+(err_max+err_min) / 2, Where err_min and err_max are the minimum and maximum residual errors of the traditional model in the trained chip dataset, respectively; err_pred is the residual error predicted by the output neural network; and err_norm is the normalized error.

[0040] S3.2 Construct the total loss function L of the neural network. This function integrates the data fitting loss, second-order smoothing regularization loss, physical monotonicity constraint loss, and first-order gradient smoothing loss. The expression for the total loss function is: L = L_data + λ_smooth·L_smooth + λ_mono·L_mono + λ_first·L_first. Wherein, λ_smooth, λ_mono, and λ_first are the balancing weight coefficients corresponding to the smoothing term, monotonicity constraint term, and first-order gradient term, respectively; L_data, L_smooth, L_mono, and L_first are the data loss, second-order smoothing regularization term, monotonicity constraint term, and first-order gradient regularization term, respectively. Similarly, the data fitting effect based on the error distribution is gradually adjusted to a suitable λ weight parameter; in this invention, the final weight values ​​used by the sensor are 0.0002, 0.00025, and 0.0003, respectively. For the calculation of data loss L_data, the Mean Absolute Error (MAE) is used to measure the difference between the network's predicted values ​​and the actual residual errors on the training set (traditional calibration temperature and pressure points). The formula is: L_data = mean(|err_pred - err_trad|). The training dataset can be distributed across the entire temperature range and the full operating range. Additionally, constraint domain sampling points are generated to participate in the physical loss calculation, applying monotonicity and smoothness physical constraints in the unlabeled interval to improve the stability of high and low temperature extrapolation. In one specific embodiment, the training set selects multiple batches of real chip datasets. The training temperature points include: -40℃, -20℃, 0℃, 10℃, 30℃, 90℃, 110℃, and 125℃; the training pressure points include: 104Kpa, 400Kpa, 600Kpa, and 900Kpa.

[0041] For the calculation of the second-order smoothing regularization term L_smooth, a penalty is applied to the mean square of the second-order partial derivatives of the system output with respect to the normalized input. This suppresses violent oscillations in the error output surface and improves the continuity of the model. The formula is as follows: .

[0042] For the calculation of the monotonicity constraint L_mono, based on the physical characteristics of the pressure sensor in this chip: the larger the ADC value, the greater the corresponding pressure, and the residual error increases monotonically with the ADC sampling value; ReLU hinge loss is used to penalize regions that violate the monotonic trend, and the formula is: .

[0043] For the calculation of the first-order gradient regularization term L_first, the mean square of the first-order partial derivative of the network output with respect to the normalized input is penalized. This constrains the model output to change gradually with temperature and ADC, reducing overfitting and improving generalization ability. The calculation formula is as follows: .

[0044] S4. Training the PINN Correction Model: Perform iterative training and save the model and parameters. Fit the traditional physical model parameters (not used in training) and superimpose them onto the PINN network parameters for training. Training data comes from additional calibration experiments over a wide temperature range and gradient data required for physical constraints. Model training uses the Adam adaptive optimizer to perform gradient iterative updates, configuring a fixed number of iterations and adding an early stopping mechanism to prevent overfitting. PINN network parameters are continuously iterated and updated until the total loss function converges and the early stopping condition is met. The final compensation result is observed, and the weights are gradually optimized to meet model accuracy requirements. Training is then terminated, and the optimal model parameters are saved.

[0045] Specifically, the training process continuously backpropagates to update the PINN network weight parameters, constantly minimizing the joint total loss, i.e., minimizing the sum of data fitting loss and physical regularization loss. When the total loss tends to converge and the early stopping condition is met, the final compensation result is observed until the model accuracy meets expectations, triggering the training termination logic, saving the optimal weights and the complete set of normalized parameters, and completing the model training. In the specific tests of this invention, the expected model accuracy includes: under a pressure range of 100-500 kPa, the error limit is ±5 kPa in the 0℃~70℃ range, and ±10 kPa in the high and low temperature ranges of -40℃~0℃ and 70℃~125℃; under a pressure range of 500-900 kPa, the error limit is ±7 kPa in the 0℃~70℃ range, and ±15 kPa in the high and low temperature ranges of -40℃~0℃ and 70℃~125℃.

[0046] Furthermore, a layered coupled training method is adopted, which uses traditional physical model parameters and only iteratively optimizes the PINN network parameters, to achieve a deep integration of physical mechanism constraints and measured data correction. The specific training process is as follows: This model abandons the traditional full-parameter training mode and solidifies the traditional physical model parameters after basic fitting calibration, so that they remain constant in value during the overall model training iteration process and do not participate in gradient backpropagation and parameter updates. This preserves the inherent physical mechanism and basic fitting accuracy of the traditional physical model and ensures that the model as a whole conforms to the general physical laws of the industry. At the same time, dynamic training and iterative optimization are performed only on all network parameters of the PINN neural network, relying on the nonlinear fitting capability of the neural network to make up for the mechanism defects and temperature range adaptation errors of the traditional physical model.

