Intelligent testing method fusing experimental measurement data and physical prior knowledge

Abstract

This invention relates to the field of intelligent testing technology, specifically to an intelligent testing method that integrates experimental measurement data and prior physical knowledge. The method includes: acquiring particular solutions of a corresponding physical field under different boundary value conditions based on a physical model; determining the spatiotemporal distribution information and interaction relationships of parameters in the physical field; and converting the particular solution information into structured data as training data; fitting the joint probability distribution of the training data to a Gaussian distribution based on a generative model; randomly sampling from the Gaussian distribution as prior distribution information of the physical field; acquiring measurement data of relevant parameters under the corresponding physical field based on an experimental measurement device as posterior distribution information; and fusing the prior and posterior distribution information of the physical field to obtain complete physical field information. This invention organically combines theoretical analysis and calculation with experimental measurement, solving the problems of traditional testing instruments being unable to measure or measuring completely.

Description

Technical Field

[0001] This invention relates to the field of intelligent testing technology, and more specifically to an intelligent testing method that integrates experimental measurement data and prior physical knowledge. Background Technology

[0002] Our understanding of natural physical phenomena such as flow, combustion, and electromagnetism currently relies primarily on experimental measurements and theoretical analysis. At present, sensor-based or optical diagnostic testing techniques are the main methods for obtaining physical quantities. However, real-world testing environments are often quite harsh and extreme, such as the confined spaces, high temperatures, high pressures, and high flow rates encountered by engines. Existing testing methods face the problem of being unable to measure or fully measure real-world environments. Specifically, current testing instruments can only measure single parameters, and are limited by operating temperatures and confined spaces, limiting their measurement to localized measurable areas and preventing the acquisition of complete physical fields. Therefore, despite extensive previous research on the same physical phenomena, the testing scope is extremely limited, and the measured physical quantities are relatively singular, resulting in a lack of comprehensive understanding of these phenomena.

[0003] Although theoretical analysis can fully describe the information of each parameter of the physical field and their interaction relationship under given boundary conditions by solving the governing equations of the physical field, physical models and mathematical descriptions are always simplified to a certain extent, which leads to theoretical results deviating from reality. Therefore, the validity of theoretical results needs to be verified or corrected by experimental measurement results.

[0004] In the fusion of measurement data and theoretical models, data assimilation, as a method to integrate measurement data to improve the accuracy and consistency of model predictions, is widely used. Among these, variational data assimilation utilizes the posterior iteration of measurement data to optimize relevant variational parameters, such as boundary conditions, making the numerical solutions closer to the observation results. This paradigm is widely used in fields such as climate prediction. However, traditional variational data assimilation methods require optimization of non-convex objective functions, are highly dependent on initial value settings, and are prone to unstable results. Furthermore, because they require extensive numerical solutions to the target physical field under different boundary conditions, these methods are very computationally and time-consuming.

[0005] Embedding machine learning methods can improve the performance of data assimilation methods. For optimization problems, existing methods improve the performance of the optimization step in data assimilation by improving the properties of the objective function or introducing priors to obtain better initial values. However, these methods are often designed for specific tasks, and well-trained neural networks lack generalization ability and still cannot avoid high computational costs.

[0006] To address the massive computational demands of numerical computation, mainstream methods employ neural network surrogate models. This involves using neural networks to replace the dynamics of physical fields and performing autoregressive forward predictions instead of numerical solutions. However, since the prediction model is not constrained, this autoregressive black-box prediction model often leads to error accumulation and unstable solution results, ultimately producing non-physical results. Furthermore, black-box models lack generalization ability and interpretability.

[0007] Physical Information Neural Networks (PINNs) can optimize the residuals of physical equations using surrogate and automatic differentiation methods for the solution function. Due to the favorable functional properties of the neural network solution function, it can effectively integrate measurement data and physical constraints. However, the trained neural network solution function is only effective under specific boundary conditions and parameters; changes to boundary conditions and parameters require retraining, which is very time-consuming. Furthermore, for some strongly nonlinear systems such as turbulence, the optimization objective derived from the physical equations is severely nonconvex, making optimization difficult when measurement data is significantly lacking.

[0008] Therefore, there is an urgent need for a testing method to overcome hardware limitations and solve the problems that traditional testing instruments cannot detect or cannot fully measure. Summary of the Invention

[0009] In view of this, the present invention provides an intelligent testing method that integrates experimental measurement data and physical prior knowledge, organically combining theoretical analysis and calculation with experimental measurement to solve the problems of traditional testing instruments being unable to measure or measuring incompletely.

