Spatiotemporal dynamic system soft measurement method for automatically determining partial differential equation structure

Abstract

A spatiotemporal dynamic system soft measurement method for automatically determining a partial differential equation structure, belonging to the technical field of the soft measurement of neural networks. The method comprises: first, constructing a loss function used for training a coupled physics-informed neural network having a recurrent prediction mechanism, so as to obtain solutions and source items that meet a partial differential equation used for describing a spatiotemporal industrial process; second, using an automatic differential method to obtain a differential operator candidate library, and selecting an appropriate partial differential equation structure from the differential operator candidate library to accurately describe the spatiotemporal industrial process; and finally, using a heat diffusion phenomenon and an actual vibration process to verify a soft measurement result. The CPINNRP-AIC provided in the present invention is applicable to a soft measurement method for multiple types of dynamic systems having spatiotemporal dependency features, comprising a heat diffusion phenomena in an actual industrial process, a fluctuation phenomena in an actual vibration process, etc. The present invention can realize the effective acquisition of key variable values during the operation of high-end complex equipment, such as an aero-engine, thereby providing assurance for a good control performance and safe operation stability of high-end industrial equipment, and has practical engineering application significance.

Description

A soft sensing method for spatiotemporal dynamic systems to automatically determine the structure of partial differential equations Technical Field

[0001] The invention belongs to the technical field of soft measurement based on neural networks, and relates to a soft measurement method for a spatiotemporal dynamic system for automatically determining a partial differential equation structure. Background Art

[0002] As industrial standards continue to improve, high-end industrial equipment such as aircraft engines is becoming increasingly complex and large-scale, increasing the likelihood of failure. The safety and reliability of industrial equipment are therefore crucial. During actual operation, complex industrial equipment contains numerous key variables that reflect the equipment's health status. Obtaining the values ​​of these variables is essential for ensuring the safe and stable operation of the equipment. However, due to the extreme operating environment constraints imposed by sensors in actual applications, such as high temperature, high pressure, and mounting location, it is difficult to obtain the measured values ​​of these key variables, posing a significant threat to the reliable, safe, and stable operation of high-end industrial equipment.

[0003] Soft sensing, as a virtual sensing technology, estimates key variables by leveraging easily measurable process variables and mathematical models. Therefore, for critical process variables in high-end industrial equipment, soft sensing plays a crucial role in ensuring the safety and reliability of equipment when sensors are unavailable. Classic observer-based approaches use time-series data to implement soft sensing. However, these methods cannot easily handle spatial information. Partial differential equations are used to describe industrial processes involving multiple variables. For example, the heat conduction equation models the diffusion of heat through a given area; the wave equation describes various wave phenomena; Maxwell's equations describe how changes in charge, current, and field generate electric and magnetic fields; and the Navier-Stokes (NS) equations describe the motion of viscous fluids. Therefore, partial differential equations (PDEs) are a natural candidate for soft sensing mathematical models for industrial processes with spatiotemporal dependencies. Appropriate PDE structures are crucial for optimization, control, and prediction in real-world industrial applications. Note that the idealized PDEs described above focus on a single phenomenon or mechanism. However, complex industrial processes often involve multiple disciplines. For example, the internal combustion engine involves thermodynamics and fluid mechanics. Integrating several idealized partial differential equations into a system of equations is a purely physical approach to industrial processes. However, the predetermined structure of partial differential equations in physical approaches does not always describe actual industrial processes well. Therefore, constructing partial differential equations that approximate the appropriate structure of actual industrial processes is crucial for soft sensing.

[0004] In recent years, the use of machine methods to develop data-physics hybrid models to solve the forward and inverse problems in partial differential equations is an effective way to solve the above problems. Physical information neural networks (PINN) have been introduced to solve partial differential equations through sparse measurements and prior physical knowledge (Raissi M, Perdikaris P, Karniadakis G E.Physics-informed neural networks:A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[J].Journal of Computational physics,2019,378:686-707.). Within the PINN framework, physical information loss is used to regularize the data loss part, so when the PDE structure including differential operator terms or unknown source terms is unknown, it will make it infeasible to use PINN to solve the PDE method. Therefore, in recent years, some research work has been devoted to the learning of PDE and solving PDE under unknown source terms. According to surveys and studies, some recent progress in solving partial differential equations with unknown source terms within the PINN framework has been made. The literature (M.Yang and JTFoster, “Multi-output physics-informed neural networks for forward and inverse pde problems with uncertainties,” Computer Methods in Applied Mechanics and Engineering, p.115041, 2022.) shows that the functional forms of both the solution and the source term are assumed to be unknown, where the measurement of the source term can be obtained separately from the measurement of the solution. A recent literature (H.Gao, MJZahr, and J.-X.Wang, “Physics-informed graph neural galerkin networks: A unified framework for solving pde-governed forward and inverse problems,” Computer Methods in Applied Mechanics and Engineering, vol.390, p.114502, 2022) proposed a graph neural network to solve partial differential equations with unmeasurable source terms, where the source term is assumed to be a constant.Although the above research works have achieved some success, independent measurement of source terms and the use of constant source terms to describe dynamic systems are not feasible for actual industrial processes. Other progress shows that PINN is a promising framework for identifying partial differential equations from spatiotemporal data. Recent research works (H.Xu, H.Chang, and D.Zhang, “Dl-pde: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data,” arXiv preprint arXiv:1908.04463, 2019. and Z.Chen, Y.Liu, and H.Sun, “Physics-informed learning of governing equations from scarce data,” Nature communications, vol.12, no.1, p.6136, 2021.) combine deep learning through neural networks and data-driven partial differential equation identification through sparse regression. The paper (Z. Long, Y. Lu, and B. Dong, “Pde-net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network,” Journal of Computational Physics, vol. 399, p. 108925, 2019) identifies PDEs using a combination of convolutional numerical approximation of differential operators and symbolic multilayer neural networks for model recovery. Although these studies have achieved some success in identifying PDEs, they do not consider the form of partial differential equations driven by unknown sources. The literature (Sun Ximing, Wang Eina, Qin Pan. Coupled physical information neural network for solving the displacement distribution of a bounded vibrating rod under unknown external driving force [P]. Liaoning Province: CN116050247A, 2023-05-02) uses idealized partial differential equations to achieve effective soft measurement of dynamic industrial processes with spatiotemporal dependence under the conditions where the form of the source term with spatiotemporal dependence is unknown and its independent measurement value is unavailable. However, there is still a gap between idealized partial differential equations and actual industrial processes. Therefore, the development of effective soft measurement algorithms for dynamic industrial processes with spatiotemporal dependence with completely unknown partial differential equation structures and source terms is an understudied problem.