[0047] The dataset required for model training is divided into two core categories, which jointly support the high-precision convergence of the model and physical compliance constraints: the first is wide-temperature-range measured calibration data, which is collected by setting multiple sets of gradient temperature conditions in the entire operating temperature range and conducting multiple batches of additional calibration experiments at multiple points, covering the entire operating range of low temperature, normal temperature, and high temperature. This effectively makes up for the deficiency of insufficient fitting accuracy of traditional models in the extreme temperature range and improves the model's wide-temperature-range adaptive capability; the second is physical constraint gradient data, which is based on the model expression, boundary conditions, and initial conditions of the traditional physical model. Gradient feature data of each dimension is calculated through numerical differentiation and equation derivation and is embedded as hard constraint conditions in the PINN network training process.

[0048] During training, the output of the traditional physical model with fixed parameters is superimposed and fused with the real-time prediction results of the PINN network to construct a coupled loss function. This loss function takes into account both the fitting error of the measured data and the constraint error of the physical equation. The weights and bias parameters of the PINN network are continuously corrected through the backpropagation algorithm until the loss function converges to the expected value. Finally, the training of the PINN corrected model is completed, so that the model can not only meet the constraints of the basic physical mechanism, but also accurately adapt to the measured characteristics of complex working conditions in a wide temperature range, which greatly improves the prediction accuracy and generalization ability of the model in the entire temperature range.

[0049] S5, parameter organization and embedded deployment.

[0050] The parameters and normalized parameters of the polynomial to be stored in the chip are organized and stored in the chip's designated storage area. Then, the trained network parameters and inference logic are encapsulated into low-level code that can run on the embedded chip and deployed to the pressure sensing terminal. The device collects raw ADC values ​​and temperature data in real time, first solves for the initial pressure value using a traditional quadratic polynomial model, then calls the PINN model to predict the residual error and complete the compensation correction, finally outputting a high-precision pressure measurement result with error correction over a wide temperature range. The PINN model is an existing known model and will not be described in detail in this application.

[0051] In specific embodiments, such as Figure 2-6As shown, each chart corresponds to the error distribution of a pressure (unit: kPa) test point at different temperatures. Each broken line in the chart represents the data of one sensor. The horizontal axis is temperature (unit: ℃), and the vertical axis is the error value (unit: kPa). The upper and lower horizontal lines represent the maximum acceptable error range of the sensor. Regarding the selection of the calibration group (3P2T), the temperature is determined by considering the sensor's operating temperature range and the configuration of the calibration equipment. The pressure is determined by considering the sensor's range; generally, one point is selected from the low-pressure, medium-pressure, and high-pressure ranges. After verification, a set of points with better performance is selected. Here, the selected temperatures are 25℃ and 70℃, and the pressures are 104 kPa, 400 kPa, and 700 kPa, with a range of 900 kPa. In addition, the operating conditions of the real training dataset for the PINN network model are: temperatures -40℃, -20℃, 0℃, 10℃, 30℃, 90℃, 110℃, and 125℃; pressures of 104 kPa, 400 kPa, 600 kPa, and 900 kPa, with a range of 900 kPa. Figure 2-4 The results are from tests conducted using the method of this invention.

[0052] like Figure 2 The figure shows the characteristic curve of sensor pressure error versus ambient temperature under 104 kPa operating conditions; the red and green dashed lines represent the upper and lower limits of pressure error, respectively, and the multiple colored curves represent the measured pressure errors of multiple groups of tested samples. The test temperature range is -40℃ to 125℃, with an error limit of ±5 kPa set in the 0℃ to 70℃ range, and an error limit of ±10 kPa in the high and low temperature ranges of -40℃ to 0℃ and 70℃ to 125℃. The measured pressure error of the batch of samples is stable within -2 kPa to +2 kPa, meeting the accuracy requirements across the entire temperature range.

[0053] like Figure 3 The figure shows the characteristic curve of sensor pressure error versus ambient temperature under 300 kPa operating conditions; the red and green dashed lines represent the upper and lower limits of pressure error, respectively, and the multiple colored curves represent the measured pressure errors of multiple groups of tested samples. The test temperature range is -40℃ to 125℃, with an error limit of ±5 kPa set in the 0℃ to 70℃ range, and an error limit of ±10 kPa in the high and low temperature ranges of -40℃ to 0℃ and 70℃ to 125℃. The measured pressure error of the batch of samples is stable within -2 kPa to +2 kPa, meeting the accuracy requirements across the entire temperature range.