[0010] To achieve the above objectives, the present invention adopts the following technical solution:

[0011] An intelligent testing method that integrates experimental measurement data and prior physical knowledge includes the following steps:

[0012] S1. Based on the physical model, obtain the particular solution information of the corresponding physical field under different boundary conditions, determine the spatiotemporal distribution information and interaction relationship of the parameters in the physical field, and convert the particular solution information into structured data information as training data;

[0013] S2. Fit the joint probability distribution of the training data based on the generative model, encode the joint probability distribution of the data into a Gaussian distribution, and randomly sample from the Gaussian distribution as the prior distribution information of the physical field.

[0014] S3. Obtain measurement data of relevant parameters under the corresponding physical field based on the experimental measurement device, and use it as posterior distribution information;

[0015] S4. The prior and posterior distribution information of the physical field are fused to obtain complete physical field information.

[0016] Furthermore, S1 includes:

[0017] S11. Determine the governing equations of the physical field, which are expressed in the following form:

[0018]

[0019]

[0020] Among them, the operator Acting on the solution function x, representing the entire governing equation, the solution function x maps from the solution domain Ω to the d-dimensional physical parameter space; when the governing equation remains unchanged, the solution function x is determined by the boundary conditions. The domain Ω, initial conditions, boundary conditions, and physical property parameters are determined by providing a series of boundary conditions.

[0021] S12. Determine the boundary conditions for the governing equations;

[0022] S13. Sample a series of solutions in the solution function space. Given the spatial and temporal coordinates and physical field parameters, interpolate the numerical solutions onto the corresponding coordinates to form structured training data of [coordinates, physical parameter 1, physical parameter 2, ...].

[0023] Furthermore, in S12, if a theoretical solution is required, its analytical solution is obtained through formula derivation, and the specific spatial and temporal location points are substituted to obtain usable numerical data; if a numerical solution is required, the solution domain is discretized to obtain a series of grids, and the equations are discretized and transformed into a system of linear equations about the parameter variables to be solved at the grid points using numerical calculation methods, and then the system of linear equations is solved using iterative methods.

[0024] Furthermore, in S1, each sample in the training data contains numerical information of all important parameters of the physical field at different discrete spatiotemporal locations within a given region under a definite solution condition. Different samples correspond to the particular solution information of the physical field under different definite solution conditions.

[0025] Furthermore, S2 includes:

[0026] S21. Construct a corresponding neural network-based generative model based on the structure of the physical field data; when the generative model adopts a denoising diffusion model, the process for determining its fitting function is as follows:

[0027] Construct a forward noise-adding process that continuously adds noise to the training data so that its joint probability density function exhibits a Gaussian distribution;

[0028] The reverse denoising process is performed based on the scores of the variables during the noise addition process.

[0029] Determine the function to fit the denoising diffusion model:

[0030] s θ (x(τ),τ)≈s(x(τ),τ)

[0031] Where θ represents the learnable parameters of the denoising diffusion model, s(x(τ),τ) represents the score obtained at each step through fitting the denoising diffusion model, and τ represents the pseudo-time step. Let x(0) be the mean and σ(τ) be the mean. 2 Artificially noisy samples with variance;

[0032] S22. Randomly sample a pseudo-time τ from [0,1] and add a given variance σ(τ) to the input data. 2 Random noise is added to obtain noisy samples:

[0033]

[0034] x(0) is a single training sample, and ε is standard Gaussian noise. It is a Gaussian distribution with a mean of 0 and a standard identity matrix equal to the covariance matrix;

[0035] The noisy sample is input into the denoising diffusion model for forward propagation calculation to obtain the output s. θ (x(τ),τ);

[0036] S23. Calculate the loss function using the score matching algorithm;

[0037] S24. Calculate the gradient of the denoising diffusion model parameters using the backpropagation algorithm, and iteratively update the model parameters accordingly.

[0038] S25. Sampling from a Gaussian distribution Performing the denoising process in S21 yields a random physical field sample that randomly falls within the solution function space; x(1) is a single sample that satisfies a Gaussian distribution. The mean is 0 and the variance is σ(τ). 2 The Gaussian distribution.

