Method and device for predicting density virial coefficient of helium isotope gas

Abstract

The present application relates to the technical field of gas thermophysical property modeling and precision measurement, and provides a density virial coefficient prediction method and device for helium isotope gas, wherein the method comprises: inputting a sampling temperature sample into a rational function model to obtain a labeled density virial coefficient value, so that stable and continuous data reconstruction and large-batch sample expansion can be performed through the rational function model, thereby providing sufficient sample support when training the density virial coefficient prediction model, and guaranteeing the data reliability of continuous calling in the whole temperature range; and inputting the obtained property prior parameters into the model, so that the model can learn and perceive the inherent physical characteristics of different helium isotope gases, thereby being compatible with the physical property differences of different isotope gases in a unified analysis dimension, and finally realizing unified and high-precision prediction of the density virial coefficients of helium-3, helium-4 and other isotope gases.

Description

Technical Field

[0001] This invention relates to the field of gas thermophysical property modeling and precision measurement technology, and in particular to a method and apparatus for predicting the density virial coefficient of helium isotope gas. Background Technology

[0002] Real gases deviate from ideal gas behavior under finite temperature and density conditions; this non-ideality is typically described by the virial equation of state. The density virial coefficient is a crucial fundamental data point for describing the thermodynamic properties of rarefied gases and supporting high-precision gas metrology and temperature measurement. With the increasing demands for precision in fields such as main gas metrology, accurately and continuously obtaining the density virial coefficients of isotopes such as helium over a wide temperature range has become a critical need that urgently needs to be addressed in current engineering metrology and scientific computing.

[0003] To meet these requirements, existing conventional techniques primarily rely on high-precision first-principles calculations or precise experimental measurements to obtain individual discrete virial coefficient data points. In practical engineering applications and metrology software, spline interpolation, piecewise interpolation, or direct table lookup are typically used to numerically reconstruct these discrete, high-precision data points. Furthermore, some existing techniques also attempt to fit the virial coefficient curves of rare gases using traditional single empirical formulas or basic rational functions, aiming to reproduce the functional relationship between the virial coefficient and temperature.

[0004] However, when reconstructing discrete data using interpolation or lookup tables, the propagation of system errors is difficult to strictly control, especially in low-temperature regions, areas with significant curvature changes, and boundary regions. This can easily introduce additional computational errors exceeding the original theoretical uncertainty, making it difficult to guarantee the reliability of continuously retrieved data across the entire temperature range. Furthermore, due to extremely high acquisition costs, the number of high-precision gold-standard data samples is extremely limited. Directly using this sparse discrete data to drive and train complex data-driven models often results in severe sample scarcity, leading to weak generalization ability and poor robustness. In addition, for isotopes such as helium-3 and helium-4, although they belong to the same element, they differ fundamentally in inherent physical properties such as quantum statistical types. Traditional empirical formulas and fitting methods can usually only provide isolated mathematical descriptions for a single gas, making it difficult to accommodate the differences in physical properties of different isotopes under a unified analytical dimension. Therefore, it is impossible to achieve unified and high-precision prediction of the virial coefficients of isotopic gas density. Summary of the Invention

[0005] This invention provides a method and apparatus for predicting the density virial coefficient of helium isotope gases, which solves the shortcomings of existing technologies such as the easy introduction of extra errors by interpolation / table lookup methods, the scarcity of high-precision gold standard samples leading to insufficient generalization of data-driven models, and the difficulty of unifying and compatibility with the quantum statistical differences between helium-3 and helium-4, thus failing to achieve high-precision consistent prediction across isotopes.

[0006] This invention provides a method for predicting the density virial coefficient of helium isotope gas, comprising the following steps.

[0007] To obtain the predicted temperature and prior physical property parameters of helium isotope gas under target operating conditions; The temperature to be predicted and the prior physical property parameters are input into the density virial coefficient prediction model to obtain the target density virial coefficient value output by the density virial coefficient prediction model. The density virial coefficient prediction model is trained based on the sampling temperature samples and prior property parameter samples of helium isotope gas, as well as the labeled density virial coefficient values; the labeled density virial coefficient values ​​are obtained by inputting the sampling temperature samples into the rational function model of the helium isotope gas.

[0008] According to the present invention, a method for predicting the density virial coefficient of a helium isotope gas is provided. The density virial coefficient prediction model is obtained by iteratively executing the following steps until a preset iteration termination condition is met: Obtain the initial density virial coefficient prediction model; The sampled temperature sample and the sampled prior physical property parameters are input into the initial density virial coefficient prediction model to obtain the predicted density virial coefficient value output by the initial density virial coefficient prediction model. Based on the predicted density virial coefficient value and the label density virial coefficient value, the prediction residual is determined; Based on the predicted residuals, the target loss is determined, and the model parameters of the initial density virial coefficient prediction model are updated based on the target loss.

[0009] According to the present invention, a method for predicting the density virial coefficient of a helium isotope gas, wherein determining the target loss based on the prediction residual includes: The sampled temperature is input into the rational function model of the helium isotope gas to obtain the reference uncertainty output by the rational function model. Based on the predicted residual and the reference uncertainty, the standardization error is determined; The extreme value penalty term is determined based on the maximum value in the standardized error. The target loss is determined based on the predicted residual and the extreme value penalty term.

[0010] According to the present invention, a method for predicting the density virial coefficient of a helium isotope gas, wherein determining the target loss based on the prediction residual and the extreme value penalty term includes: Based on the standardization error corresponding to each of the sampled temperature samples, a weighting factor corresponding to each of the sampled temperature samples is determined; wherein, the weighting factor corresponding to each of the sampled temperature samples increases as the standardization error increases; The prediction residuals of each of the sampled temperature samples are weighted based on the weighting factors to obtain the weighted basic loss. The target loss is determined based on the weighted basic loss and the extreme value penalty term.

[0011] According to the method for predicting the density virial coefficient of a helium isotope gas provided by the present invention, the determination step of the rational function model includes: Obtain a standard dataset of the helium isotope gas; the standard dataset includes standard temperature, standard density virial coefficient, and standard uncertainty. Obtain the set of the number of terms in the numerator polynomial and the set of the number of terms in the denominator polynomial; Construct candidate rational function models with each numerator term in the set of terms of the numerator polynomial and each denominator term in the set of terms of the denominator polynomial as structural parameters; Using the standard temperature as the independent variable and the standard density virial coefficient as the target value, parameter fitting is performed on each of the candidate rational function models to obtain each fitted candidate rational function model and the fitted density virial coefficient value output by each fitted candidate rational function model. Based on the fitted density virial coefficients, the standard density virial coefficients, and the standard uncertainty, the evaluation index corresponding to each fitted candidate rational function model is determined. Based on the evaluation index, the candidate rational function models are jointly screened to obtain the rational function model.

[0012] According to the present invention, a method for predicting the density virial coefficient of helium isotope gas is provided, wherein the evaluation index includes root mean square error, Akaike information criterion and maximum standardization error; The step of determining the evaluation index corresponding to each of the fitted candidate rational function models based on the fitted density virial coefficient value, the standard density virial coefficient, and the standard uncertainty includes: The root mean square error is determined based on the fitted density virial coefficient value and the standard density virial coefficient value. The fitting residual is determined based on the difference between the fitted density virial coefficient and the standard density virial coefficient, and the Akaike information criterion is determined based on the fitting residual and the standard uncertainty. The maximum standardized error is determined based on the fitted density virial coefficient, the standard density virial coefficient, and the standard uncertainty.

[0013] According to the present invention, a method for predicting the density virial coefficient of a helium isotope gas, wherein the method involves jointly screening each of the fitted candidate rational function models based on the evaluation index to obtain the rational function model, including: From the candidate rational function models for fitting, a target rational function model is selected where the root mean square error is less than a first preset value and the maximum standardized error is less than a second preset value. The objective rational function model that minimizes the Akaike information criterion value is selected as the rational function model.