[0054] Figure 4The curves show the characteristic curves of sensor pressure error versus ambient temperature under 900 kPa operating conditions. The red and green dashed lines represent the upper and lower limits of pressure error, respectively, and multiple colored curves represent the measured pressure errors of multiple groups of tested samples. The test temperature range is -40℃ to 125℃. An error limit of ±7 kPa is set for the 0℃ to 70℃ range, and an error limit of ±15 kPa is set for the high and low temperature ranges of -40℃ to 0℃ and 70℃ to 125℃. The measured pressure error of batch samples is stable between -4 kPa and +5 kPa, meeting the accuracy requirements across the entire temperature range.

[0055] Figure 5 This is a curve showing the characteristic curve of sensor pressure error versus ambient temperature under a 900 kPa operating condition using only traditional polynomial compensation. The red and green dashed lines represent the upper and lower limits of pressure error, respectively, and multiple colored curves represent the measured pressure errors of multiple sets of tested samples. The test temperature range is -40℃ to 125℃. An error limit of ±7 kPa is set for the 0℃ to 70℃ range, and an error limit of ±15 kPa is set for the high and low temperature ranges of -40℃ to 0℃ and 70℃ to 125℃. The measured pressure error of batch samples is relatively large in the low and high temperature ranges, and there are critical situations at individual points in the normal temperature range.

[0056] Figure 6 This chart shows the characteristic curves of sensor pressure error versus ambient temperature under a 900 kPa operating condition, based on the same training conditions and neural network levels, using only PINN compensation. The red and green dashed lines represent the upper and lower limits of the pressure error, respectively, and multiple colored curves represent the measured pressure errors of multiple groups of tested samples. The test temperature range was -40℃ to 125℃, with an error limit of ±7 kPa set for the 0℃ to 70℃ range, and an error limit of ±15 kPa for the high and low temperature ranges of -40℃ to 0℃ and 70℃ to 125℃. However, the measured pressure errors of the batch of samples exhibited overall divergence, especially at temperature points without training, making them susceptible to noise. Compared to the superposition compensation method of this invention, polynomial fitting lays a good foundation for the smoothness of extrapolation of the overall fitting effect. If better results are desired by using neural network fitting alone, it is necessary to increase the number of layers and neurons and a large amount of training data to approximate the real error. However, this will increase the training time and the difficulty of embedded deployment, which does not meet the cost requirements of mass production. Moreover, completely bypassing the physical characteristics and directly fitting the data may learn spurious correlations in the data rather than real physical laws, and the generalization ability is also poor in scenarios outside the range of training data.

[0057] Therefore, after comparing the different testing methods described above, the pressure accuracy compensation method based on traditional polynomial fitting superimposed with PINN in this invention significantly optimizes the measurement accuracy and error distribution. Specifically, the pressure error after superimposed compensation is reduced from a maximum error of ±14 to ±4 compared to the measured pressure error after compensation by traditional polynomial fitting alone, with a relative decrease in error amplitude of 71%, and the ultimate measurement accuracy is improved to 3 times the original. At the same time, the error distribution is significantly optimized, with the positive and negative error distributions being basically symmetrical without significant systematic shifts, the data dispersion is greatly reduced, and the consistency of error performance under various working conditions is significantly improved. Overall measurement accuracy, stability, and global consistency are simultaneously and comprehensively improved.

[0058] In summary, the present invention provides a pressure accuracy compensation method based on traditional polynomial fitting superimposed with PINN, which effectively combines the physical prior advantages of traditional polynomial models with the nonlinear fitting capability of PINN networks. Through multiple physical constraints, it ensures that the output conforms to the inherent physical laws of the sensor. While reducing calibration costs, it achieves fine compensation for pressure measurement errors over a wide temperature range. Furthermore, the model is easy to deploy in a lightweight manner to embedded terminals, has good engineering practicality, and demonstrates significant progress.

[0059] The above embodiments are merely illustrative of the technical concept of the present invention and should not be construed as limiting the scope of protection of the present invention. Any modifications made to the technical solutions based on the technical concept proposed in this invention shall fall within the scope of protection of this invention.

Claims

1. A pressure accuracy compensation method based on traditional polynomial fitting and superposition of PINN, characterized in that, Includes the following steps: S1. Data Acquisition: Record the sensor's original analog-to-digital conversion count value, actual temperature value, and actual pressure value at each operating point; S2. Calculate the residual error of the traditional model: Based on the physical priors provided by the traditional quadratic polynomial model, calculate the polynomial coefficients and residual error of the traditional model. S3. Construct the PINN correction model: Set training parameters and generate physical constraint points; S4. Train the PINN correction model: Perform iterative training and save the model and parameters; S5, parameter organization and embedded deployment.

