A dynamic measurement uncertainty quantification method based on physical information and evidence learning

Abstract

The present invention discloses a method for quantifying dynamic measurement uncertainty based on physical information and evidence learning, comprising the following steps: Step 1: Obtaining an objective function of the output result of a constrained neural network based on a target dynamic system, and obtaining a training data set and a test data set; Step 2: Based on an evidence theory uncertainty quantification method, constructing and generating four independent feedforward neural networks, training the feedforward neural networks using the training data set, and obtaining a dynamic system model consisting of the four trained feedforward neural networks; Step 3: Inputting the test data set into the dynamic system model, and calculating the total uncertainty of the prediction result based on the tensor of the evidence distribution output by the four trained feedforward neural networks. The present invention improves the generalization ability and physical interpretability of the neural network, and avoids the situation in traditional methods where the final uncertainty change trend is the same as the model input due to the dependence of uncertainty propagation on the model input.

Description

Technical Field

[0001] The present invention relates to the field of dynamic measurement uncertainty estimation methods, and in particular to a dynamic measurement uncertainty quantification method based on physical information and evidence learning. Background Art

[0002] Dynamic measurement uncertainty is an important factor in measurement uncertainty, which itself depends on the dynamic characteristics of the measurement system and the frequency spectrum of the input signal. With the increasing importance of dynamic measurements, the evaluation of dynamic measurement uncertainty is considered to be an important scientific issue in metrology.

[0003] The GUM (Guide to the Expression of Uncertainty in Measurement) series of documents establishes a system for expressing uncertainty, providing a comprehensive framework for evaluating static measurement uncertainty. However, they fail to include reasonable estimates of measurement uncertainty in certain special cases, particularly dynamic measurements. Because the state of a dynamic system changes over time, and the measured quantity also changes continuously over time, the GUM no longer satisfies the requirements for evaluating uncertainty in dynamic systems.

[0004] Currently, several different emerging mathematical methods are being used to assess the uncertainty of dynamic measurements. Commonly used methods based on dynamic measurement error theory are the Monte Carlo method and grey system theory. The Monte Carlo method is an improved statistical simulation method based on Bayesian statistical theory. It uses random sampling to simulate the dynamic measurement process to assess the uncertainty of dynamic measurements. This method requires estimating the distribution of the measured quantity using a large number of samples or the distribution of the measured quantity is known. The huge computational effort and the uncertainty introduced by repeated sampling increase the overall uncertainty of the final result. The applicability of grey system theory, however, relies on the correct distribution of the sample data.

[0005] In recent years, deep neural networks have been widely used in system modeling and identification due to their exceptional capabilities for extracting nonlinear features and identifying complex models. Existing approaches for assessing uncertainty in deep neural networks include network ensembles, Bayesian methods, test-time data augmentation, and single deterministic methods. However, these existing neural network uncertainty assessment methods suffer from poor generalization and physical interpretability when applied to dynamic systems. Furthermore, due to the dependence of uncertainty propagation on model inputs, the resulting uncertainty trend can be identical to the model input. Summary of the Invention

[0006] The present invention provides a dynamic measurement uncertainty quantification method based on physical information and evidence learning to solve the problems of poor generalization ability and physical interpretability, and the same uncertainty change trend as the model input in the existing deep neural network uncertainty assessment method in the prior art.

[0007] In order to achieve the above object, the technical solution adopted by the present invention is:

[0008] The dynamic measurement uncertainty quantification method based on physical information and evidence learning includes the following steps:

[0009] Step 1: Based on the structure and physical information of the target dynamic system, a partial differential equation of the output result of the target dynamic system is established, and the objective function used to constrain the output result of the subsequent neural network is obtained according to the partial differential equation;

[0010] Furthermore, a simulation experiment is conducted on the target dynamic system, and a training data set is constructed using an input signal input into the target dynamic system during the simulation experiment and a corresponding output signal of the target dynamic system;

[0011] Furthermore, a real experiment is conducted on the target dynamic system, and a test data set is constructed using the input signal input into the target dynamic system during the real experiment and the corresponding output signal of the target dynamic system;

[0012] Step 2: Based on the method of quantifying uncertainty based on evidence theory, four independent feedforward neural networks are constructed. Then, each feedforward neural network is trained using the training data set constructed in step 1. When the training data set is input into each feedforward neural network, each feedforward neural network outputs a tensor of predicted values ​​and a tensor of evidence distribution parameters, and the dimensions of the tensors output by the four feedforward neural networks are the same.