[0014] According to the present invention, a method for predicting the density virial coefficient of a helium isotope gas is provided, the method further includes: In the case where there are multiple objective rational function models with the same Akaike information criterion value, the objective rational function model with the smallest sum of the number of terms in the numerator and the number of terms in the denominator is selected as the rational function model.

[0015] According to the present invention, a method for predicting the density virial coefficient of helium isotope gas is provided, wherein the density virial coefficient prediction model includes an input layer, multiple cascaded residual blocks, and an output layer. The input layer is used to map the temperature to be predicted and the prior parameters of the physical properties into an initial feature vector, and input the initial feature vector into the first residual block in the plurality of cascaded residual blocks; The multiple cascaded residual blocks are used to take the previous enhancement feature output by the previous residual block as the input feature of the current residual block to obtain the current enhancement feature output by the current residual block; The enhanced features output by the last residual block are used as fusion features, and the fusion features are input into the output layer to obtain the target density virial coefficient value output by the output layer. Each residual block includes a first fully connected layer, a second fully connected layer, an activation layer, and a cross-layer connection channel.

[0016] The present invention also provides a device for predicting the density virial coefficient of helium isotope gas, comprising the following units: The acquisition unit is used to acquire the predicted temperature and prior physical property parameters of helium isotope gas under the target operating conditions. The input unit is used to input the temperature to be predicted and the prior physical property parameters into the density virial coefficient prediction model to obtain the target density virial coefficient value output by the density virial coefficient prediction model. The density virial coefficient prediction model is trained based on the sampling temperature samples and prior property parameter samples of helium isotope gas, as well as the labeled density virial coefficient values; the labeled density virial coefficient values ​​are obtained by inputting the sampling temperature samples into the rational function model of the helium isotope gas.

[0017] The present invention also provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the density virial coefficient prediction method for any of the helium isotope gases described above.

[0018] The present invention also provides a non-transitory computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the density virial coefficient prediction method for helium isotope gas as described above.

[0019] The present invention also provides a computer program product, including a computer program that, when executed by a processor, implements the density virial coefficient prediction method for any of the helium isotope gases described above.

[0020] The density virial coefficient prediction method and apparatus for helium isotope gases provided by this invention are based on inputting sampled temperature samples into a rational function model to obtain labeled density virial coefficient values. This allows for stable and continuous data reconstruction and large-scale sample amplification through the rational function model, providing sufficient sample support when training the density virial coefficient prediction model. This avoids the problem of easily introducing additional calculation errors in low-temperature regions, regions with significant curvature changes, and boundary regions when reconstructing discrete data using interpolation or table lookup methods, ensuring the reliability of continuously retrieved data across the entire temperature range. Furthermore, by acquiring prior physical property parameters and inputting them into the model, the model can learn and perceive the inherent physical characteristics of different helium isotope gases, thus accommodating the differences in physical properties of different isotope gases under a unified analytical dimension. Ultimately, this achieves unified and high-precision prediction of the density virial coefficients of helium-3, helium-4, and other isotope gases. Attached Figure Description

[0021] To more clearly illustrate the technical solutions in this invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0022] Figure 1 This is a flowchart illustrating the method for predicting the density virial coefficient of helium isotope gas provided by the present invention.

[0023] Figure 2This is a flowchart illustrating the training steps of the density virial coefficient prediction model provided by the present invention.

[0024] Figure 3 This is a flowchart illustrating the process of determining a rational function model provided by the present invention.

[0025] Figure 4 This is a schematic diagram of the density virial coefficient prediction device for helium isotope gas provided by the present invention.

[0026] Figure 5 This is a schematic diagram of the structure of the electronic device provided by the present invention. Detailed Implementation

[0027] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0028] In related technologies, real gases deviate from ideal gas behavior under finite temperature and density conditions; this non-ideality is typically described by the virial equation of state. For rarefied gases, the virial equation of state can be written as: ; in, For pressure, molar density, The molar gas constant, For sampling temperature samples, The second density virial coefficient, The density virial coefficient is the third density virial coefficient, which is only related to temperature. The virial equation of state, starting from microscopic intermolecular interactions, establishes the connection between molecular potential functions and macroscopic thermodynamic quantities, and is a crucial theoretical foundation for describing the thermodynamic properties of rarefied gases and supporting high-precision gas metrology and temperature measurement. In the revised International System of Units (SI), after Kelvin was defined by fixing the Boltzmann constant, the dependence of the master thermometer method based on gas thermophysical properties on high-precision density virial coefficients significantly increased. Therefore, accurately obtaining the density virial coefficient of inert gases, especially helium, is of great significance for master gas metrology.

[0029] This invention provides a method for predicting the density virial coefficient of helium isotope gas. The subject executing this method can be an electronic device with data processing capabilities, such as a server, workstation, or personal computer, or a property calculation module integrated into a temperature measuring instrument, state equation calculation software, or high-precision property database system. This invention does not limit the specific implementation of this method.

[0030] Figure 1 This is a flowchart illustrating the method for predicting the density virial coefficient of helium isotope gas provided by the present invention, as shown below. Figure 1 As shown, the method includes the following: Step 110: Obtain the predicted temperature and prior physical property parameters of the helium isotope gas under the target operating conditions.

[0031] Specifically, firstly, the a priori parameters of the temperature and physical properties of the helium isotope gas under the target operating conditions are obtained.

[0032] Among them, helium isotope gas refers to gas composed of isotopes of the element helium, specifically including helium-3 gas ( 3 He) and helium-4 gas ( 4 At least one of He). It should be noted that although helium-3 and helium-4 both belong to the element helium, their inherent physical properties, such as molar mass and nuclear spin, are different. For example, helium-3 is a fermion while helium-4 is a boson. The different quantum statistical types result in different changes in their density virial coefficients with temperature. Therefore, traditional single empirical formulas are still insufficient for unified modeling of isotopic gases such as helium-3 and helium-4.

[0033] Here, the target operating condition refers to the working conditions corresponding to the actual application scenario where the density virial coefficient of helium isotope gas needs to be predicted. For example, the target operating condition could be the main gas metering condition based on a revised fixed value of the Boltzmann constant under the International System of Units (SI), the acoustic gas thermometer condition, the dielectric constant gas thermometer condition, the refractive index gas thermometer condition, or the operating conditions of equation-of-state modeling, high-precision property database construction, and the property calculation modules in related instrument software. Under different target operating conditions, the temperature conditions of the helium isotope gas are different, and the required predicted density virial coefficient value will also be different.

[0034] Here, the temperature to be predicted refers to the thermodynamic temperature of the helium isotope gas under the target operating conditions, i.e., the temperature value corresponding to the density virial coefficient that needs to be predicted. The temperature to be predicted can be a single temperature point or a temperature sequence consisting of multiple temperature points within a wide temperature range. The temperature to be predicted can be obtained by actually measuring the helium isotope gas under the target operating conditions using a temperature measuring instrument, or it can be a target temperature value input by the user and issued by a host computer or metering software, or it can be read from a pre-stored temperature configuration file. This embodiment of the invention does not limit the specific methods used.

[0035] The a priori physical properties are used to reflect the inherent physical properties of helium isotope gases. In a specific embodiment, the a priori physical properties may include at least one of atomic number, molar mass, polarizability, ionization energy, potential well depth, collision diameter, dispersion coefficient, critical temperature, critical pressure, eccentricity factor, nuclear spin, and Bose / Fermi statistics type.

[0036] It should be noted that the prior physical property parameters can be used to characterize the gas categories of helium-3 and helium-4, that is, to distinguish the types of helium isotope gas to be predicted. On the other hand, they can be used as feature extensions, together with the temperature to be predicted, to form input features to support the collaborative prediction of density virial coefficients of different helium isotope gases in a unified framework, or to impose physical constraints on the prediction process.

[0037] Since the prior physical parameters are inherent physical properties of helium isotope gases, they can be obtained and stored in advance from property data manuals, verified literature data, or pre-built property databases, and can be directly read when performing predictions.