2. The pressure accuracy compensation method based on traditional polynomial fitting and superposition of PINN as described in claim 1, characterized in that, In step S2, the traditional quadratic polynomial model uses data from 3 pressure points and 2 temperature points for data fitting, and the model expression is: P code (T,ADC)=(Ma×ΔT+A)×ADC 2 +(Mb×ΔT+B)×ADC+(Mc×ΔT+C); Among them, P code (T, ADC) represents the ADC digital value calculated using traditional polynomial fitting; T represents the measured temperature under current operating conditions; ADC represents the raw ADC count value output by the pressure sensor; ΔT represents the temperature change, satisfying ΔT = TT. ref T ref The reference temperature is used for calibration; Ma, Mb, and Mc are temperature coupling correction coefficients, which are used to compensate for the drift error of the ADC square term, ADC first term, and constant term caused by temperature changes, respectively; A, B, and C are the basic coefficients for the pressure and ADC second-order fitting at the reference temperature, where A corresponds to the basic coefficient of the ADC square term, B corresponds to the basic coefficient of the ADC first term, and C is the zero-point offset constant term.

3. The pressure accuracy compensation method based on traditional polynomial fitting and superposition of PINN as described in claim 1, characterized in that, In step S3, the training parameters include: optimizer, initial learning rate, learning decay rate, batch size, training epochs, early stopping patience value, and various λ weights of the loss function.

4. The pressure accuracy compensation method based on traditional polynomial fitting and superposition of PINN as described in claim 1, characterized in that, In step S3, the model building process is as follows: S3.1 Construct a fully connected neural network; S3.2 Construct the total loss function L of the neural network, which combines the data fitting loss, second-order smooth regularization loss, physical monotonicity constraint loss, and first-order gradient smoothing loss to construct the total loss function.

5. The pressure accuracy compensation method based on traditional polynomial fitting and superposition of PINN according to claim 4, characterized in that, In step S3.1, the structure of the fully connected neural network is set as follows: the input layer has 2 nodes, and the input variables are the measured temperature T under the working condition and the original analog-to-digital conversion value ADC of the sensor. Both the temperature and ADC values ​​are pre-normalized using min-max. The hidden layer has 2 layers, each containing 64 neurons. tanh is selected as the activation function to ensure that the network output is smooth and differentiable, meeting the physical constraint differentiation requirements. The output layer has 1 node, which outputs the residual error err_pred predicted by the neural network.

6. The pressure accuracy compensation method based on traditional polynomial fitting and superposition of PINN according to claim 4, characterized in that, In step S3.2, the total loss function is constructed, with the expression: L = L_data + λ_smooth·L_smooth + λ_mono·L_mono + λ_first·L_first. Where λ_smooth, λ_mono, and λ_first are the balancing weight coefficients corresponding to the smoothing term, monotonicity constraint term, and first-order gradient term, respectively; L_data, L_smooth, L_mono, and L_first are the data loss, second-order smoothing regularization term, monotonicity constraint term, and first-order gradient regularization term, respectively.

7. The pressure accuracy compensation method based on traditional polynomial fitting and superposition of PINN as described in claim 6, characterized in that, For the calculation of data loss L_data, the mean absolute error (MAE) is used to measure the difference between the network prediction and the actual residual error. The formula is: L_data=mean(|err_pred-err_trad|). The formula for calculating the second-order smoothness regularization term L_smooth is as follows: ； For the calculation of the monotonicity constraint L_mono, the ReLU hinge loss is used to penalize the region where the monotonically increasing condition is not met. The formula is as follows: ； The formula for calculating the first-order gradient regularization term L_first is: ； Where err_pred is the residual error obtained from the neural network prediction, and err_trad is the residual error calculated by the traditional model; T_norm and ADC_norm are the normalized temperature and ADC value, respectively.

8. The pressure accuracy compensation method based on traditional polynomial fitting and superposition of PINN according to claim 1, characterized in that, In step S4, the process of training the PINN correction model is as follows: the residual error after fitting the parameters of the traditional physical model is superimposed on the PINN network parameters for training. The training data comes from additional calibration experiments in a wide temperature range and gradient data required for physical constraints.

9. The pressure accuracy compensation method based on traditional polynomial fitting and superposition of PINN according to claim 3, characterized in that, In step S4, the model training uses the Adam adaptive optimizer to complete gradient iteration updates, configures a fixed number of iteration rounds and superimposes an early stopping mechanism, and continuously iterates and updates the PINN network parameters until the total loss function converges and the early stopping condition is met. After optimization to meet the model accuracy, the training is finally terminated and the optimal model parameters are saved.

10. The pressure accuracy compensation method based on traditional polynomial fitting and superposition of PINN according to claim 1, characterized in that, In step S5, the polynomial coefficients of the traditional model and the PINN corrected model and normalized parameters after training convergence are organized and embeddedly deployed.

Images