[0013] Furthermore, a data-driven loss function is designed for each feedforward neural network training based on evidence-based regression theory. A partial differential loss is constructed by minimizing the mean squared error loss of the objective function obtained in step 1. The total loss function for each feedforward neural network training is constructed from the data-driven loss function and the partial differential loss. During each feedforward neural network training, the network parameters are updated and optimized based on the calculation results of the total loss function. This results in a dynamic system model consisting of four trained feedforward neural networks.

[0014] Step 3: Input the test data set obtained in step 1 into the dynamic system model obtained in step 2, and calculate the total uncertainty of the prediction results of the dynamic system model based on the tensor of the evidence distribution output by the four trained feedforward neural networks in the dynamic system model.

[0015] In a further step 1, the input signals constituting the test data set are not included in the input signals constituting the training data set.

[0016] In a further step 1, the signal lengths in the test dataset and the training dataset are kept consistent.

[0017] In the further step 1, for the long sequence input signals in the training dataset and the test dataset, the time dimension slices of the long sequence input signals are reconstructed, and the one-dimensional long sequence input signals with a length of N are reconstructed into a tensor with a shape of (1×H×W).

[0018] In the further step 2, the four tensors of the same dimension of the four evidence distributions output by the four feedforward neural networks are used to approximate the mean μ and variance σ of the output signal respectively. 2 The distribution parameters of , where the mean μ follows a normal distribution and the variance σ 2 Obeys the normal inverse gamma distribution.

[0019] In the further step 2, the data-driven loss function includes three parts of loss, where:

[0020] Estimate the predictive distribution of the network output by minimizing the negative log-marginal likelihood function (NLL) as the first part of the loss.

[0021] The second part of the loss uses an evidence regularizer to adjust the first part of the loss for training by increasing the prediction uncertainty where the prediction error occurs;

[0022] The third part of the loss is the MSE loss function, which is used to improve the prediction accuracy of the neural network and uses the gradient descent method for optimization approximation.

[0023] In the further step 3, the total uncertainty calculated includes epistemic uncertainty and arbitrary uncertainty.

[0024] This invention applies the powerful modeling and uncertainty quantification capabilities of feedforward neural networks to dynamic systems, while integrating the physical information of the dynamic system. This improves the generalization and physical interpretability of the model, and fills the gap in the use of neural networks to model the uncertainty of dynamic systems. Compared with the existing technology, the advantages of this invention are:

[0025] 1. This paper uses the physical information of the dynamic system as part of the constraints to guide neural network modeling, and integrates it with the loss function based on uncertainty evidence-aware learning, which supplements the missing knowledge and improves the generalization ability and physical interpretability of the neural network.

[0026] 2. The present invention uses the deep evidence regression method to quantify the uncertainty of the estimated value of the neural network output, avoiding the situation in traditional methods where the final uncertainty change trend is the same as the model input due to the dependence of uncertainty propagation on the model input, and improving the efficiency of uncertainty modeling and quantification. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1 It is a principle diagram of the method of an embodiment of the present invention.

[0028] Figure 2 These are the output result diagrams of the simulation model and the actual experiment obtained by five groups of voltage input signals at a frequency of 20 Hz in an embodiment of the present invention, where (a) is the output result diagram of the simulation model and (b) is the output result diagram of the actual experiment.

[0029] Figure 3 This is a performance diagram of the dynamic system model trained in an embodiment of the present invention on a test data set.

[0030] Figure 4 is an error and uncertainty diagram of the prediction results of the trained dynamic system model according to an embodiment of the present invention. DETAILED DESCRIPTION

[0031] The present invention will be further described below with reference to the accompanying drawings and examples.

[0032] like Figure 1 As shown, this embodiment discloses a dynamic measurement uncertainty quantification method based on physical information and evidence learning, including the following steps:

[0033] Step 1: Based on the structure and physical information of the target dynamic system, a partial differential equation of the output result of the target dynamic system is established, and the objective function used to constrain the subsequent neural network output result is obtained according to the partial differential equation.

[0034] The target dynamic system in this embodiment can be a nonlinear system or a linear system. Taking the piezoelectric ceramic drive system as an example, the piezoelectric ceramic drive system is approximated as a spring-mass-damper system based on the structure and physical information of the piezoelectric ceramic drive system. The partial differential equation of the output result of the piezoelectric ceramic drive system is shown in the following formula:

[0035]

[0036] Where y(x, t) is the output signal of the dynamic system, which is determined by the input signal x and time t; c and k are the damping coefficient and the elastic coefficient of the spring, respectively.

[0037] The partial differential equation shown in the above formula is expressed as the objective function used to constrain the output of the subsequent neural network, and y(x, t) is used as the predicted output of the network, as shown in the following formula:

[0038]

[0039] Among them, g(x, t) is the objective function of establishing the network model.