[0005] Summary of the Invention

[0006] In response to the current problems, this application proposes a soft measurement technology (CPINNRP-AIC) based on a coupled physical information neural network (CPINN) with a cyclic prediction (RP) mechanism combined with the Akaike Information Criterion (AIC), which is used to obtain the partial differential equation structure that satisfies the description of dynamic industrial processes with spatiotemporal dependence characteristics, and then obtain solutions and source terms. Through the method provided by the present invention, appropriate soft measurement results can be obtained to ensure system optimization, control and prediction, which is of great significance for ensuring the actual safe and stable operation of high-end industrial equipment such as aircraft engines. The present invention applies the proposed CPINNRP-AIC to soft measurement of dynamic industrial processes with spatiotemporal dependence characteristics with completely unknown PDE structures, and verifies it using the heat conduction equation describing the heat diffusion phenomenon and data collected from the aircraft engine spline test bench. The effectiveness of the proposed method is verified using the evaluation indicators root mean square error (RMSE) and Pearson correlation coefficient (CC).

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

[0008] A soft measurement method for spatiotemporal dynamic systems for automatically determining the structure of partial differential equations is a soft measurement method that combines a coupled physical information neural network with a cyclic prediction mechanism with the Akaike information criterion. First, a loss function is constructed for training the coupled physical information neural network with a cyclic prediction mechanism to obtain solutions and source terms that satisfy the partial differential equations used to describe spatiotemporal industrial processes. Second, an automatic differentiation technique is used to obtain a candidate library of differential operators. Third, the Akaike information criterion is used to select appropriate partial differential equation structures from the candidate library of differential operators to accurately describe spatiotemporal industrial processes. Finally, the thermal diffusion phenomenon and actual vibration process are used to verify that the method proposed by the present invention can obtain feasible soft measurement results. Specifically, the method includes the following steps:

[0009] The partial differential equations used to describe dynamic industrial processes with time-space dependence have the following general form: t (x,t)+N[u(x,t)]=g(x,t),x∈Ω,t∈[0,T] (1)

[0010] x is a spatial variable, t is a time variable, t=0 is the initial state, is the solution of the equation, i.e. the variable to be measured in the industrial process with spatiotemporal dependence characteristics, and the variable has spatiotemporal dependence; is a source term with a general form, that is, the driving source has time and space dependence, including linear, nonlinear, and steady-state or dynamic. Ω is the open set space to which the spatial variable belongs, is its boundary; N is a series of differential operators.

[0011] When the exact PDE structure of an industrial process with spatiotemporal dependencies is unknown, it can be described by the following general partial differential equation. The proposed CPINNRP-AIC is used to solve the following partial differential equation:

[0012] Ф[·] is a series of candidate differential operator terms for partial differential operators, such as u t ,u x ,u xx ,u tx , which is assumed to be sufficient to fully include all differential operator terms obtained using automatic differentiation, that is, a series of states of a dynamic industrial process with time and space dependence characteristics that changes over time and space.

[0013] The present invention is to learn to obtain the solution u and source term g that can approximately satisfy the formula (1) from the formula (2); the formula (2) can be written in the form of the following residual function: f N (x,t):=Ф[u(x,t)]·λ-g(x,t). (3)

[0014] When the external driving force g(x,t) is completely known, f N (x, t) can be directly obtained by sparse regression method. However, the unknown external driving force g(x, t), that is, the dynamic industrial process under the unknown external driving force, will lead to the sparse regression method to obtain f N (x,t) method is not feasible.

[0015] Therefore, the present invention proposes CPINNRP-AIC to obtain a neural network-based solution to the partial differential equation in formula (1). The proposed CPINNRP-AIC includes two stages: 1) CPINNRP is used to approximate u and g that satisfy the partial differential equation in formula (1); 2) AIC is used to determine an appropriate combination of differential operators of the partial differential equation to satisfy the partial differential equation in formula (1). CPINNRP consists of NetU, NetG and NetU-RP, where NetU is used to approximate u that satisfies (1); NetG is used to approximate g that satisfies (1), and NetU-RP is used to compensate for the information deficiency caused by the discretization sampling strategy for t. The specific steps are as follows:

[0016] Step 1: Construct the loss function for training CPINNRP.