[0039] Furthermore, in S23, the expression for the loss function is:

[0040]

[0041] in, Let λ(τ) be the expected value of a given distribution, λ(τ) be a given series of weighting coefficients, and τ ~ U(0,1) be the pseudo-time step, which is sampled from the uniform distribution U(0,1) during training and determines the magnitude of the Gaussian noise added to the sample. p0(x(0)) is the probability density function of the distribution followed by the training data sample, and p0τ(x(τ)|x(0)) is the probability density function of the conditional distribution of the noisy sample x(τ) given the training data sample x(0). The operator for calculating the gradient of a noisy sample x(τ).

[0042] Furthermore, S3 includes:

[0043] S31. For the physical field of interest, use the corresponding experimental measurement device to measure the specific parameters of the physical field;

[0044] S32. Convert the raw measurement data into a storable digital signal;

[0045] S33. Perform dimensionless transformation and coordinate transformation on the digital signal to obtain the final measurement data.

[0046] Furthermore, in S4, Bayes' theorem is used to obtain the joint expression of the posterior distribution and the prior distribution:

[0047]

[0048] Where y is the observation data, considered as part of the complete physical field information; p(y) is the posterior distribution; p(x) is the prior distribution; p(y|x) is the probability density function of the conditional distribution satisfied by the incomplete observation information y given the complete physical field information x; and p(x|y) is the final desired complete physical field conditional distribution.

[0049] Furthermore, in S4, when the generative model is a denoising diffusion model, after obtaining the scores of the final desired complete physical field condition distribution p(x|y) at each pseudo-time step, the sampled samples are then corrected posteriorly based on the scores; wherein, the scores of the final desired complete physical field condition distribution p(x|y) at each pseudo-time step are calculated as follows:

[0050]

[0051] The first term within the square brackets on the right-hand side of the above equation represents the guidance of experimental observation data, and the second term represents the prior information of the physical field. The residuals representing the governing equations of the physical field are used to guide the generation process of prior information about the physical field to gradually satisfy the constraints of the physical equations; s θ (x|y,τ) is the approximate score of the conditional distribution p(x|y,τ) related to the neural network parameters at time τ; s θ(x(τ),τ) represents the score of the probability distribution of the noisy training samples approximated by the neural network at time τ; := is the assignment symbol; α(τ) is an adjustable weight parameter; The operator for calculating the gradient of a noisy sample x(τ); The observed values ​​are calculated from the samples estimated by the neural network; ||| is the vector 2 norm; β(τ) is an adjustable weight parameter; This is the estimate of the original sample obtained by the neural network in one step of denoising at time τ.

[0052] Furthermore, the posterior correction of the sampled samples based on the scores of the desired complete physical field condition distribution p(x|y) at each pseudo-time step includes:

[0053] Random noise is sampled from a Gaussian distribution. As an initial condition for the denoising process, the following reverse denoising is performed:

[0054]

[0055] in, Let x(0) be the mean and σ(τ) be the mean. 2 For artificially noisy samples with variance; σ(τ) 2 The magnitude of the variance; s θ (x|y,τ) is the approximate score of the conditional distribution p(x|y,τ) related to the neural network parameters at time τ; w is a standard Brownian process;

[0056] The stochastic differential equation solver based on the prediction correction scheme performs an inverse denoising process on the sampled initial conditions:

[0057]

[0058] Where ∫ is Itō's random integral; x(0)|y is the physical field sample under the required experimental observation conditions; and x(1) is a single sample that satisfies a Gaussian distribution.

[0059] As can be seen from the above technical solution, compared with the prior art, the present invention has the following beneficial effects:

[0060] This invention uses a generative model to fit all relevant physical quantities in the natural phenomenon under study as a joint distribution, making it a flexible and usable prior information about the target physical field, greatly improving the ability and efficiency of fusing observational data. It organically combines theoretical analysis and experimental measurement, obtaining spatiotemporal location points and parameters that cannot be measured by testing instruments from limited measurement data through posterior sampling. It can reconstruct the complete physical field using a small amount of local experimental measurement data, greatly expanding the capabilities of traditional testing instruments and providing real-time, complete, and multi-parameter experimental measurements. Specifically, it is reflected in:

[0061] 1. This invention utilizes artificial intelligence methods to construct physical field parameters and their interaction relationships. Instead of directly modeling conditional distributions for specific problems, it directly uses generative models to model the joint probability distribution of the spatiotemporal distributions of all relevant parameters. It can obtain prior knowledge of human physics in digital form and has good generalization ability for different tasks.