[0040] Furthermore, in this embodiment, the target dynamic system is simulated using the Simulink toolbox of MATLAB, and a training data set is constructed using input signals of the target dynamic system and corresponding output signals of the target dynamic system during the simulation experiment.

[0041] Specifically, a simulation model of the target dynamic system is established using the Simulink toolbox of MATLAB. K groups of dynamic time-related signals are input into the simulation model in batches. The duration of each group of dynamic time-related signals is t, and the frequencies of each group of dynamic time-related signals are different. Each group of dynamic time-related signals contains M input signals X1, X2, X3...X with the same frequency and different amplitudes. M , each input signal X of the dynamic measurement system i (i=1, 2, ...M) are all functions x of time t t (t=1, 2, ...N).

[0042] Each set of dynamic time-correlated signals contains the input signal X i Under this condition, the simulation model of the target dynamic system generates M corresponding output signals Y1, Y2, Y3...Y M , each output signal Y i are all functions y of time t t (t=1,2,..,.N) The function y at each time t t All obey the normal distribution, and the probability density function is η t (t=1,2,...,N), the joint probability density function of the entire time series is encoded as Then the probability distribution P(Y|X) of the output sequence of the simulation model of the dynamic system is as follows:

[0043]

[0044] Match each group of dynamic time-correlated signals with the output signal of the corresponding target dynamic system simulation model as a training data set.

[0045] Taking the piezoelectric ceramic drive system as the target dynamic system, 10 groups (i.e. K = 10) of sawtooth wave voltage signals with different frequencies (excluding 20 Hz) from 5 Hz to 40 Hz are used as dynamic time-related signals. Each group of sawtooth wave voltage signals has 40 (i.e. M = 40) input voltages with different amplitudes (i.e. input signal X i ), the amplitude increases from 30V to 120V, and 10 (ie N = 10) cycles of the signal are taken at each amplitude (ie x t), the duty cycle is 0.9999, 10 groups of sawtooth wave voltage signals are input into the simulation model of the piezoelectric ceramic drive system, and the output signals are obtained from the simulation model of the piezoelectric ceramic drive system. The 10 groups of sawtooth wave voltage signals and the corresponding output signals are used as training data sets.

[0046] Furthermore, this embodiment conducts a real experiment on the target dynamic system, constructing a test data set using the input signals input to the target dynamic system and the corresponding output signals of the target dynamic system during the real experiment. The signal length of the test data set is consistent with the signal length of the training data set, but the input signals constituting the test data set are not included in the input signals constituting the training data set.

[0047] Taking the piezoelectric ceramic drive system as the target dynamic system, in the actual experiment of the piezoelectric ceramic drive system, 10 sawtooth voltages with different voltage amplitudes at a frequency of 20 Hz that are not included in the input signal of the simulation model are used as input signals to obtain the corresponding output signals. The sawtooth voltages with 10 different voltage amplitudes at a frequency of 20 Hz and the corresponding output signals are used as the test set. In order to compare the performance of the network model in the test set, the output signals of the simulation model and the real experiment at 20 Hz are shown as follows. Figure 2 shown.

[0048] After obtaining the training dataset and the test dataset, the signals in the training dataset and the test dataset are preprocessed to adjust each set of input and the corresponding output signal to the same dimension and size, and the time dimension slices of the long sequence input signal are reconstructed, and the one-dimensional long sequence input signal of length N is reconstructed into a tensor with a shape of (1×H×W).

[0049] Taking the piezoelectric ceramic drive system as the target dynamic system as an example, in the training data set and the test data set, the original input signal is an ultra-long time series signal x t (t=1, 2, ..., N, N=2000), by changing the input dimension through slicing reconstruction in the time dimension, the one-dimensional long sequence of input signals is reconstructed into a tensor with a shape of (1×250×8). The subsequent neural network inputs the signal values ​​at each of the eight time points until the signal ends, thus avoiding the huge computational complexity caused by processing each time point, and improving the operation speed.

[0050] Step 2: Based on the uncertainty quantification method of evidence theory, four independent feedforward neural networks are constructed. Each feedforward network consists of four hidden layers, each containing 256 neurons. The output layer is activated using the softplus function, limiting the output range to positive numbers. Each feedforward neural network is then trained using the training dataset constructed in Step 1. A dynamic system model consisting of the four trained feedforward neural networks is obtained, thus completing the modeling of the dynamic system measurement process.