[0038] Step 120: Input the temperature to be predicted and the prior physical property parameters into the density virial coefficient prediction model to obtain the target density virial coefficient value output by the density virial coefficient prediction model. The density virial coefficient prediction model is trained based on the sampling temperature samples and prior property parameter samples of helium isotope gas, as well as the labeled density virial coefficient values; the labeled density virial coefficient values ​​are obtained by inputting the sampling temperature samples into the rational function model of the helium isotope gas.

[0039] Specifically, after obtaining the prior parameters of the temperature and physical properties to be predicted, the prior parameters of the temperature and physical properties to be predicted can be input into the density virial coefficient prediction model to obtain the target density virial coefficient value output by the density virial coefficient prediction model.

[0040] The density virial coefficient prediction model is a pre-trained machine learning model that takes prior parameters of temperature and physical properties as input and outputs density virial coefficient values. In specific implementation, the density virial coefficient prediction model can be implemented using a neural network model, such as a fully connected neural network model, a convolutional neural network model, or a neural network model with residual connections, etc. The embodiments of this invention do not specifically limit this.

[0041] When inputting the temperature to be predicted and prior physical property parameters into the density virial coefficient prediction model, the temperature to be predicted and the prior physical property parameters can be concatenated to form an input feature vector. Optionally, to enhance the ability of the density virial coefficient prediction model to represent the variation law of density virial coefficient over a wide temperature range, the temperature to be predicted can be transformed before input. For example, the logarithm or inverse temperature value of the temperature to be predicted can be used as temperature input features, so that the density virial coefficient prediction model learns the law of density virial coefficient variation with temperature, and is not affected by the difference in numerical magnitude between different gases.

[0042] Here, the target density virial coefficient value refers to the density virial coefficient prediction result output by the density virial coefficient prediction model for helium isotope gas at the temperature to be predicted. It should be noted that the density virial coefficient in this embodiment of the invention can be the second density virial coefficient in the virial equation of state. It can also be the third density virial coefficient. Higher-order density virial coefficients are also discussed. This embodiment uses the second density virial coefficient as an example, but this does not constitute a limitation on the scope of protection of this invention. The obtained target density virial coefficient value can be directly provided to downstream applications such as main gas metering, acoustic gas thermometers, and equation of state calculations.

[0043] The density virial coefficient prediction model is trained based on the sampling temperature samples and prior property parameter samples of helium isotope gas, as well as the labeled density virial coefficient values. The labeled density virial coefficient values ​​are obtained by inputting the sampling temperature samples into the rational function model of helium isotope gas.

[0044] Here, the sampled temperature sample refers to the temperature value obtained by sampling within a preset temperature range. Specifically, high-density sampling can be performed on helium-3 and helium-4 gases within the preset temperature range, such as a wide temperature range covering the low-temperature region to the high-temperature region. The sampling method can be uniform sampling at equal intervals, logarithmic scaling sampling, or random sampling, etc. Optionally, in order to provide sufficient sample support for training the density virial coefficient prediction model, the number of generated sampled temperature samples can be no less than 20,000 sample points.

[0045] Here, the prior parameter sample refers to the prior parameters of the helium isotope gas corresponding to each sampling temperature sample. Its meaning is consistent with the aforementioned prior parameters of the physical properties. For example, it includes at least one of the following: atomic number, molar mass, polarizability, ionization energy, potential well depth, collision diameter, dispersion coefficient, critical temperature, critical pressure, eccentricity factor, nuclear spin, and Bose / Fermi statistics type, which are used to identify the isotope category and provide physical property information in the training samples.

[0046] The rational function model is a pre-constructed mathematical model in the form of a rational function, with temperature as the independent variable and the density virial coefficient as the dependent variable. Compared to ordinary polynomials, rational functions have advantages in fitting over wide intervals, describing curvature changes, and controlling boundary behavior, and can more stably reproduce the functional relationship between the density virial coefficient and temperature. In a specific embodiment, for each helium isotope gas, the rational function model for the second density virial coefficient is... It can be represented as: ; in, Represents the basic functions. The sampled temperature can be represented using functions such as logarithmic, exponential, sine, or cosine functions. and The definition is as follows: ; in, The coefficients of the numerator polynomial, The coefficients of the denominator polynomial, The number of numerator terms. Let be the number of terms in the denominator, and The constant term is fixed at 1 to ensure parameter identifiability and to prevent non-physical singularities from appearing near the origin.

[0047] Here, the coefficients of the rational function model can be determined in advance based on high-precision experimental measurement data of helium isotope gas, verified first-principles calculation data, or a combination of both.

[0048] For example, the second density virial coefficient of helium-3 gas can be modeled using a rational function with the fundamental function ln(T): (1) Unfolding format: ; in, This represents a sampled temperature; its corresponding coefficients can be, for example: , Similarly, for helium-4 gas, a rational function model with the same structure but different highest powers and coefficients can be used. It is only necessary to determine the corresponding number of numerator terms, number of denominator terms, and coefficients based on the high-precision data of helium-4 gas.

[0049] For example, the second density virial coefficient of helium-4 gas can be modeled using a rational function with the fundamental function ln(T): (2) Unfolding format: ; in, This represents a sampled temperature; its corresponding coefficients can be, for example: , .

[0050] Here, the label density virial coefficient refers to the density virial coefficient output value corresponding to each sampled temperature sample, calculated by substituting each sampled temperature into the rational function model of helium isotope gas. The label density virial coefficient value is used as supervision information during model training.

[0051] Understandably, due to the high cost and limited sample size of standard high-precision density virial coefficient data, it is difficult to support robust training of machine learning models by directly using limited discrete gold standard data. However, the embodiments of this invention generate labels for sampled temperature samples within a preset temperature range by using a pre-constructed rational function model, which essentially expands the training samples. That is, it uses the rational function model to generate expanded training samples. Each expanded training sample may include a sampled temperature sample, a corresponding prior physical property parameter sample, and a label density virial coefficient value generated by the rational function model. This effectively expands the training set under the condition of small sample standard data, and the expanded samples inherit the high-precision reconstruction capability of the rational function model for standard data, maintaining physical consistency.

[0052] During training, sampled temperature samples and prior physical property parameter samples can be used as inputs to the density virial coefficient prediction model. The density virial coefficient prediction results output by the model are compared with the labeled density virial coefficient values ​​to construct a loss function. Based on this loss function, a gradient descent-type optimization algorithm is used to iteratively update the parameters of the density virial coefficient prediction model until the model converges, thus obtaining the trained density virial coefficient prediction model. Since the training samples for helium-3 and helium-4 gases each carry their own prior physical property parameter samples, the trained density virial coefficient prediction model can achieve collaborative prediction of the density virial coefficients of helium-3 and helium-4 gases within a unified framework, resulting in a density virial coefficient prediction model capable of unified prediction for one or more target gases.

[0053] The method provided in this invention is based on inputting sampled temperature samples into a rational function model to obtain the label density virial coefficient value. This allows for stable and continuous data reconstruction and large-scale sample amplification through the rational function model, providing sufficient sample support when training the density virial coefficient prediction model. This avoids the problem of easily introducing additional calculation errors in low-temperature regions, regions with significant curvature changes, and boundary regions when reconstructing discrete data using interpolation or table lookup methods. It ensures the reliability of continuously retrieved data across the entire temperature range and also solves the problem of severe sample scarcity, weak generalization ability, and poor robustness in data-driven models due to the extremely limited availability of high-precision standard data samples. Furthermore, by acquiring prior physical property parameters and inputting them into the model, the model can learn and perceive the inherent physical characteristics of different helium isotope gases. This allows for compatibility with the differences in physical properties of different isotope gases under a unified analytical dimension, breaking the limitation of traditional empirical formulas and fitting methods that can only provide isolated mathematical descriptions for single gases. Ultimately, this achieves unified and high-precision prediction of the density virial coefficients of helium-3, helium-4, and other isotope gases.