[0017] Use the full runspace The installable hardware sensors provide a training data set (x, t, u)∈D, where the installable hardware sensors include the installable sensors inside the operating space Ω and the operating space boundary The dataset is divided into D B ∪DI and D B and D I D is obtained from the boundary and initial conditions of Ω and the internal random sampling, respectively. B Indicates that the run space boundary Installable sensors and initial sampled training set, D I represents the training set sampled from the sensors that can be installed in the operating space Ω; x, t, u represent the spatial variable, time variable, and variable to be measured with spatiotemporal dependency, respectively; D represents the training data set. The corresponding configuration point set E = E is obtained according to the positions of the available sensors. B ∪E I , where (x, t)∈E, E represents the set of configuration points in the entire space; E B Represents the corresponding training data set D B The set of configuration points E I Represents the corresponding training data set D I The data-physics hybrid loss function shown in formula (4) is used to train CPINNRP-AIC. MSE = MSE DN +MSE PN (4)

[0018] Among them, MSE DN and MSE PN They represent the data loss function and physical information loss function of the given formula (2) that does not determine the exact PDE structure, where the physical information loss function contains the physical information part of the combination of the undetermined differential operator terms.

[0019] The MSE DN It is obtained from the following formula:

[0020] in, is a function of the network NetU, whose parameter set is Θ U , which is used to approximately satisfy formula (1) u, ​​card(·) is the cardinality of the set ·.

[0021] The MSE PN It is obtained from the following formula:

[0022] Furthermore, based on the combination of undetermined differential operators get As shown in formula (7):

[0023] in, is a function of the network NetG, whose parameter set is Θ G, which is used to approximately satisfy g of formula (1); λ represents the parameter vector; It means that formula (3) is (Θ U ,λ) parameterized residual function form.

[0024] Step 2: Optimize the coupled CPINNRP-AIC with an undetermined exact PDE structure using a hierarchical training strategy.

[0025] Considering the interdependence between the network NetU and the network NetG in formula (4), a hierarchical training strategy is proposed to train CPINNRP-AIC by iterative parameter transmission, where and are used to approximate u and g respectively. Here, k represents the number of iterations. The purpose of the hierarchical training strategy is to solve the following two coupled optimization problems.

[0026] and

[0027] in, is the physical information part containing the combination of undetermined differential operators, where is based on obtained; is based on obtained;

[0028] In order to make up for the information loss problem caused by time discretization sampling, the output of CPINN The loop delay output of the hardware sensor is used as part of the NetU-RP input, that is, the input of NetU-RP consists of three parts: x, t and Or the loop-delayed output of a hardware sensor. The output of the hardware sensor is selected in an either-or manner, that is, when the hardware sensor can be installed at the configuration point (x, t), the output delay of its measured value is used as the input of the configuration point; when the hardware sensor data cannot be obtained at the configuration point (x, t), the output delay of the measured value is used as the input of the configuration point. The delay is used as the input of the configuration point. Therefore, the undetermined differential combination term is obtained. and g that satisfies formula (1).

[0029] Step 3: Use AIC to select an appropriate differential operator combination that satisfies formula (1).

[0030] Idealized models, such as the heat conduction equation, wave equation, Maxwell equation, and NS equation, are involved in complex industrial processes with many interference factors. Therefore, there is often a gap between the ideal situation and the actual situation. For this reason, it is considered to use AIC to evaluate and select the differential combination terms for evaluation and selection.

[0031] Among them, p is the number of differential operator terms included in the evaluation , n is the size of the data set, The selected combination of differential terms realizes the correct PDE structure, that is, the combination with the minimum AIC value is selected from all candidates as the final PDE model for approximating formula (1). The hierarchical optimization coupling strategy of CPINNRP-AIC and the details of model selection are shown in Algorithm 1.

[0032] Algorithm 1: Hierarchical optimization coupling strategy and model selection of CPINNRP-AIC:

[0033] (1) Initialization: Let k = 0, initialize the network NetU and the network NetG, and set the maximum number of iteration steps k as M, which is used as the loop stop condition;

[0034] 1.1) Obtain the training data set (x, t, u) ∈ D and the collocation points (x, t) ∈ E;

[0035] 1.2) Randomly generate the initialization parameter sets of the network NetU and the network NetG and

[0036] 1.3) When k < M, execute the following loop:

[0037] (a1) Obtain by solving the optimization problem (8), where the MSE PN in<​​​​​​​​​​​​​​​​​​​​

[0041] (2) Initialization: Let k = 0, and use to initialize the network NetU-RP;

[0042] (2.1) Except for x and t, the output of CPINN and the measured values are input into NetU-RP in an either-or manner through a cyclic delay of time according to the availability of the hardware sensors;

[0043] (2.2) Execute the following loop when k < M:

[0044] (a2) Train NetU-RP using formula (4);

[0045] (2.3) Determine whether the loop stop condition is satisfied: If yes, end the loop and return the candidate differential terms obtained using automatic differentiation If no, return to step (a2).

[0046] (2.4) Calculate the AIC value from the candidate differential term combinations using formula (10);

[0047] (2.5) Select the differential term combination with the minimum AIC as the final model for approximating formula (1).

[0048] Step 4: Evaluate the performance of the CPINNRP-AIC method in the soft sensing method for spatio-temporal dynamic systems that automatically determines the partial differential equation structure, that is, evaluate the and performance of u and g that approximate (1) obtained by training CPINNRP-AIC.

[0049] Use the root mean square error (RMSE) and Pearson correlation coefficient as evaluation indicators to evaluate the performance of CPINNRP-AIC.

[0050] Among them, RMSE is used to evaluate the distance between the actual value and the predicted value, |T| is the cardinality of the set of test configuration points (x, t) ∈ T, u(x, t) and respectively represent the actual measured value and the neural network-based solution corresponding thereto. The smaller the value of RMSE, the better the performance of CPINNRP-AIC.