[0062] 2. This invention combines physical equation information for posterior sampling, which can effectively integrate experimental observation data and prior physical information. By manipulating the generation process of the trained generative model and applying observation constraints and control equation constraints, the final generated samples not only conform to the prior information of the parameters of the physical field and their interaction relationships, but also well satisfy the experimental observation data.

[0063] 3. This invention organically combines measurement equipment hardware and artificial intelligence model software, breaking away from the traditional model of independent development of hardware and software. It uses software to extract prior physical knowledge to expand the measurement capabilities of the hardware, enabling the acquisition of spatiotemporal location information and parameter information that were previously unmeasurable by measuring instruments. This intelligent measurement system can effectively improve the efficiency of scientific research and engineering practice while reducing costs. Attached Figure Description

[0064] To more clearly illustrate the technical solutions in the embodiments of the present 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 only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0065] Figure 1 This is a schematic diagram illustrating the problem that existing experimental measurement devices provided by this invention cannot measure or cannot fully measure physical fields.

[0066] Figure 2 The flowchart of the intelligent testing method that integrates experimental measurement data and prior physical knowledge provided by the present invention. Detailed Implementation

[0067] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0068] like Figure 1 As shown, taking the flow field measurement of a rocket engine nozzle as an example, due to the opaque nature of the nozzle material and the extreme physical environment inside, no form of measurement can be performed in its internal region. Furthermore, in the downstream region of the nozzle exit, sensors can only simultaneously measure a small portion of the numerous physical parameters, resulting in incomplete measurements. It should be emphasized that this diagram is only used as an example of a nozzle; this invention is applicable to the measurement of any physical field.

[0069] like Figure 2 As shown, this embodiment of the invention discloses an intelligent testing method that integrates experimental measurement data and prior physical knowledge, including the following steps:

[0070] S1. Based on the physical model, obtain the particular solution information of the corresponding physical field under different boundary conditions, determine the spatiotemporal distribution information and interaction relationship of the parameters in the physical field, and convert the particular solution information into structured data information as training data;

[0071] S2. Fit the joint probability distribution of the training data based on the generative model, encode the joint probability distribution of the data into a Gaussian distribution, and randomly sample from the Gaussian distribution as the prior distribution information of the physical field.

[0072] S3. Obtain measurement data of relevant parameters under the corresponding physical field based on the experimental measurement device, and use it as posterior distribution information;

[0073] S4. The prior and posterior distribution information of the physical field are fused to obtain complete physical field information.

[0074] Specifically, S1 corresponds to a reliable theoretical result of the physical field, providing a specific physical field, the governing equations describing it, and the corresponding boundary conditions (such as boundary conditions and equation parameter settings), generating a particular solution under given conditions. Through theoretical methods such as formula derivation and numerical solutions, the spatiotemporal distribution information and interaction relationships of the parameters in the studied physical field can be obtained and transformed into structured data information for subsequent training of the generative model. In the output training data, each sample contains numerical information of all important parameters of the physical field at different discrete spatiotemporal locations within a given region under a specific boundary condition. Different samples correspond to particular solutions of the physical field under different boundary conditions.

[0075] In S2, a generative model is used to fit the joint distribution of relevant parameters of the physical field rather than learning a specific mapping relationship. The training data is the specific physical field information obtained in S1. The model can use existing generative models such as variational autoencoders (VAE), generative adversarial networks (GAN), denoising diffusion models, etc. Samples of the physical field under specific boundary conditions are sampled from the trained distribution as the prior distribution information of the physical field, so as to be used in subsequent fusion with experimental data.

[0076] S3 corresponds to the experimental measurement device for the physical field. The specific device depends on the physical field to be measured, or it can be historical information about experimental measurements of that physical field. The measuring instrument consists of some kind of sensor, such as a speed, temperature, or pressure sensor, or an image sensor. This step measures the target physical field to obtain the corresponding raw data.

[0077] S4 uses a pre-trained generative model as a plug-and-play prior and uses the incomplete physical field information obtained from the measurement as a posterior condition for correction, quickly obtaining complete physical field information that conforms to the experimental measurement data. Moreover, this physical field information is closer to the experimental measurement results than the numerical solution.

[0078] The following provides further explanation of each of the above steps.

[0079] S1 Acquisition of physical prior information.

[0080] S11. Define the solution function space of the physical model (i.e., the governing equations). Give the governing equations of the physical field, expressed in the following form:

[0081]

[0082]

[0083] Among them, the operator Acting on the solution function x, representing the entire governing equation, the solution function x maps from the solution domain Ω to the d-dimensional physical parameter space; when the governing equation remains unchanged, the solution function x is determined by the boundary conditions. The domain Ω, initial conditions, boundary conditions, and physical property parameters are determined by providing a series of boundary conditions.