[0054] Based on the above embodiments, the density virial coefficient prediction model is obtained by iteratively executing the following steps until a preset iteration termination condition is met: Step 21: Obtain the initial density virial coefficient prediction model; Step 22: Input the sampled temperature sample and the sampled prior physical property parameters into the initial density virial coefficient prediction model to obtain the predicted density virial coefficient value output by the initial density virial coefficient prediction model; Step 23: Determine the prediction residual based on the predicted density virial coefficient value and the label density virial coefficient value; Step 24: Based on the predicted residual, determine the target loss, and update the model parameters of the initial density virial coefficient prediction model based on the target loss.

[0055] Specifically, Figure 2 This is a flowchart illustrating the training steps of the density virial coefficient prediction model provided by the present invention, as shown below. Figure 2 As shown, the density virial coefficient prediction model is obtained by iteratively executing the following steps until a preset iteration termination condition is met: First, we can obtain the initial density virial coefficient prediction model, which refers to a neural network model that has not yet been trained and whose internal network weights, biases, and other model parameters are in a randomly initialized or pre-trained state.

[0056] Here, the preset iteration termination condition may be one or more of the following: the total number of training rounds reaches a preset upper limit, the target loss converges to below a preset threshold, the target loss no longer decreases significantly for several consecutive rounds, or the performance index of the density virial coefficient prediction model on the validation set no longer shows a significant improvement. This embodiment of the invention does not limit this.

[0057] Then, the sampled temperature samples and prior physical property parameter samples can be input into the initial density virial coefficient prediction model to obtain the predicted density virial coefficient value output by the initial density virial coefficient prediction model. The predicted density virial coefficient value refers to the estimated density virial coefficient output by the initial density virial coefficient prediction model in the current training iteration, based on the input sampled temperature samples and prior physical property parameter samples.

[0058] Furthermore, the prediction residuals can be determined based on the predicted density virial coefficients and the label density virial coefficients. The prediction residuals refer to the initial loss term calculated solely from the numerical deviation between the predicted density virial coefficients and the label density virial coefficients, and are used to measure the overall average error of the model's predictions.

[0059] To mitigate the problem of excessively large standardization errors for individual samples, this invention employs a composite loss consisting of prediction residuals and an extremum penalty term, ranging from mean squared error to extremum perception. In traditional standard regression loss, mean squared error (MSE) is typically used directly. The formula for mean squared error is: ; in, This represents the mean square error. For the first The predicted density virial coefficient value for each sample. For the first The label density virial coefficient value of each sample. This represents the total number of samples.

[0060] However, this embodiment of the invention considers that if only the average error is minimized, individual points with extremely large errors will not be distinguished, and the problem of large errors in individual samples cannot be addressed. Therefore, in a specific embodiment, considering that directly using the mean squared error may not distinguish individual points with extremely large deviations, the prediction residual can be calculated using Huber loss. Huber loss uses a squared penalty for small errors and a linear penalty for large errors, which can suppress the excessive pull of outliers on the gradient and is more robust than the traditional MSE. The formula for the prediction residual can be expressed as: ; in, Indicates the predicted residual; This represents the set of samples in the current training batch. Indicates the total number of samples; , For the first The predicted density virial coefficient value for each sample. For the first The label density virial coefficient value of each sample. This is the preset smoothing threshold parameter.

[0061] Finally, the target loss can be determined based on the prediction residuals, and the model parameters of the initial density virial coefficient prediction model can be updated based on the target loss. The target loss reflects the final optimization objective used in this iteration to guide the model parameter updates during backpropagation. In the basic implementation scenario, the target loss can be directly equivalent to the basic prediction loss, or a simple scaling of the basic prediction loss.

[0062] Here, an optimizer, such as the AdamW optimizer, is used to calculate the gradient of each model parameter relative to the initial density virial coefficient prediction model based on the target loss, and the model parameters are updated along the gradient descent direction. This process is iteratively executed for different batches of samples until a preset iteration termination condition is met, ultimately resulting in a converged density virial coefficient prediction model.

[0063] One approach is to use a learning rate scheduling strategy based on the maximum standardized error of the validation set, where the initial learning rate can be set to... The weight decays to The learning rate scheduler uses ReduceLROnPlateau, and the monitoring metric is the maximum standardized error on the validation set. If the monitoring metric does not improve within a preset number of rounds, the learning rate is automatically reduced. Furthermore, to avoid gradient explosion caused by extreme value penalties, gradients are pruned during training.

[0064] The method provided in this invention enables the density virial coefficient prediction model to robustly learn the intrinsic mapping law of the density virial coefficient under the drive of a large number of amplified samples, thus ensuring the generalization ability and prediction accuracy of the final prediction model.

[0065] Based on the above embodiments, step 24, determining the target loss based on the predicted residual, includes: Step 241: Input the sampled temperature sample into the rational function model of the helium isotope gas to obtain the reference uncertainty output by the rational function model; Step 242: Determine the standardized error based on the predicted residual and the reference uncertainty; Step 243: Determine the extreme value penalty term based on the maximum value in the standardized error; Step 244: Determine the target loss based on the predicted residual and the extreme value penalty term.

[0066] Specifically, firstly, the sampled temperature can be input into a rational function model of the helium isotope gas to obtain the reference uncertainty output by the rational function model. This rational function model includes not only the rational formula model used to generate the density virial coefficients, but also a rational function model of the uncertainty with respect to the sampled temperature, pre-established based on high-precision standard uncertainty data.

[0067] Here, reference uncertainty refers to the corresponding uncertainty limit generated after substituting the sampled temperature sample into the rational function model of uncertainty with respect to temperature.

[0068] Then, the standardized error can be determined based on the predicted residuals and the reference uncertainty. The standardized error reflects the relative deviation of the predicted residuals from the reference uncertainty scale.

[0069] Here, the formula for standardized error is as follows: ; in, Indicates the first The standardized error corresponding to each sample; Indicates the first The predicted density virial coefficient values ​​for each sample; For the first The label density virial coefficient value of each sample; This indicates the reference uncertainty.

[0070] Furthermore, an extremum penalty term can be determined based on the maximum value of the standardized errors. This extremum penalty term applies an additional penalty constraint to the worst-fitting sample in the current training batch. Since minimizing only the average error cannot constrain the extreme errors of individual samples, the conventional loss may still be small when a small number of samples have standardized errors far exceeding the limit (e.g., >1 / 3). Therefore, an extremum penalty term is constructed based on the maximum value of the standardized errors. Specifically, the extremum penalty term is implemented using the square of the maximum standardized error, and its formula can be expressed as: ; ; in, This represents the extreme value penalty term. This represents the set of standardized errors of all samples in the current training batch; This represents the square of the Chebyshev norm (infinite norm), which characterizes the square of the maximum value of the standardized error set. It is the weighting coefficient of the maximum error of a single sample; Indicates the first The standardized error corresponding to each sample.

[0071] Understandably, using the squared form not only smooths the gradient field but also strongly compels the density virial coefficient prediction model to focus on the most difficult-to-fit samples.

[0072] Finally, based on the prediction residuals and the extreme value penalty term, the target loss is determined, and the formula for the target loss is as follows: ; in, Indicates target loss. Indicates the predicted residual. This indicates the extreme value penalty term.

[0073] The method provided in this invention combines the predicted residual and the reference uncertainty generated based on the rational function model to calculate the standardized error, and extracts the maximum standardized error to construct an extreme value penalty term. This term is then fused with the predicted residual to obtain the target loss, thereby forming an extreme value-aware collaborative loss function that includes a worst-case penalty term. This allows the density virial coefficient prediction model to focus on the worst-case sample while optimizing the overall average error, effectively suppressing the extreme value error of outlier samples. The maximum standardized error is strictly controlled within the requirements of econometric applications, overcoming the limitation of traditional loss functions that cannot constrain individual difficult samples.

[0074] Based on the above embodiments, step 244 includes: Step 2441: Based on the standardization error corresponding to each of the sampled temperature samples, determine the weighting factor corresponding to each of the sampled temperature samples; wherein, the weighting factor corresponding to each of the sampled temperature samples increases as the standardization error increases; Step 2442: Based on the weighting factor, the prediction residuals of each of the sampled temperature samples are weighted to obtain the weighted basic loss; Step 2443: Determine the target loss based on the weighted basic loss and the extreme value penalty term.