[0051] Among them, CC is used to evaluate the similarity with the predicted value. CC is the correlation coefficient between u(x, t) and is the covariance between u(x, t) and Var u(x, t) and They are u(x,t) and The closer the CC value is to 1, the better the CPINNRP-AIC performance. Both RMSE and AIC provide snapshot graphs and evaluation indicator values ​​for the entire time period.

[0052] Without loss of generality, any industrial process with time-space dependence and unknown precise PDE structure and driving source can be described by formula (2), which typically includes: (a) the heat conduction equation describing the heat diffusion phenomenon: the heat inside the bounded rod is diffused under the drive of the heat source, and the active heat conduction equation describing the actual temperature distribution inside the bounded rod. For example, the amount of heat generated by the internal aero-engine during actual operation cannot be measured, and the specific heat diffusion phenomenon cannot be described by the idealized heat conduction equation. To obtain the temperature distribution at any point, the heat conduction equation with the unknown precise PDE structure is solved; (b) the wave equation describing the wave phenomenon: the vibration displacement distribution of the bounded rod is described by the wave equation. For example, the spline shaft of the aero-engine vibrates during operation and is affected by multiple driving sources, and involves multiple dynamic processes with time-space dependence. The idealized wave equation cannot accurately describe the vibration process, and to obtain the vibration displacement distribution at any point of the object under study, the wave equation with the unknown precise PDE structure is required.

[0053] The beneficial effects of the present invention are:

[0054] This paper proposes a new PINN, called CPINNRP-AIC, for approximating the solutions and source terms of partial differential equations used to describe real dynamic industrial processes with spatiotemporal dependencies. In this paper, first, CPINNRP is used to approximate the solutions to the partial differential equations and obtain the source terms that satisfy the partial differential equations. Then, we propose a data-physics hybrid loss function for training CPINNRP, where the physical information part involves the undetermined combination of differential operators. Therefore, AIC is used to determine the appropriate combination of differential operators. The proposed CPINNRP-AIC is a data-driven method for achieving appropriate PDE structures and neural network-based solutions that satisfy the PDEs. Finally, the feasibility and effectiveness of CPINNRP-AIC for soft sensing of dynamic industrial processes with spatiotemporal dependencies with completely unknown PDE structures are verified using the heat conduction equation and an aviation spline vibration test bench. The results validated the performance of CPINNRP-AIC using RMSE and CC, with the results approaching 0 and 1, respectively. This demonstrates that CPINNRP-AIC performs well as a soft-sensing method for dynamic systems with spatiotemporal dependencies and unknown PDE structures. Furthermore, the proposed CPINNRP-AIC is applicable to soft-sensing techniques for a wide range of dynamic systems with spatiotemporal dependencies, including those associated with thermal diffusion in real industrial processes and fluctuations in vibrations. The proposed CPINNRP-AIC can effectively capture the values ​​of key variables during the operation of high-end, complex equipment, such as aircraft engines, thereby ensuring the excellent control performance and stable operation of high-end industrial equipment, and possesses both practical engineering applications and economic value. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 is a histogram of the AIC size of the number of typical model terms in the heat conduction equation describing heat diffusion.

[0056] Figure 2 shows the predicted and actual values ​​corresponding to the snapshot at t = 3;

[0057] Figure 3 shows the predicted and actual values ​​corresponding to the snapshot at t = 7;

[0058] Figure 4 is the original time domain displacement signal sampled from sensor 1;

[0059] Figure 5 is the original time domain displacement signal sampled from sensor 2;

[0060] FIG6 is the original time domain displacement signal sampled from sensor 3;

[0061] FIG7 is the original time domain displacement signal sampled from sensor 4.

[0062] Figure 8 is a histogram of the AIC size of the number of terms in the typical model of the wave equation describing the actual vibration process. DETAILED DESCRIPTION

[0063] The present invention will be further described below with reference to specific embodiments.

[0064] The present invention provides a soft-sensing method for spatiotemporal dynamic systems for automatically determining the structure of partial differential equations. The method utilizes a coupled physical information neural network with a cyclic prediction mechanism combined with the Akaike Information Criterion to solve partial differential equations. The partial differential equations are used to describe dynamic industrial processes with spatiotemporal dependence characteristics, and their precise PDE structure, including differential operator combinations and source terms, is unknown. The specific embodiments discussed are merely illustrative of the embodiments of the present invention and do not limit the scope of the invention. The embodiments of the present invention are described in detail below with reference to the accompanying drawings, specifically including the following steps:

[0065] In order to theoretically illustrate the effectiveness of CPINNRP-AIC, Example 1 first uses a numerical simulation of a one-dimensional heat conduction equation to illustrate the embodiment of the present invention.

[0066] Example 1: In an actual industrial system, the thermal diffusion phenomenon of a one-dimensional bounded rod under the influence of an unknown heat source with one end adiabatic and the other heat sink, i.e., the measured temperature distribution at any (x, t) point, is investigated. Therefore, this Example 1 utilizes CPINNRP-AIC to solve the one-dimensional heat conduction equation for an unknown precise model structure with Neumann boundary conditions, obtaining soft measurement results.

[0067] To obtain the training set and the test set, the following partial differential equation is given to describe the heat diffusion phenomenon of a one-dimensional bounded rod driven by a heat source with one end as an adiabatic end and the other end as a heat dissipation end.