[0051] When the training data set is input into each feedforward neural network, each feedforward neural network outputs a tensor of predicted value and evidence distribution, and the dimensions of the tensors output by the four feedforward neural networks are the same. The four feedforward neural networks output a total of four tensors of the same dimension of the evidence distribution, which are used to approximate the mean μ and variance σ of the output signal respectively. 2 The distribution parameters of , where the mean μ follows a normal distribution Variance σ 2 Obeys normal inverse gamma distribution (NIG) Γ -1 (α,β), is the conjugate prior of the normal distribution.

[0052] In this embodiment, the total loss function L during training of each feedforward neural network includes the data-driven loss function L data and partial differential loss L pde , where the data-driven loss function L data It is designed for each feedforward neural network based on the evidence regression theory. When training each feedforward neural network, the network parameters are updated and optimized based on the calculation results of the total loss function, thereby obtaining a trained feedforward neural network.

[0053] Among them, the data-driven loss function L in the total loss function L data The losses are composed of the following three parts:

[0054] A. The parameters of the evidence distribution output by each feedforward neural network are m = {γ,υ,α,β}. Then the conditional probability distribution of the predicted value y output by each feedforward neural network is p(y|m) as follows:

[0055]

[0056] Where: γ,υ,α,β represent mean μ and variance σ 2 The conditional probability distribution p(y|m) follows the Student t distribution, as shown below:

[0057]

[0058] By minimizing the negative log marginal likelihood function NLL, as the data-driven loss function L dataThe first part of the loss function L NLL , as shown below:

[0059]

[0060] Wherein, Ω=2β(1+υ); Γ() represents the gamma distribution.

[0061] B. Data-driven loss function L data The second part of the loss is the evidence regularizer L R , by increasing the prediction uncertainty where the prediction error occurs to adjust the NLL loss of the first part for training, as shown in the following formula:

[0062] L R =|γ-y|(2υ+α)

[0063] C. Data-driven loss function L data The third part of the loss is the mean square error loss L MSE During training, the prediction accuracy of the feedforward neural network is improved based on the mean square error loss, and then the gradient descent method is used for optimization approximation. The calculation formula is:

[0064]

[0065] where N pde is the total number of PDE sampling points.

[0066] Therefore, the data-driven loss function L is composed of three parts of loss data As shown in the following formula:

[0067]

[0068] Among them, λ R is the regularization parameter; N data is the total number of training data points.

[0069] Partial differential loss L in the total loss function pde , is constructed by minimizing the mean square error (MSE) loss of the objective function g(x, t) obtained in step 1. During the specific calculation, the partial differential of y(x, t) is calculated through PyTorch's automatic differentiation technology and then inserted into g(x, t).

[0070] In this embodiment, the total loss function L during training of each feedforward neural network is the data-driven loss function L data and partial differential loss L pde The weighted sum of is as follows:

[0071] L=λ data L data +λ pde Lpde

[0072] Among them, λ data and λ pde are the weight coefficients for balancing the interaction between the two losses.

[0073] Step 3: Input the test data set obtained in step 1 into the dynamic system model composed of the four trained feedforward neural networks obtained in step 2. The dynamic system model outputs the prediction results and the four sets of evidence distribution tensors, such as Figure 3 shown.

[0074] The variance of the evidence distribution is calculated based on the tensor of the evidence distribution output by the four trained feedforward neural networks, and then the epistemic uncertainty U of the prediction results of the dynamic system model is evaluated based on the variance. epi , arbitrary uncertainty U ale ,like Figure 4 shown.

[0075] Among them, epistemic uncertainty U epi As shown in the following formula:

[0076]

[0077] Where: Var[μ] represents the variance of the mean μ, that is, the epistemic uncertainty; p(μ) represents the distribution of the mean μ; E[μ] represents the expectation of the mean μ; μ represents the mean of the output value; α, β, υ are the parameters of the evidence distribution.

[0078] Arbitrary uncertainty U ale As shown in the following formula:

[0079]

[0080] Where: E[σ 2 ] represents the variance σ 2 The mean of p(σ 2 ) represents the variance σ 2 distribution; σ represents the standard deviation of the output value; α, β, υ are the parameters of the evidence distribution.

[0081] The total uncertainty U of the prediction result of the dynamic system model is the epistemic uncertainty U epi , arbitrary uncertainty U ale The sum is as follows:

[0082] U=U epi +U ale .

[0083] Although there are significant differences between the uncertainty quantification based on deep neural networks and the uncertainty assessment method of GUM, and there is no clear uncertainty propagation process, compared with traditional nonlinear dynamic system modeling, deep neural networks have good recognition and modeling capabilities for complex input-output mapping relationships, and the integration of physical information improves the interpretability and generalization ability of neural networks.