[0075] Specifically, firstly, the weighting factor corresponding to each sampled temperature sample can be determined based on the standardization error corresponding to each sampled temperature sample; wherein, the weighting factor corresponding to each sampled temperature sample increases as the standardization error increases.

[0076] The weighting factor reflects the information weights assigned to samples of different difficulty levels when calculating the weighted basic loss. Since the density virial coefficient prediction model fits each sample at different paces during training, the standardization error of some samples may be larger; therefore, these samples are assigned a larger weighting factor.

[0077] Here, weighted basic loss refers to the loss value obtained by weighting and summing the corresponding prediction residuals according to the weight factors of each sample instead of using a simple arithmetic mean when calculating the prediction residuals of the batch.

[0078] After obtaining the weighted basic loss, the weighted basic loss is added to the extreme value penalty term to obtain the final target loss. It is understandable that by introducing this uncertainty weighting and single-sample error supervision, dynamic weighting can be applied to samples with large standardized errors within a batch, with the maximum single-sample error weighting coefficient... The ratio used to balance the weighted basic loss and the extreme value penalty term globally.

[0079] The method provided in this invention enables dynamic adaptive attention to samples with large standardization errors, preventing gradient explosion caused by excessively large penalty terms in the early stages of training, making the model training process more stable, and further enhancing the fitting ability and stability of individual difficult samples.

[0080] Based on the above embodiments, the steps for determining the rational function model include: Step 31: Obtain the standard dataset of the helium isotope gas; the standard dataset includes standard temperature, standard density virial coefficient, and standard uncertainty; Step 32: Obtain the set of terms in the numerator polynomial and the set of terms in the denominator polynomial; Step 33: Construct candidate rational function models with each numerator term in the set of terms in the numerator polynomial and each denominator term in the set of terms in the denominator polynomial as structural parameters; Step 34: Using the standard temperature as the independent variable and the standard density virial coefficient as the target value, perform parameter fitting on each of the candidate rational function models to obtain each fitted candidate rational function model and the fitted density virial coefficient value output by each fitted candidate rational function model. Step 35: Based on the fitted density virial coefficient value, the standard density virial coefficient, and the standard uncertainty, determine the evaluation index corresponding to each fitted candidate rational function model; Step 36: Based on the evaluation index, the candidate rational function models are jointly screened to obtain the rational function model.

[0081] Specifically, Figure 3 This is a flowchart illustrating the process of determining a rational function model provided by the present invention, as shown below. Figure 3 As shown, in order to provide high-quality amplified training samples for the density virial coefficient prediction model, a high-precision rational function model needs to be established in advance. Specifically, the steps for determining the rational function model include: First, a standard dataset of helium isotope gases can be obtained, which includes standard temperature, standard density virial coefficient, and standard uncertainty.

[0082] Here, a standard dataset refers to baseline thermophysical property data of helium isotope gases under known, high-precision conditions. In practical applications, standard datasets are typically derived from high-precision experimental measurements, rigorously validated first-principles calculations, or a combination of both.

[0083] Here, standard temperature refers to the discrete temperature node corresponding to the measurement or calculation in the standard dataset. Standard density virial coefficient refers to the highly accurate measured or theoretically calculated density virial coefficient exhibited by helium isotope gas at the corresponding standard temperature. Standard uncertainty refers to the theoretical or experimental uncertainty that matches the standard density virial coefficient; standard uncertainty characterizes the confidence interval and error tolerance of the reference data.

[0084] Then, we can obtain the sets of polynomial terms in the numerator and the denominator. These sets refer to the traversal range of the polynomial order (number of polynomial terms) contained in the numerator and denominator when constructing a rational function. For example, the number of numerator terms *m* can be a set of integers from 1 to a preset maximum value, and the number of denominator terms *n* can also be a set of integers.

[0085] Furthermore, candidate rational function models are constructed with each numerator term in the set of numerator polynomial terms and each denominator term in the set of denominator polynomial terms as structural parameters.

[0086] Here, structural parameters refer to the information that determines the mathematical framework of a specific rational function expression. Specifically, a set of structural parameters is formed by selecting a specific number of numerator terms m from the set of numerator polynomial terms and a specific number of denominator terms n from the set of denominator polynomial terms.

[0087] Here, each candidate rational function model refers to all possible rational function expressions whose internal polynomial coefficients are yet to be determined, generated by traversing the different combinations of structural parameters mentioned above. By traversing the number of numerator and denominator terms, multiple sets of candidate rational function models with different complexities can be constructed.

[0088] After obtaining the candidate rational function models, the parameters of each candidate rational function model can be fitted using standard temperature as the independent variable and standard density virial coefficients as the objective value. This yields the fitted candidate rational function models and the fitted density virial coefficients output by each model. In a specific embodiment, the fitting process can employ a nonlinear least squares optimization algorithm, such as the Levenberg-Marquardt method, to find the optimal combination of coefficients that minimizes the sum of squared residuals by solving the following optimization problem. : ; in, Denotes the polynomial coefficients to be optimized, i.e., in formula (1) and , or in formula (2) and ; For the first The label density virial coefficient value of each sample. This indicates that the candidate rational function model is given by parameters. The predicted value is below.

[0089] Here, each fitted candidate rational function model refers to the model entity whose internal polynomial coefficients have been optimized after being solved by the above optimization algorithm. The fitted density virial coefficient value refers to the predicted value output by these fitted candidate rational function models when they are recalculated for the standard temperature.

[0090] Furthermore, based on the fitted density virial coefficients, standard density virial coefficients, and standard uncertainty, evaluation metrics are determined for each candidate rational function model. These evaluation metrics reflect the goodness of fit and usability of each candidate rational function model across multiple dimensions. Based on the fitted density virial coefficients, standard density virial coefficients, and standard uncertainty obtained above, a set of quantitative indicators can be calculated. Finally, based on these evaluation metrics, all candidate rational function models are jointly cross-selected, eliminating unsuitable models, and ultimately determining the optimal rational function model.

[0091] The method provided in this invention obtains a standard dataset including temperature, density virial coefficients, and uncertainty, and constructs candidate models by traversing the number of numerator and denominator terms for least squares parameter fitting. Finally, it performs joint screening of the fitted models based on multi-dimensional evaluation indicators, thereby realizing automatic order search for rational function models. This avoids the drawbacks of relying on manual experience to determine the function order and manual trial and error in traditional methods. While ensuring the advantages of curvature change description and boundary behavior control, it significantly improves modeling efficiency and the scientific nature of the model structure.

[0092] Based on the above embodiments, the evaluation metrics include root mean square error, Akaike information criterion, and maximum standardized error; Step 35 includes: Step 351: Determine the root mean square error based on the fitted density virial coefficient value and the standard density virial coefficient value; Step 352: Determine the fitting residual based on the difference between the fitted density virial coefficient value and the standard density virial coefficient value, and determine the Akaike information criterion based on the fitting residual and the standard uncertainty; Step 353: Determine the maximum standardized error based on the fitted density virial coefficient value, the standard density virial coefficient, and the standard uncertainty.

[0093] Specifically, the evaluation metrics include root mean square error (RMSE), the Akaike information criterion, and the maximum standardized error. RMSE reflects the average absolute deviation of each candidate rational function model across all standard temperature sample points. RMSE focuses on evaluating the overall average fitting accuracy of the model; a smaller RMSE indicates a better overall macroscopic fit of the candidate rational function model to the standard dataset.

[0094] Here, the fitting residuals can be determined based on the fitted density virial coefficients and the standard density virial coefficients, and the root mean square error can be calculated based on the fitting residuals.