[0068] Thermal diffusivity a = 1, u(x,t) is the temperature at any position (x,t), the length of the bounded rod L = π, and the initial temperature heat source The analytical expression of temperature distribution is

[0069] Considering the heat diffusion phenomenon in actual industrial systems, the exact partial differential equation structure and heat source are unknown, and the heat source measurement value cannot be measured. The heat conduction equation describing the unknown exact model and source term of the one-dimensional bounded rod temperature distribution is converted into a PDE equation in the residual form as shown in Equation (3). N (x,t):=Ф[u(x,t)]·λ-g(x,t). (14)

[0070] To this end, CPINNRP-AIC consists of two stages: 1) CPINNRP is used to approximate u and g that satisfy (13); 2) AIC is used to determine the appropriate differential operator term ut ,u xx CPINNRP consists of NetU, NetG, and NetU-RP, where NetU is used to approximately satisfy u in (13); NetG is used to approximately satisfy u in (13), and NetU-RP is used to compensate for the information deficiency caused by the discretization sampling strategy for t.

[0071] 1. Construct loss function:

[0072] The training set is obtained by uniform random sampling from the system controlled by Equation (13). In this embodiment, the training set is obtained by random uniform sampling in [0,π]×[0,10], including 130 boundaries and initial training data (x,t,u)∈D B , there are 10 initial condition training data, 60 left boundary condition training data and 60 right boundary condition training data, 20 internal training data (x, t, u) ∈ D I . 20 collocation points (x, t) ∈ E collocation point set, using collocation points to ensure the structure of formula (13). Use the loss function of formula (4) to train CPINNRP-AIC. MSE DN and MSE PN They represent data loss (5) and physical information loss (6) respectively. DN From (5) we get is a function of the network NetU, whose training parameter set is Θ U .MSE PN From formula (6), we can get is a function of the network NetG, whose training parameter set is Θ G .MSE PN corresponds to the physical information loss of (13) on a finite set of collocation points (x, t) ∈ E, and It involves differential operator terms with undetermined combinations, which are used to regularize u in the network NetU to satisfy (13).

[0073] 2. Optimizing coupled CPINNRP-AIC using a hierarchical training strategy with an undetermined exact PDE structure;

[0074] Considering the interdependence between the network NetU and the network NetG in the loss function MSE, a hierarchical training strategy is proposed. In many cases of practical applications, such as the heat conduction equation used to describe the internal heat diffusion phenomenon of an aircraft engine and the exact expression of the heat source g(x,t), even sparse measurements are not available. However, CPINNRP-AIC can be used to obtain the heat source g(x,t) based on the sparse measurement data D. I get and are used to approximately satisfy u and g in (13). Considering ΘU and Θ G The interdependent relationship is estimated by means of interactive iteration. Assume that k is the number of steps of the current iteration. The purpose of the hierarchical training strategy is to solve the solutions of the two optimization problems in (8) and (9).

[0075] 3. Use AIC to select an appropriate combination of partial differential operators that satisfy formula (13);

[0076] There is often a gap between the idealized heat conduction equation model describing the heat diffusion phenomenon and the actual situation. Therefore, consider using AIC to implement model evaluation and selection. For the heat diffusion phenomenon, we consider the first-order derivative with respect to the time variable. The candidate library is constructed as follows: Ф = [u t , u x , u xx , u xt . The AIC in (10) is then used to evaluate and select a suitable combination from Ф. Figure 1 shows a histogram of the AIC magnitudes of the number of terms in the typical heat conduction equation model describing the heat diffusion phenomenon. The results show that the combination of u t and u xx is the correct combination of heat differential operators to describe this heat diffusion phenomenon. The hierarchical optimization coupling strategy and model selection details of CPINNRP-AIC are shown in Algorithm 1.

[0077] Algorithm 1: Hierarchical Optimization Coupling Strategy and Model Selection of CPINNRP-AIC

[0078] (1) Initialization: Let k = 0, initialize the network NetU and the network NetG, and set the maximum number of iteration steps k as M, which is used as the loop stop condition;

[0079] 1.1) Obtain the training data set (x, t, u) ∈ D and the collocation points (x, t) ∈ E;

[0080] 1.2) Randomly generate the initialization parameter sets of the network NetU and the network NetG and<00003X09>

[0081] 1.3) When k < M, execute the following loop:

[0082] (a1) Obtain at this time MSE[[ID=4F]] PN in from the iteration result of the previous step

[0083] (b1) Obtain which is called

[0084] (c1) k = k + 1;

[0085] Judge whether the loop stop condition is satisfied: if yes, end the loop and return if no, return to step (a1) and execute parameter Θ U and Θ G Iteratively transmit the process and solve the optimization problems (8) and (9);

[0086] (2) Initialization: Let k = 0 and use Initialize the network NetU-RP;

[0087] 2.1) Except for x and t, the output of CPINN and the measured values are input into NetU-RP in an either-or manner through the cyclic delay of time according to whether the hardware sensors are available;

[0088] 2.2) Execute the following loop when k < M:

[0089] (a2) Train NetU-RP using formula (4);

[0090] 2.3) Judge whether the loop stop condition is satisfied: if yes, end the loop and return the candidate differential terms obtained by automatic differentiation if no, return to step (a2).

[0091] 2.4) Calculate the AIC value from the candidate differential term combinations using formula (10) ;

[0092] 2.5) Select the differential term combination with the minimum AIC as the final model for approximating (13).

[0093] 4. Evaluate the performance

[0094] Use (11) RMSE to evaluate the soft measurement error of the proposed CPINNRP-AIC method in the heat diffusion system where one end of the one-dimensional bounded rod is an adiabatic end and the other end is a heat dissipation end, that is, evaluate the performance of the CPINNRP-AIC method in solving the one-dimensional heat conduction equation with unknown exact PDE structure under Neumann boundary conditions, for f over the entire time period N (x,t) RMSE = 5.658725e-03, u(x,t) and respectively represent the actual temperature value and the corresponding soft measurement temperature value. To further verify the performance of CPINNRP-AIC, use (12) CC to further illustrate the similarity between the soft measurement temperature and the actual temperature of the CPINNRP-AIC method.