[0084] S12. Determine the boundary conditions of the governing equations. If a theoretical solution is required, obtain the analytical solution through formula derivation and substitute it with specific spatial and temporal locations to obtain usable numerical data. If a numerical solution is required, discretize the solution domain to obtain a series of grids. Use numerical calculation methods (finite difference, finite element, etc.) to discretize the equations and transform them into a system of linear equations about the parameters to be solved at the grid points. Then, use the Gauss-Seidel iterative method to solve the system of linear equations.

[0085] S13. Sample a series of solutions in the solution function space. Given the spatial and temporal coordinates and physical field parameters, interpolate the numerical solutions onto the corresponding coordinates to form structured training data of [coordinates, physical parameter 1, physical parameter 2, ...].

[0086] S2. Fitting the prior distribution of physical field parameters based on a generative model. The input to the generative model includes: the training dataset (structured data about each coordinate of the physical field obtained in S1), model training hyperparameters (learning rate, model structure parameters), and model parameters (convolution kernel, bias, weight matrix, bias vector). Specifically, it includes:

[0087] In this step, a corresponding neural network structure is constructed based on the structure of the physical field data. For example, for uniformly distributed data in 4-dimensional spacetime, a neural network structure based on 3D U-Net can be used. The input and output of this neural network have the same shape and consist of a series of downsampling and upsampling modules. Each module consists of several residual blocks and self-attention blocks.

[0088] Unlike previous input-output-based physical field modeling and neural network training methods, this invention models the spatiotemporal distribution of all relevant physical field parameters as a joint probability distribution, which requires no label data and is more flexible in use.

[0089] The following section uses a denoising diffusion model as an example to explain in detail the fitting of the prior distribution of the physical field.

[0090] S21. Determine the function to fit the denoising diffusion model:

[0091] The final function to be obtained is the logarithmic gradient of the joint probability distribution of the physical parameters, called the score.

[0092]

[0093] Since directly constructing the score is difficult, the denoising diffusion model constructs a forward denoising process, which is implemented through the following stochastic differential equation:

[0094]

[0095] Where τ is the pseudo-time step, which measures the variance of the noise, and w is the standard Brownian process.

[0096] This forward noise-adding process continuously adds noise to the training data, causing its joint probability density function to exhibit a Gaussian distribution.

[0097] Based on the scores of the variables during the noise addition process, the following reverse noise reduction process is performed:

[0098]

[0099] Where s(x(τ),τ) represents the score obtained at each step through fitting the denoising diffusion model, and x(τ) represents the noisy sample; σ(τ) 2 Given variance, which is the variance of the Gaussian noise artificially added to the training data samples.

[0100] By fitting the score s(x(τ),τ) at each step using the denoising diffusion model, we can sample from a simple Gaussian distribution, perform the above denoising process, and obtain a physical field sample that conforms to the data distribution. Here, the function fitted by the denoising diffusion model is determined as:

[0101] s θ (x(τ),τ)≈s(x(τ),τ)

[0102] Where θ is a learnable parameter of the denoising diffusion model.

[0103] S22. Sample input from training data for denoising diffusion model forward propagation: Randomly sample a pseudo-time τ from [0,1] and add a given variance σ(τ) to the input data. 2 Random noise is added to obtain noisy samples:

[0104]

[0105] x(0) is a single training sample, and ε is standard Gaussian noise. It is a Gaussian distribution with a mean of 0 and a standard identity matrix equal to the covariance matrix;

[0106] The noisy sample is input into the denoising diffusion model for forward propagation calculation to obtain the output s. θ (x(τ),τ);

[0107] S23. Calculate the loss function using the score matching algorithm:

[0108]

[0109] in, Given the expectation of the distribution, λ(τ) represents a given series of weighting coefficients, the subscript τ ~ U(0,1) represents the pseudo-time step, which is sampled from the uniform distribution U(0,1) during training and determines the magnitude of the Gaussian noise added to the samples; the subscript p0(x(0)) represents the probability density function of the distribution followed by the training data samples, p 0τ (x(τ)|x(0)) is the probability density function of the conditional distribution of the noisy sample x(τ) given the training data sample x(0). The operator for calculating the gradient of a noisy sample x(τ).