[0095] Here, the Akaike Information Criterion (AIC) is used to reflect the balance between the goodness of fit and the complexity of the candidate rational function model. Since a higher order of a rational function model generally results in a smaller root mean square error, it is also more prone to overfitting, such as causing severe oscillations between sample points. The AIC criterion, by penalizing the number of structural parameters (the sum of the number of numerator and denominator terms) in the candidate rational function model, can find the optimal solution between fitting accuracy and model simplicity, avoiding parameter redundancy.

[0096] Here, the fitting residual can be determined first by the difference between the fitted density virial coefficient and the standard density virial coefficient. Then, the fitting residual of each sample point is divided by the standard uncertainty corresponding to that sample point to obtain the normalized relative deviation. The natural logarithm of the sum of squares of the normalized relative deviations of all sample points is then multiplied by the total number of samples. Finally, twice the number of parameters to be estimated in the fitted candidate rational function model is added as a complexity penalty term to obtain the Akaike information criterion.

[0097] Here, the maximum standardized error is used to reflect the worst-case scenario where the predicted extreme value of a single sample deviates from the theoretically permissible error range. Traditional fitting methods often only focus on the average fitting accuracy, while rarely considering the extreme value control of single-sample errors, leading to extremely large interpolation or fitting errors for individual sample points. This invention introduces the maximum standardized error of a single sample, with uncertainty as the normalization scale, as the core evaluation index. In a specific embodiment, the maximum standardized error... The calculation is defined as follows: ; in, Represents a function that takes a maximum value. To fit the output of the candidate rational function model, the first... The fitted density virial coefficient values ​​for each sample point For the first The standard density virial coefficients of each sample point For the first The standard uncertainty of a sample point.

[0098] The method provided in this invention uses root mean square error, Akaike information criterion, and maximum standardized error as evaluation indicators. This not only takes into account the global average fitting error and model complexity during the model selection stage, but more importantly, it introduces a worst-case error assessment with uncertainty as a constraint. This ensures that the rational function model selected in the end can strictly reduce the worst-case error to within the tolerance range of high-precision metrology applications, significantly improving the reliability of the model reconstruction data at the physical application level.

[0099] Based on the above embodiments, step 36 includes: Step 361: From each of the fitted candidate rational function models, select the target rational function model whose root mean square error is less than a first preset value and whose maximum standardized error is less than a second preset value. Step 362: Select the objective rational function model with the smallest Akaike information criterion value as the rational function model.

[0100] Specifically, firstly, from each candidate fitted rational function model, a target rational function model with a root mean square error less than a first preset value and a maximum standardized error less than a second preset value can be selected.

[0101] Here, the first preset value refers to the maximum tolerance limit for the global average fitting deviation. In a specific embodiment, the first preset value can be set to 0.003 or 0.01, etc., to ensure that the root mean square error is controlled at an extremely low level.

[0102] Here, the second preset value refers to the mandatory constraint threshold for the deviation of the predicted extreme value of a single sample. In a specific embodiment, the second preset value can be set to 1 / 3, and in engineering implementation, it can even be further preferred to be less than 1 / 6, that is, requiring that the absolute fitting error of all sample points must not exceed one-third of its own standard uncertainty.

[0103] Here, the objective rational function model is used to reflect the set of all qualified candidates that meet the basic requirements of metrological applications, initially screened under accuracy constraints and extreme value control constraints. After filtering with the preconditions RMSE < 0.003 and MaxNormErr < 1 / 3, the obtained objective rational function models all have high-precision reconstruction capabilities.

[0104] Based on this, since there may be multiple objective rational function models that satisfy the above threshold, the objective rational function model with the smallest Akaike Information Criterion value is selected as the rational function model in order to seek the optimal solution in terms of accuracy and complexity.

[0105] The method provided in this invention sets hard constraints of first and second preset values ​​to screen out target rational function models, and further selects the model with the smallest Akaike information criterion. This can minimize the complexity of the model while satisfying global high accuracy and strict extreme value error control, thereby making the finally determined rational function model easy to deploy and run stably. It solves the problem that conventional fitting methods are difficult to balance average fitting error, model complexity and maximum standardization error of a single sample. At the same time, it avoids the problem that when using interpolation or table lookup methods to reconstruct discrete data, additional calculation errors exceeding the original theoretical uncertainty are easily introduced in areas with obvious curvature changes and boundary regions.

[0106] Based on the above embodiments, the method further includes: In the case where there are multiple objective rational function models with the same Akaike information criterion value, the objective rational function model with the smallest sum of the number of terms in the numerator and the number of terms in the denominator is selected as the rational function model.

[0107] Specifically, when multiple objective rational function models with the same Akaike Information Criterion value exist, the objective rational function model with the smallest sum of the number of terms in the numerator and the number of terms in the denominator is selected as the rational function model. In other words, the objective rational function model with the lowest order is preferentially selected as the final rational function model.

[0108] Based on the above embodiments, the density virial coefficient prediction model includes an input layer, multiple cascaded residual blocks, and an output layer; The input layer is used to map the temperature to be predicted and the prior parameters of the physical properties into an initial feature vector, and input the initial feature vector into the first residual block in the plurality of cascaded residual blocks; The multiple cascaded residual blocks are used to take the previous enhancement feature output by the previous residual block as the input feature of the current residual block to obtain the current enhancement feature output by the current residual block; The enhanced features output by the last residual block are used as fusion features, and the fusion features are input into the output layer to obtain the target density virial coefficient value output by the output layer. Each residual block includes a first fully connected layer, a second fully connected layer, an activation layer, and a cross-layer connection channel.

[0109] Specifically, the density virial coefficient prediction model includes an input layer, multiple cascaded residual blocks, and an output layer. The input layer maps the predicted temperature and physical property prior parameters to an initial feature vector, which is then input into the first residual block of the cascaded residual blocks. The cascaded residual blocks use the previous enhancement feature output from the previous residual block as the input feature of the current residual block, resulting in the current enhancement feature output by the current residual block. Finally, the enhancement feature output from the last residual block is used as a fusion feature, which is input into the output layer to obtain the target density virial coefficient value.

[0110] Here, multiple cascaded residual blocks refer to the core backbone structure within the density virial coefficient prediction model that performs deep nonlinear feature extraction. Cascading means that multiple residual blocks are connected end-to-end and work sequentially. Based on data complexity and computational resources, the number of residual blocks can be preset.

[0111] In a cascaded structure, the previous residual block refers to the network module preceding the current residual block in the data flow sequence. The previous augmented feature refers to the high-dimensional feature tensor output by the previous residual block after processing. The current augmented feature refers to the feature tensor output by the current residual block after further extraction of nonlinear mapping relationships after combining with the previous augmented feature.

[0112] Each residual block is processed based on the previous enhanced feature, until the last residual block outputs the final fused feature.

[0113] Here, each residual block may include a first fully connected layer, a second fully connected layer, an activation layer, and cross-layer connection channels. Regarding the internal structure of each residual block, the first and second fully connected layers refer to network layers that perform linear affine transformations on the feature vectors to discover linear combinations between features. The activation layer is a module that introduces a non-linear mapping after the fully connected layers. In practical implementations, the activation layer can use the SiLU (Sigmoid Linear Unit) activation function, which is not only smooth but also maintains good gradient flow in deep networks.

[0114] Among them, the Skip Connection is used to reflect a fast data path that directly skips the intermediate fully connected layer and activation layer to merge and add the original input features of the current residual block with the output result after activation of the second fully connected layer.

[0115] The density virial coefficient prediction device for helium isotope gas provided by the present invention will be described below. The density virial coefficient prediction device for helium isotope gas described below can be referred to in correspondence with the density virial coefficient prediction method for helium isotope gas described above.

[0116] Based on any of the above embodiments, the present invention provides a device for predicting the density virial coefficient of helium isotope gas. Figure 4 This is a schematic diagram of the density virial coefficient prediction device for helium isotope gas provided by the present invention, as shown in the figure. Figure 4 As shown, the device includes: Acquisition unit 410 is used to acquire the predicted temperature and prior physical property parameters of helium isotope gas under target operating conditions; Input unit 420 is used to input the temperature to be predicted and the prior parameters of the physical properties into the density virial coefficient prediction model to obtain the target density virial coefficient value output by the density virial coefficient prediction model. The density virial coefficient prediction model is trained based on the sampling temperature samples and prior property parameter samples of helium isotope gas, as well as the labeled density virial coefficient values; the labeled density virial coefficient values ​​are obtained by inputting the sampling temperature samples into the rational function model of the helium isotope gas.