[0095] In this example, the CPINNRP-AIC algorithm was configured with three hidden layers and 30 neurons per layer. The prediction scale is shown in Figure 2 . Specifically, the comparisons between the predicted and actual values ​​for the snapshots at t = 3 and t = 7 are shown in Figures 3 and 4 , respectively. Table 1 shows the prediction performance evaluation metrics for t = 3, t = 7, and the entire time period.

[0096] Table 1 Soft sensing performance of CPINNRP-AIC based on unknown exact heat conduction equation in thermal diffusion of one-dimensional bounded rod

[0097] As shown in Table 1, RMSE is close to 0 and CC is close to 1, indicating that the CPINNRP-AIC of this embodiment has good soft sensing performance based on the unknown exact heat conduction equation in the one-dimensional bounded rod heat diffusion system.

[0098] Example 2: The vibration phenomenon of a one-dimensional bounded rod under the action of an unknown driving force in an actual industrial system, that is, the measurement result is the vibration displacement distribution of any (x, t). Therefore, this Example 2 is to use CPINNRP-AIC to solve the one-dimensional wave equation of the unknown precise model structure, and then obtain the soft measurement result. The feasibility and effectiveness of CPINNRP-AIC soft measurement are verified using actual data collected from the micro-wear test platform of the involute spline coupling of an aircraft engine. The training and test data sets are sampled from displacement sensors 1-4 installed on the micro-wear test platform of the involute spline coupling of an aircraft engine. The sampling conditions of the data sets shown in Figures 4-7 are as follows, wherein Figure 4 is the time domain data of the vibration displacement measured from sensor 1, Figure 5 is the time domain data of the vibration displacement measured from sensor 2, Figure 6 is the time domain data of the vibration displacement measured from sensor 3, and Figure 7 is the time domain data of the vibration displacement measured from sensor 4.

[0099] 1) The operating speed of the motor driver is 3000r / min.

[0100] 2) The sampling frequency is 2048 Hz, with 4096 sampling points.

[0101] Considering the vibration processes in real industrial systems, the exact partial differential equation structure used to describe the vibration process is unknown, and the driving force cannot be measured. Therefore, a two-stage CPINNRP-AIC soft sensing method is used: 1) CPINNRP is used to approximate the displacement u and driving force g that satisfy the partial differential equations; 2) AIC is used to determine the appropriate partial differential equation structure. CPINNRP consists of NetU, NetG, and NetU-RP. NetU approximates the displacement u that satisfies the partial differential equation; NetG approximates the driving force g that satisfies the partial differential equation; and NetU-RP compensates for the information gap caused by the discretized sampling strategy for t.

[0102] 1. Construct a loss function.

[0103] MSE DN and MSE PN Data loss (5) and physical information loss (6) respectively. is a function of the network NetU, whose training parameter set is Θ U .MSE PN From formula (5), we can get is a function of the network NetG, whose training parameter set is Θ G .MSE PN corresponds to the loss of physical information on a finite set of collocation points (x, t)∈E, and It involves differential operator terms with undetermined combinations, which are used to regularize the displacement of u in the network NetU to satisfy the partial differential equation describing the vibration process.

[0104] 2. Optimizing the coupled CPINNRP-AIC using a hierarchical training strategy with an undetermined exact PDE structure

[0105] Considering the interdependence between the network NetU and the network NetG in the loss function MSE, a hierarchical training strategy is proposed. The wave equation structure that actually describes the vibration phenomenon of the involute spline shaft of the aircraft engine and even the sparse measurement of the driving force cannot be obtained. However, CPINNRP-AIC can be used to obtain the sparse measurement data D I get and are used to approximate the vibration displacement and driving force respectively. Considering Θ U and Θ G The interdependent relationships are estimated using interactive iterations. Assuming k is the number of steps in the current iteration, the purpose of the hierarchical training strategy is to solve the two optimization problems (8) and (9).

[0106] 3. Use AIC to select appropriate differential operator combinations that describe the PDE structure of the actual vibration process;

[0107] There is often a gap between the idealized wave equation model that describes vibration phenomena and the actual situation. To this end, we consider using AIC to implement model evaluation and selection. For wave phenomena, we consider the second-order derivative with respect to the time variable. The candidate library is constructed as follows: Φ = [u tt ,u x ,u xx ,u xt ]. The AIC in (10) is then used to evaluate and select the appropriate combination from Φ. Figure 8 shows a histogram of the AIC size of the number of terms in the typical wave equation model that describes the wave phenomenon. The results show that u tt and uxx The combination is a differential operator combination that correctly describes the wave phenomenon. The hierarchical optimization coupling strategy and model selection details of CPINNRP-AIC are shown in Algorithm 1.

[0108] Algorithm 1: Hierarchical Optimization Coupling Strategy and Model Selection of CPINNRP-AIC:

[0109] (1) Initialization: Let k = 0, initialize network NetU and network NetG, and set the maximum number of iteration steps k as M, which serves as the loop termination condition;

[0110] (1.1) Obtain the training data set (x, t, u) ∈ D and the collocation points (x, t) ∈ E;

[0111] (1.2) Randomly generate the initial parameter sets of network NetU and network NetG and

[0112] (1.3) When k < M, execute the following loop:

[0113] (a1) Obtain At this time, MSE PN in from the iterative result of the previous step [[ID=,29]]

[0114] (b1) Obtain <, which is called

[0115] (c1) k = k + 1;

[0116] Judge whether the loop termination condition is satisfied: If yes, end the loop and return If no, return to step (a1) and execute the parameter Θ U and Θ G iterative transmission process, and solve the optimization problems (8) and (9);

[0117] (2) Initialization: Let k = 0, and use to initialize network NetU-RP;

[0118] (2.1) Except for x and t, the output of CPINN and the measured value are input into NetU-RP in an either-or manner through the cyclic delay of time according to whether the hardware sensor is available;

[0119] (2.2) When k < M, execute the following loop:

[0120] (a2) Train NetU-RP using formula (4)

[0121] 2.3) Determine whether the loop stop condition is met: If yes, end the loop and return the candidate differential term obtained by automatic differentiation If not, return to step (a2).