[0084] The preferred embodiments of the present invention are described in detail above with reference to the accompanying drawings. The embodiments described in the present invention are merely descriptions of the preferred embodiments of the present invention and do not limit the concept and scope of the present invention. The various specific technical features described in the above specific embodiments can be combined in any suitable manner unless there is any contradiction. Such combinations should also be regarded as the contents disclosed in this disclosure as long as they do not violate the concept of the present invention. In order to avoid unnecessary repetition, the present invention will not further describe various possible combinations.

[0085] The present invention is not limited to the specific details of the above-mentioned embodiments. Within the scope of the technical concept of the present invention and without departing from the design concept of the present invention, various modifications and improvements made to the technical solution of the present invention by those skilled in the art should fall within the scope of protection of the present invention. The technical contents for which protection is sought in the present invention have been fully recorded in the claims.

Claims

1. A dynamic measurement uncertainty quantification method based on physical information and evidence learning, characterized by: The following steps are involved: Step 1: Based on the structure and physical information of the target dynamic system, a partial differential equation of the output result of the target dynamic system is established, and the objective function used to constrain the output result of the subsequent neural network is obtained according to the partial differential equation; Furthermore, a simulation experiment is conducted on the target dynamic system, and a training data set is constructed using an input signal input into the target dynamic system during the simulation experiment and a corresponding output signal of the target dynamic system; Furthermore, a real experiment is conducted on the target dynamic system, and a test data set is constructed using the input signal input into the target dynamic system during the real experiment and the corresponding output signal of the target dynamic system; Step 2: Based on the method of quantifying uncertainty based on evidence theory, four independent feedforward neural networks are constructed. Then, each feedforward neural network is trained using the training data set constructed in step 1. When the training data set is input into each feedforward neural network, each feedforward neural network outputs a tensor of predicted values ​​and a tensor of evidence distribution parameters, and the dimensions of the tensors output by the four feedforward neural networks are the same. Furthermore, a data-driven loss function is designed for each feedforward neural network training based on evidence-based regression theory. A partial differential loss is constructed by minimizing the mean squared error loss of the objective function obtained in step 1. The total loss function for each feedforward neural network training is constructed from the data-driven loss function and the partial differential loss. During each feedforward neural network training, the network parameters are updated and optimized based on the calculation results of the total loss function. This results in a dynamic system model consisting of four trained feedforward neural networks. Step 3: Input the test data set obtained in step 1 into the dynamic system model obtained in step 2, and calculate the total uncertainty of the prediction results of the dynamic system model based on the tensor of the evidence distribution output by the four trained feedforward neural networks in the dynamic system model.

2. The dynamic measurement uncertainty quantification method based on physical information and evidence learning according to claim 1 is characterized in that: In step 1, the input signals constituting the test data set are not included in the input signals constituting the training data set.

3. The dynamic measurement uncertainty quantification method based on physical information and evidence learning according to claim 1 is characterized in that: In step 1, the signal lengths in the test dataset and the training dataset are kept consistent.

4. The dynamic measurement uncertainty quantification method based on physical information and evidence learning according to claim 1 is characterized in that: In step 1, for the long sequence input signals in the training dataset and the test dataset, the time dimension slices of the long sequence input signals are reconstructed, and the one-dimensional long sequence input signals with a length of N are reconstructed into a tensor with a shape of (1×H×W).

5. The dynamic measurement uncertainty quantification method based on physical information and evidence learning according to claim 1 is characterized in that: In step 2, the four feedforward neural networks output four tensors of the same dimension for the evidence distribution, which are used to approximate the mean μ and variance σ of the output signal respectively. 2 The distribution parameters of , where the mean μ follows a normal distribution and the variance σ 2 Obeys the normal inverse gamma distribution.

6. The dynamic measurement uncertainty quantification method based on physical information and evidence learning according to claim 1 is characterized in that: In step 2, the data-driven loss function includes three parts of loss, where: Estimate the predictive distribution of the network output by minimizing the negative log-marginal likelihood function (NLL) as the first part of the loss. The second part of the loss uses an evidence regularizer to adjust the first part of the loss for training by increasing the prediction uncertainty where the prediction error occurs; The third part of the loss is the MSE loss function, which is used to improve the prediction accuracy of the neural network and uses the gradient descent method for optimization approximation.

7. The dynamic measurement uncertainty quantification method based on physical information and evidence learning according to claim 1 is characterized in that: In step 3, the calculated total uncertainty includes epistemic uncertainty and arbitrary uncertainty.

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