[0117] The apparatus provided in this invention obtains the label density virial coefficient value by inputting the sampled temperature sample into a rational function model. This allows for stable and continuous data reconstruction and large-scale sample amplification through the rational function model, providing sufficient sample support when training the density virial coefficient prediction model. This avoids the problem of easily introducing additional calculation errors in low-temperature regions, regions with significant curvature changes, and boundary regions when reconstructing discrete data using interpolation or table lookup methods. It ensures the reliability of continuously retrieved data across the entire temperature range and also solves the problem of severe sample scarcity, weak generalization ability, and poor robustness in data-driven models due to the extremely limited availability of high-precision standard data samples. Furthermore, by acquiring prior physical property parameters and inputting them into the model, the model can learn and perceive the inherent physical characteristics of different helium isotope gases. This allows for compatibility with the differences in physical properties of different isotope gases under a unified analytical dimension, breaking the limitation of traditional empirical formulas and fitting methods that can only provide isolated mathematical descriptions for single gases. Ultimately, this achieves unified and high-precision prediction of the density virial coefficients of isotope gases such as helium-3 and helium-4.

[0118] Based on any of the above embodiments, a training unit is further included, wherein the training unit specifically includes: Obtain model units to obtain the initial density virial coefficient prediction model; The prediction unit is used to input the sampled temperature sample and the sampled prior physical property parameters into the initial density virial coefficient prediction model to obtain the predicted density virial coefficient value output by the initial density virial coefficient prediction model. A prediction residual unit is defined to determine the prediction residual based on the prediction density virial coefficient value and the label density virial coefficient value. A target loss unit is defined to determine the target loss based on the prediction residual and to update the model parameters of the initial density virial coefficient prediction model based on the target loss.

[0119] Based on any of the above embodiments, the target loss determination unit specifically includes: A reference uncertainty unit is determined, which is used to input the sampled temperature sample into the rational function model of the helium isotope gas to obtain the reference uncertainty output by the rational function model; An error unit is defined to determine the standardized error based on the predicted residual and the reference uncertainty. A penalty term determination unit is used to determine an extreme value penalty term based on the maximum value in the standardized error; A target loss subunit is determined to determine the target loss based on the prediction residual and the extreme value penalty term.

[0120] Based on any of the above embodiments, the determination of the target loss subunit is specifically used for: Based on the standardization error corresponding to each of the sampled temperature samples, a weighting factor corresponding to each of the sampled temperature samples is determined; wherein, the weighting factor corresponding to each of the sampled temperature samples increases as the standardization error increases; The prediction residuals of each of the sampled temperature samples are weighted based on the weighting factors to obtain the weighted basic loss. The target loss is determined based on the weighted basic loss and the extreme value penalty term.

[0121] Based on any of the above embodiments, the system further includes a rational function model determination unit, which specifically includes: The dataset acquisition unit is used to acquire a standard dataset of the helium isotope gas; the standard dataset includes standard temperature, standard density virial coefficient, and standard uncertainty. The set acquisition unit is used to obtain the set of the number of terms in the numerator polynomial and the set of the number of terms in the denominator polynomial. The construction unit is used to construct candidate rational function models with each numerator term in the set of terms of the numerator polynomial and each denominator term in the set of terms of the denominator polynomial as structural parameters. The parameter fitting unit is used to perform parameter fitting on each of the candidate rational function models with the standard temperature as the independent variable and the standard density virial coefficient as the target value, so as to obtain each fitted candidate rational function model and the fitted density virial coefficient value output by each fitted candidate rational function model. The evaluation index unit is determined based on the fitted density virial coefficient value, the standard density virial coefficient, and the standard uncertainty to determine the evaluation index corresponding to each fitted candidate rational function model. The screening unit is used to jointly screen each of the fitted candidate rational function models based on the evaluation index to obtain the rational function model.

[0122] Based on any of the above embodiments, the evaluation metrics include root mean square error, Akaike information criterion, and maximum standardized error; The unit for determining evaluation indicators is specifically used for: The root mean square error is determined based on the fitted density virial coefficient value and the standard density virial coefficient value. The fitting residual is determined based on the difference between the fitted density virial coefficient and the standard density virial coefficient, and the Akaike information criterion is determined based on the fitting residual and the standard uncertainty. The maximum standardized error is determined based on the fitted density virial coefficient, the standard density virial coefficient, and the standard uncertainty.

[0123] Based on any of the above embodiments, the filtering unit is specifically used for: From the candidate rational function models for fitting, a target rational function model is selected where the root mean square error is less than a first preset value and the maximum standardized error is less than a second preset value. The objective rational function model that minimizes the Akaike information criterion value is selected as the rational function model.

[0124] Based on any of the above embodiments, a selection unit is further included, wherein the selection unit is specifically used for: In the case where there are multiple objective rational function models with the same Akaike information criterion value, the objective rational function model with the smallest sum of the number of terms in the numerator and the number of terms in the denominator is selected as the rational function model.

[0125] Based on any of the above embodiments, the density virial coefficient prediction model includes an input layer, multiple cascaded residual blocks, and an output layer; The input layer is used to map the temperature to be predicted and the prior parameters of the physical properties into an initial feature vector, and input the initial feature vector into the first residual block in the plurality of cascaded residual blocks; The multiple cascaded residual blocks are used to take the previous enhancement feature output by the previous residual block as the input feature of the current residual block to obtain the current enhancement feature output by the current residual block; The enhanced features output by the last residual block are used as fusion features, and the fusion features are input into the output layer to obtain the target density virial coefficient value output by the output layer. Each residual block includes a first fully connected layer, a second fully connected layer, an activation layer, and a cross-layer connection channel.

[0126] Figure 5 An example is a schematic diagram of the physical structure of an electronic device, such as... Figure 5As shown, the electronic device may include: a processor 510, a communications interface 520, a memory 530, and a communication bus 540, wherein the processor 510, communications interface 520, and memory 530 communicate with each other via the communication bus 540. The processor 510 can call logical instructions in the memory 530 to execute a density virial coefficient prediction method for helium isotope gas. This method includes: acquiring the temperature to be predicted and prior physical property parameters of the helium isotope gas under target operating conditions; inputting the temperature to be predicted and the prior physical property parameters into a density virial coefficient prediction model to obtain a target density virial coefficient value output by the density virial coefficient prediction model; the density virial coefficient prediction model is trained based on sampled temperature samples and prior physical property parameter samples of the helium isotope gas, as well as labeled density virial coefficient values; the labeled density virial coefficient values ​​are obtained by inputting the sampled temperature samples into a rational function model of the helium isotope gas.

[0127] Furthermore, the logical instructions in the aforementioned memory 530 can be implemented as software functional units and, when sold or used as independent products, can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, or the part that contributes to the prior art, or a part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0128] On the other hand, the present invention also provides a computer program product, which includes a computer program that can be stored on a non-transitory computer-readable storage medium. When the computer program is executed by a processor, the computer can execute the density virial coefficient prediction method for helium isotope gas provided by the above methods. The method includes: obtaining the temperature to be predicted and prior physical property parameters of the helium isotope gas under target operating conditions; inputting the temperature to be predicted and the prior physical property parameters into a density virial coefficient prediction model to obtain a target density virial coefficient value output by the density virial coefficient prediction model; the density virial coefficient prediction model is trained based on sampled temperature samples and prior physical property parameter samples of the helium isotope gas, as well as labeled density virial coefficient values; the labeled density virial coefficient values ​​are obtained by inputting the sampled temperature samples into a rational function model of the helium isotope gas.