[0122] 2.4) Use formula (10) to combine the candidate differential terms Calculate the AIC value;

[0123] 2.5) The differential term combination with the smallest AIC is selected as the final model to approximate the PDE describing the actual vibration process.

[0124] 4. Evaluate Performance

[0125] The measurement error of the proposed CPINNRP-AIC method in the soft measurement of the vibration displacement distribution of the involute spline shaft of an aero-engine is evaluated by using (11) RMSE, that is, the performance of the CPINNRP-AIC method in solving the unknown exact one-dimensional wave equation structure is evaluated. The f N The RMSE of (x, t) is 1.053161e-04, the training set and the test set f N The RMSE of u(x,t) are RMSE=9.945489e-05 and RMSE=1.059776e-04 respectively. Represent the measured value of the actual hardware vibration displacement sensor and the corresponding soft-measured vibration displacement value, respectively. In order to further verify the performance of CPINNRP-AIC, the CC of (12) is used to further illustrate the similarity between the soft-measured vibration displacement of the CPINNRP-AIC method and the measured value of the actual hardware vibration displacement sensor. In this embodiment, the relevant settings of CPINNRP-AIC are that the number of hidden layers is 3, and each layer has 10 neurons. Table 2 shows the soft-measurement performance of the model obtained using CPINNRP-AIC for the actual aviation spline shaft vibration displacement.

[0126] Comparative example: Early work of this research group (Sun Ximing, Wang Eina, Qin Pan. Coupled physical information neural network for solving the displacement distribution of bounded vibrating rod under unknown external driving force [P]. Liaoning Province: CN116050247A, 2023-05-02)

[0127] The following idealized wave equation is used to describe the actual vibration process:

[0128] Assume that the vibration wave velocity a = 1, length L = π, and vibration wave propagation time t = 6. The driving force in the (x, t) region is The vibration displacement is The wave equation representing the displacement distribution is converted into a PDE equation in the residual form of Equation (3):

[0129] Therefore, this comparative example uses the idealized wave equation (15) to describe the vibration process of the aero-engine spline shaft. The comparative example does not involve the evaluation and selection process of the undetermined differential terms using the AIC of formula (10). This comparative example uses a coupled physical information neural network with a cyclic prediction mechanism to solve the idealized wave equation (15), and the solution obtained is This is the soft-sensing result of the vibration displacement to be obtained. Table 2 shows the soft-sensing performance comparison of the ideal wave equation describing the vibration displacement distribution of the aero-engine involute spline and the model obtained using CPINNRP-AIC. The best value in Table 2 is shown in bold.

[0130] Table 2 Comparison of soft sensing performance of the ideal wave equation and the model obtained by CPINNRP-AIC

[0131] As shown in Table 2, the RMSE is close to 0 and the CC is close to 1. Because the ideal wave equation is not fully suitable for controlling the actual vibration process, the soft measurement results of the comparative example are inferior to those of the CPINNRP-AIC method proposed in this paper. The reason is that our research group's earlier work used an idealized wave equation to describe the actual vibration process. However, due to the uncertainty involved in the actual vibration process, the idealized wave equation cannot fully and accurately describe the actual vibration process. The CPINNRP-AIC method proposed in this paper uses vibration displacement data sampled from the actual vibration process and evaluates the Akaike Information Criterion of the differential operator combination to obtain a wave equation model that can accurately describe the actual vibration process. Therefore, the soft measurement accuracy of the CPINNRP-AIC method proposed in this paper is higher.

[0132] The above-described embodiments merely express the implementation methods of the present invention, but should not be understood as limiting the scope of the patent of the present invention. It should be pointed out that for those skilled in the art, several variations and improvements can be made without departing from the concept of the present invention, and these all fall within the scope of protection of the present invention.

Claims

1. A soft-sensing method for spatiotemporal dynamic systems for automatically determining the structure of partial differential equations, characterized in that: It includes the following steps: First, construct a loss function for training a neural network with a cyclic prediction mechanism coupled with physical information to obtain a structure that satisfies the partial differential equation (PDE) describing the spatio-temporal industrial process, and then obtain the solution and source term; second, use the automatic differentiation method to obtain a candidate library of differential operators; third, use the Akaike information criterion to select an appropriate PDE structure from the candidate library of differential operators to accurately describe the spatio-temporal industrial process; finally, utilize the heat diffusion phenomenon and the actual vibration process to verify the soft measurement results obtained based on this method.