[0110] S24. Calculate the gradient of the denoising diffusion model parameters using the backpropagation algorithm, and iteratively update the model parameters accordingly:

[0111]

[0112] Where η is the given learning rate, this process is performed until convergence.

[0113] S25. Sampling from a Gaussian distribution Performing the denoising process in S21 yields a random physical field sample that randomly falls within the solution function space; x(1) is a single sample that satisfies a Gaussian distribution. The mean is 0 and the variance is σ(τ). 2 The Gaussian distribution.

[0114] S3. Acquisition of experimental measurement data, specifically including:

[0115] S31. For the physical field of interest, use the corresponding experimental measurement device to measure the specific parameters of the physical field;

[0116] S32. Specific parameters of a physical field are measured using a measurement sensor, and the raw measurement data is converted into storable digital signals through a data acquisition system.

[0117] S33. Preprocess the collected data using a computer program: First, the physical field is dimensionless. Based on the characteristic length, characteristic time, and other characteristic quantities of the physical phenomenon to be measured, the measured data is dimensionless to obtain comparable results. The coordinate system of the measurement system is appropriately transformed to align with the coordinate system in the theoretical calculation to obtain the final measurement data, which is convenient for subsequent processing.

[0118] S4. Information Fusion: Prior information refers to the probability distribution of the complete physical field data learned by the neural network, while posterior information refers to new, incomplete measurement information about the physical field. This invention aims to obtain the complete information of the physical field by using this incomplete posterior information as a condition, specifically through Bayes' theorem, which provides the transformation relationship between the prior and posterior distributions. Specifically, it includes:

[0119] S41. Determine the posterior sampling method.

[0120] For the same prior distribution of the physical field obtained from training, the joint expression of the posterior distribution and the prior distribution can be obtained using Bayes' theorem:

[0121]

[0122] Where p(y) is the posterior distribution; p(x) is the prior distribution; p(y|x) is the probability density function of the conditional distribution satisfied by the incomplete observation information y given complete physical field information x; p(x|y) is the final desired complete physical field conditional distribution. y represents the observation data, considered as partial observation data of the complete physical field information, i.e.:

[0123]

[0124] Where H represents the observation operator, and ε represents the measurement noise. To measure the variance of noise, The mean is 0 and the variance is The Gaussian distribution.

[0125] Since the conditional distribution p(x|y) is the desired final distribution, but not every term on the right-hand side of the equation is available, different simplification methods will yield different posterior approximation methods. For the denoising diffusion model, obtaining the scores of this conditional distribution at each pseudo-time step is sufficient to sample from the posterior distribution, i.e.:

[0126]

[0127] The score s(x|y,τ) is the gradient of the conditional distribution p(x|y) with respect to x at time τ, which is the score of the left-hand side of the above joint expression. This score cannot be obtained directly and needs to be calculated using the joint expression because the right-hand side can be approximated by known information. The first term on the right-hand side of the above equation represents the score of the data distribution at time τ, which can be replaced by the function fitted by the trained neural network; the second term represents the score of the conditional distribution of the observed data y given a noisy sample x at time τ. After establishing the relationship with the data distribution x(0) through the Tweedie formula, the corresponding expression is obtained by substituting it into the observation operator, and finally a usable posterior sampling method is obtained, namely:

[0128]

[0129] The first term within the square brackets on the right-hand side of the above equation represents the guidance of experimental observation data, and the second term represents the prior information of the physical field. The residuals representing the governing equations of the physical field are used to guide the generation process of prior information about the physical field to gradually satisfy the constraints of the physical equations; s θ (x|y,τ) is the approximate score of the conditional distribution p(x|y,τ) related to the neural network parameters at time τ; s θ (x(τ),τ) represents the score of the probability distribution of the noisy training samples approximated by the neural network at time τ; := is the assignment symbol; α(τ) is an adjustable weight parameter; The operator for calculating the gradient of a noisy sample x(τ); τ represents the observation calculated from the samples estimated by the neural network; |||| represents the vector 2 norm; β(τ) is an adjustable weight parameter; This is the estimate of the original sample obtained by the neural network in one step of denoising at time τ.