[0129] In another aspect, the present invention also provides a non-transitory computer-readable storage medium storing a computer program thereon, which, when executed by a processor, implements a method for predicting the density virial coefficient of helium isotope gas provided by the methods described above. This method includes: acquiring the temperature to be predicted and prior physical property parameters of the helium isotope gas under target operating conditions; inputting the temperature to be predicted and the prior physical property parameters into a density virial coefficient prediction model to obtain a target density virial coefficient value output by the density virial coefficient prediction model; the density virial coefficient prediction model is trained based on sampled temperature samples and prior physical property parameter samples of the helium isotope gas, as well as labeled density virial coefficient values; the labeled density virial coefficient values ​​are obtained by inputting the sampled temperature samples into a rational function model of the helium isotope gas.

[0130] The device embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this embodiment according to actual needs. Those skilled in the art can understand and implement this without any creative effort.

[0131] Through the above description of the embodiments, those skilled in the art can clearly understand that each embodiment can be implemented by means of software plus necessary general-purpose hardware platforms, and of course, it can also be implemented by hardware. Based on this understanding, the above technical solutions, in essence or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a computer-readable storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods described in the various embodiments or some parts of the embodiments.

[0132] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for predicting the density virial coefficient of a helium isotope gas, characterized in that, include: To obtain the predicted temperature and prior physical property parameters of helium isotope gas under target operating conditions; The temperature to be predicted and the prior physical property parameters are input into the density virial coefficient prediction model to obtain the target density virial coefficient value output by the density virial coefficient prediction model. The density virial coefficient prediction model is trained based on the sampling temperature samples and prior property parameter samples of helium isotope gas, as well as the labeled density virial coefficient values; the labeled density virial coefficient values ​​are obtained by inputting the sampling temperature samples into the rational function model of the helium isotope gas. The steps for determining the rational function model include: Obtain a standard dataset of the helium isotope gas; the standard dataset includes standard temperature, standard density virial coefficient, and standard uncertainty. Obtain the set of the number of terms in the numerator polynomial and the set of the number of terms in the denominator polynomial; Construct candidate rational function models with each numerator term in the set of terms of the numerator polynomial and each denominator term in the set of terms of the denominator polynomial as structural parameters; Using the standard temperature as the independent variable and the standard density virial coefficient as the target value, parameter fitting is performed on each of the candidate rational function models to obtain each fitted candidate rational function model and the fitted density virial coefficient value output by each fitted candidate rational function model. Based on the fitted density virial coefficients, the standard density virial coefficients, and the standard uncertainty, the evaluation index corresponding to each fitted candidate rational function model is determined. Based on the evaluation index, the candidate rational function models are jointly screened to obtain the rational function model.

2. The method for predicting the density virial coefficient of helium isotope gas according to claim 1, characterized in that, The density virial coefficient prediction model is obtained by iteratively executing the following steps until a preset iteration termination condition is met: Obtain the initial density virial coefficient prediction model; The sampled temperature sample and the sampled prior physical property parameters are input into the initial density virial coefficient prediction model to obtain the predicted density virial coefficient value output by the initial density virial coefficient prediction model. Based on the predicted density virial coefficient value and the label density virial coefficient value, the prediction residual is determined; Based on the predicted residuals, the target loss is determined, and the model parameters of the initial density virial coefficient prediction model are updated based on the target loss.

3. The method for predicting the density virial coefficient of helium isotope gas according to claim 2, characterized in that, The determination of the target loss based on the predicted residual includes: The sampled temperature is input into the rational function model of the helium isotope gas to obtain the reference uncertainty output by the rational function model. Based on the predicted residual and the reference uncertainty, the standardization error is determined; The extreme value penalty term is determined based on the maximum value in the standardized error. The target loss is determined based on the predicted residual and the extreme value penalty term.

4. The method for predicting the density virial coefficient of helium isotope gas according to claim 3, characterized in that, The determination of the target loss based on the predicted residual and the extreme value penalty term includes: Based on the standardization error corresponding to each of the sampled temperature samples, a weighting factor corresponding to each of the sampled temperature samples is determined; wherein, the weighting factor corresponding to each of the sampled temperature samples increases as the standardization error increases; The prediction residuals of each of the sampled temperature samples are weighted based on the weighting factors to obtain the weighted basic loss. The target loss is determined based on the weighted basic loss and the extreme value penalty term.

5. The method for predicting the density virial coefficient of helium isotope gas according to claim 1, characterized in that, The evaluation metrics include root mean square error, Akaike information criterion, and maximum standardized error. The step of determining the evaluation index corresponding to each of the fitted candidate rational function models based on the fitted density virial coefficient value, the standard density virial coefficient, and the standard uncertainty includes: The root mean square error is determined based on the fitted density virial coefficient value and the standard density virial coefficient value. The fitting residual is determined based on the difference between the fitted density virial coefficient and the standard density virial coefficient, and the Akaike information criterion is determined based on the fitting residual and the standard uncertainty. The maximum standardized error is determined based on the fitted density virial coefficient, the standard density virial coefficient, and the standard uncertainty.

6. The method for predicting the density virial coefficient of helium isotope gas according to claim 5, characterized in that, The step of jointly screening the candidate rational function models based on the evaluation index to obtain the rational function model includes: From the candidate rational function models for fitting, a target rational function model is selected where the root mean square error is less than a first preset value and the maximum standardized error is less than a second preset value. The objective rational function model that minimizes the Akaike information criterion value is selected as the rational function model.

7. The method for predicting the density virial coefficient of helium isotope gas according to claim 6, characterized in that, The method further includes: In the case where there are multiple objective rational function models with the same Akaike information criterion value, the objective rational function model with the smallest sum of the number of terms in the numerator and the number of terms in the denominator is selected as the rational function model.

8. The method for predicting the density virial coefficient of helium isotope gas according to any one of claims 1 to 4, characterized in that, The density virial coefficient prediction model includes an input layer, multiple cascaded residual blocks, and an output layer. The input layer is used to map the temperature to be predicted and the prior parameters of the physical properties into an initial feature vector, and input the initial feature vector into the first residual block in the plurality of cascaded residual blocks; The multiple cascaded residual blocks are used to take the previous enhancement feature output by the previous residual block as the input feature of the current residual block to obtain the current enhancement feature output by the current residual block; The enhanced features output by the last residual block are used as fusion features, and the fusion features are input into the output layer to obtain the target density virial coefficient value output by the output layer. Each residual block includes a first fully connected layer, a second fully connected layer, an activation layer, and a cross-layer connection channel.

9. A device for predicting the density virial coefficient of a helium isotope gas, characterized in that, include: The acquisition unit is used to acquire the predicted temperature and prior physical property parameters of helium isotope gas under the target operating conditions. The input unit is used to input the temperature to be predicted and the prior physical property parameters into the density virial coefficient prediction model to obtain the target density virial coefficient value output by the density virial coefficient prediction model. The density virial coefficient prediction model is trained based on the sampling temperature samples and prior property parameter samples of helium isotope gas, as well as the labeled density virial coefficient values; the labeled density virial coefficient values ​​are obtained by inputting the sampling temperature samples into the rational function model of the helium isotope gas. It also includes a rational function model determination unit, which is specifically used for: Obtain a standard dataset of the helium isotope gas; the standard dataset includes standard temperature, standard density virial coefficient, and standard uncertainty. Obtain the set of the number of terms in the numerator polynomial and the set of the number of terms in the denominator polynomial; Construct candidate rational function models with each numerator term in the set of terms of the numerator polynomial and each denominator term in the set of terms of the denominator polynomial as structural parameters; Using the standard temperature as the independent variable and the standard density virial coefficient as the target value, parameter fitting is performed on each of the candidate rational function models to obtain each fitted candidate rational function model and the fitted density virial coefficient value output by each fitted candidate rational function model. Based on the fitted density virial coefficients, the standard density virial coefficients, and the standard uncertainty, the evaluation index corresponding to each fitted candidate rational function model is determined. Based on the evaluation index, the candidate rational function models are jointly screened to obtain the rational function model.

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