[0130] S42. After obtaining the scores of the final complete physical field condition distribution p(x|y) at each pseudo-time step, the posterior distribution sampling that conforms to the prior physical knowledge is then implemented based on the scores. The sampling process includes:

[0131] Random noise is sampled from the Gaussian distribution of the joint probability density function of the training data. As the initial condition for the denoising process, the denoising process is determined by the inverse denoising process defined in S21. It simply involves replacing the prior score with the posterior score to fuse the observation information y, thereby performing a posterior correction on the generation process, as follows:

[0132]

[0133] in, Let x(0) be the mean and σ(τ) be the mean. 2 Artificially noisy samples with variance; s θ (x|y,τ) is the approximate score of the conditional distribution p(x|y,τ) related to the neural network parameters at time τ; w is a standard Brownian process.

[0134] S43. The stochastic differential equation solver based on the prediction correction scheme performs a reverse denoising process on the sampled initial conditions. The reverse denoising process is obtained by simultaneously integrating both sides of the equation in S42. The purpose is to solve the equation using a numerical integrator. Specifically:

[0135]

[0136] Where ∫ is Itō's random integral; x(0)|y is the physical field sample under the required experimental observation conditions.

[0137] The following example of Darcy flow illustrates the relationship between the pressure field u and porosity a when a fluid flows through a porous medium. This relationship can be described by the following equation.

[0138]

[0139]

[0140] Where p∈Ω=[0,1] 2 Represents two-dimensional spatial coordinates Given the known source terms, the physical parameters that need to be modeled are:

[0141]

[0142] The solution function space is determined by choosing different values ​​of 'a'. Given f and 'a', the equation is discretized onto a uniform grid in space (e.g., 100×100), and the physical field parameters at the discrete points can be obtained by using existing numerical methods. Different values ​​of f and a yield a series of physical field parameters, i.e., a training set:

[0143] After obtaining the training set, the joint probability distribution of the target physics parameters, i.e., p, needs to be obtained through a generative model. θ (x)←p(x)=p(u,a). Based on the characteristics of this physical field, a 2D U-Net neural network structure is designed with input of B×2×100×100 and output of the same size.

[0144] Through experiments, pressure sensors such as Pitot tubes can be used to measure the pressure of actual fluid flowing through porous media at certain spatial locations. After preprocessing operations such as dimensionless conversion and coordinate alignment, the observed data y is obtained. Substituting this as posterior information into the solution process of S4, the distribution of the real and complete pressure field u and porosity field a in this region can be obtained.

[0145] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0146] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. An intelligent testing method that integrates experimental measurement data and prior physical knowledge, characterized in that, Includes the following steps: S1. Based on the physical model, obtain the particular solution information of the corresponding physical field under different boundary conditions, determine the spatiotemporal distribution information and interaction relationship of the parameters in the physical field, and convert the particular solution information into structured data information as training data; S2. Fit the joint probability distribution of the training data based on the generative model, encode the joint probability distribution of the data into a Gaussian distribution, and randomly sample from the Gaussian distribution as the prior distribution information of the physical field. S3. Obtain measurement data of relevant parameters under the corresponding physical field based on the experimental measurement device, and use it as posterior distribution information; S4. Fuse the prior distribution information and the posterior distribution information of the physical field to obtain complete physical field information; In S4, Bayes' theorem is used to obtain the joint expression of the posterior distribution and the prior distribution: in, These are observational data, considered as partial observational data representing complete information about the physical field. It is a posterior distribution; It is the prior distribution; Given complete physical field information Incomplete observation information The probability density function of the conditional distribution; To achieve the desired complete distribution of physical field conditions; In S4, when the generative model is a denoising diffusion model, the desired complete physical field condition distribution is obtained. After scoring at each pseudo-time step, a posterior correction is performed on the sampled samples based on the scores; the final desired complete physical field condition distribution is then determined. The score is calculated at each pseudo-time step as follows: The first term within the square brackets on the right-hand side of the above equation represents the guidance of experimental observation data, and the second term represents the prior information of the physical field. The residuals represent the governing equations of the physical field, which are used to guide the generation process of prior information of the physical field to gradually satisfy the constraints of the physical equations. In order to be in Conditional distribution of time and neural network parameters Approximate score; In order to be in The score of the probability distribution of the noisy training samples approximated by the neural network at any given time. This is an assignment operator; These are adjustable weight parameters; To add noise to the samples Operator for finding gradient; These are the observations calculated from samples estimated by the neural network. It is the 2-norm of the vector; These are adjustable weight parameters; For neural networks in Estimation of the original sample obtained by one-step denoising at each time step; Based on the desired complete physical field condition distribution The posterior correction of the sampled samples based on the scores at each pseudo-time step includes: Random noise is sampled from a Gaussian distribution. As an initial condition for the denoising process, the following reverse denoising is performed: in, For mean Artificially noisy samples with variance; The magnitude of the variance; In order to be in Conditional distribution of time and neural network parameters Approximate score; This is a standard Brownian process; The stochastic differential equation solver based on the prediction correction scheme performs an inverse denoising process on the sampled initial conditions: in, For Ito's random integral; For the physical field samples under the required experimental observation conditions; For a single sample that satisfies a Gaussian distribution.

2. The intelligent testing method for integrating experimental measurement data and prior physical knowledge according to claim 1, characterized in that, S1 includes: S11. Determine the governing equations of the physical field, which are expressed in the following form: Among them, the operator Acting on the solution function Above represents the entire governing equation and the solution function. From the solution domain Mapped to 3D physical parameter space; when the governing equations remain unchanged, the solution function From the boundary conditions The solution domain is determined by providing a series of boundary value conditions. Initial conditions, boundary conditions, and physical property parameters; S12. Determine the boundary conditions for the governing equations; S13. Sample a series of solutions in the solution function space. Given the spatial and temporal coordinates and physical field parameters, interpolate the numerical solutions onto the corresponding coordinates to form structured training data of [coordinates, physical parameter 1, physical parameter 2, ...].

3. The intelligent testing method for integrating experimental measurement data and prior physical knowledge according to claim 1, characterized in that, In S12, if a theoretical solution is required, its analytical solution is obtained through formula derivation, and the specific spatial and temporal location points are substituted to obtain usable numerical data; if a numerical solution is required, the solution domain is discretized to obtain a series of grids, and the equations are discretized and transformed into a system of linear equations about the parameter variables to be solved at the grid points using numerical calculation methods, and then the system of linear equations is solved by iterative methods.

4. The intelligent testing method for integrating experimental measurement data and prior physical knowledge according to claim 1, characterized in that, In S1, each sample in the training data contains numerical information of all important parameters of the physical field at different discrete spatiotemporal locations within a given region under a definite solution condition. Different samples correspond to the particular solution information of the physical field under different definite solution conditions.

5. The intelligent testing method for integrating experimental measurement data and prior physical knowledge according to claim 1, characterized in that, S2 include: S21. Construct a corresponding neural network-based generative model based on the structure of the physical field data; when the generative model adopts a denoising diffusion model, the process for determining its fitting function is as follows: Construct a forward noise-adding process that continuously adds noise to the training data so that its joint probability density function exhibits a Gaussian distribution; The reverse denoising process is performed based on the scores of the variables during the noise addition process. Determine the function to fit the denoising diffusion model: in, These are the learnable parameters for the denoising diffusion model. To obtain the score for each step by fitting a denoising diffusion model, For pseudo time steps, For The mean, Artificially noisy samples with variance; S22, from Randomly sample a pseudo-time And add a given variance to the input data. Random noise is added to obtain noisy samples: in, For a single training sample, Standard Gaussian noise, It is a Gaussian distribution with a mean of 0 and a standard identity matrix equal to the covariance matrix; The noisy sample is input into the denoising diffusion model for forward propagation calculation to obtain the output. ; S23. Calculate the loss function using the score matching algorithm; S24. Calculate the gradient of the denoising diffusion model parameters using the backpropagation algorithm, and iteratively update the model parameters accordingly. S25. Sampling from a Gaussian distribution The denoising process in S21 is executed to obtain a random physical field sample, which randomly falls within the solution function space. For a single sample to satisfy a Gaussian distribution, The mean is 0 and the variance is The Gaussian distribution.

6. The intelligent testing method for integrating experimental measurement data and prior physical knowledge according to claim 5, characterized in that, In S23, the expression for the loss function is: in, Given the expectation of the distribution, For a given set of weighting coefficients, the subscripts... As a pseudo-time step, training starts from a uniform distribution. The sampled data determines the magnitude of the Gaussian noise added to the sample; subscript Let the probability density function be the distribution that the training data samples follow. Given a sample of training data Time-added noisy samples The probability density function of the conditional distribution, To add noise to the samples Operator for calculating gradient.

7. The intelligent testing method for integrating experimental measurement data and prior physical knowledge according to claim 1, characterized in that, S3 include: S31. For the physical field of interest, use the corresponding experimental measurement device to measure the specific parameters of the physical field; S32. Convert the raw measurement data into a storable digital signal; S33. Perform dimensionless transformation and coordinate transformation on the digital signal to obtain the final measurement data.

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