2. A spatiotemporal dynamic system soft measurement method for automatically determining the structure of a partial differential equation according to claim 1, characterized in that: It includes the following steps: The partial differential equation used to describe the dynamic industrial process with spatio-temporal dependence features has the following general form: u t (x,t)+N[u(x,t)]=g(x,t),x∈Ω,t∈[0,T] (1) Among them, x is the spatial variable, t is the time variable, t = 0 is the initial state, u: is the solution of the equation, i.e. the variable to be measured in the dynamic industrial process with time-space dependence characteristics, and the variable has time-space dependence; g: is a source term with a general form, that is, the driving source has time and space dependence; Ω is the open set space to which the spatial variable belongs, is its boundary; N is a series of differential operators; When the exact PDE structure of an industrial process with spatiotemporal dependence is unknown, the soft sensing method CPINNRP-AIC is based on the following partial differential equation approximation formula (1): Among them, Φ[·] is a candidate differential operator library with a series of partial differential operator terms, and is a series of states of the dynamic industrial process with spatio-temporal dependence features changing with time and space; The described soft measurement method is to learn to obtain from formula (2) the solution u and source term g that can approximately satisfy formula (1); formula (2) can be written in the form of the following residual function: f N (x,t):=Φ[u(x,t)]·λ-g(x,t). (3) The described CPINNRP-AIC includes two stages: 1) CPINNRP is used to approximately satisfy the u and g of the partial differential equation of formula (1); 2) AIC is used to determine an appropriate combination of differential operators to satisfy the partial differential equation of formula (1); CPINNRP consists of NetU, NetG, and NetU-RP, where NetU is used to approximately satisfy the u of (1); NetG is used to approximately satisfy the g of (1), and NetU-RP is used to compensate for the information defect caused by the discretization sampling strategy for t; the specific steps are as follows: Step 1: Construct a loss function for training CPINNRP; The training data set (x, t, u)∈D is obtained by using hardware sensors, where the hardware sensors include sensors that can be installed inside the operating space Ω and the operating space boundary. Installable sensors; the dataset is divided into D B ∪D I and D B and D I D is obtained from the boundary and initial conditions of Ω and internal random sampling, respectively. B Indicates that from the run space boundary Installable sensors and initial sampled training set, D I represents the training set sampled from the sensors that can be installed inside the operation space Ω; x, t, u are divided into represents spatial variables, temporal variables and variables to be measured with spatiotemporal dependencies; D represents the training data set; the corresponding configuration point set E=E is obtained according to the position of the sensor B ∪E I , where (x, t)∈E, E represents the set of configuration points in the entire space; E B Denotes the corresponding training data set D B The set of configuration points of E I Denotes the corresponding training data set D I The set of configuration points; Train CPINNRP-AIC using the data-physics hybrid loss function shown in formula (4); MSE=MSE DN +MSE PN (4) Among them, MSE DN and MSE PN They represent the data loss function and physical information loss function of the given formula (2) whose exact PDE structure is not determined, respectively, where the physical information loss function contains the physical information part Φ[u(x,t)]·λ of the combination of undetermined differential operators; Step 2: Optimize the coupled CPINNRP-AIC using a hierarchical training strategy under the condition of an undetermined exact PDE structure; Considering the interdependence between the network NetU and the network NetG in formula (4), a hierarchical training strategy is proposed to train CPINNRP-AIC by iterative parameter transmission, where the obtained and They are used to approximate u and g respectively; here, k is used to represent the number of iterations; the purpose of the hierarchical training strategy is to solve the following two coupled optimization problems; and in, is the physical information part containing the combination of undetermined differential operators, where is based on obtained; is based on obtained; In order to compensate for the information loss caused by time discretization sampling, the output of CPINN is The loop delay output of the hardware sensor is used as part of the NetU-RP input. That is, the input of NetU-RP consists of three parts: x, t and or loop-delayed output of a hardware sensor; The output of the hardware sensor is selected in an either-or manner, that is, when the hardware sensor can be installed at the configuration point (x, t), the delayed output of its measured value is used as the input of the configuration point; when the hardware sensor data cannot be obtained at the configuration point (x, t), the delayed output of its measured value is used as the input of the configuration point. The time delay of is used as the input of this configuration point; So far, we get the undetermined differential combination term and the g that satisfies formula (1); Step 3: Use AIC to select an appropriate combination of differential operators that satisfies formula (1); Use AIC to include undetermined differential combinations Conduct evaluation and selection; Where p is is the number of differential operators contained in , n is the size of the data set, The selected combination of differential terms achieves the correct PDE structure, that is, from all candidate terms Select the combination with the minimum AIC value in as the final PDE model for approximating formula (1); Step 4: Evaluate the performance of the soft measurement method for the spatio-temporal dynamic system that automatically determines the PDE structure by the CPINNRP-AIC method; use the root mean square error (RMSE) and Pearson correlation coefficient as evaluation indicators to evaluate the performance of CPINNRP-AIC.

3. A spatiotemporal dynamic system soft measurement method for automatically determining the structure of a partial differential equation according to claim 2, characterized in that: In formula (4) of the described Step 1: The MSE DN It is obtained from the following formula: in, is a function of the network NetU, whose parameter set is Θ U , used to approximate u that satisfies formula (1), card(·) is the cardinality of the set ·; The MSE PN It is obtained from the following formula: Furthermore, based on the undetermined operator combination get As shown in formula (7): in, is a function of the network NetG, whose parameter set is Θ G , which is used to approximately satisfy g of formula (1); λ represents the parameter vector; It means that formula (3) is (Θ U ,λ) parameterized residual function form.

4. The method for soft-sensing a spatiotemporal dynamic system for automatically determining the structure of a partial differential equation according to claim 2, characterized in that: The hierarchical training strategy for optimizing the coupled CPINNRP-AIC is shown in Algorithm 1: Algorithm 1: Hierarchical Optimization Coupling Strategy and Model Selection of CPINNRP-AIC (1) Initialization: Let k = 0, initialize the network NetU and the network NetG, and set the maximum number of iteration steps k as M as the loop stop condition; 1.1) Obtain the training dataset (x, t, u) ∈ D and the configuration points (x, t) ∈ E; 1.2) Randomly generate the initialization parameter set of network NetU and network NetG and 1.3) Execute the following loop when k < M: (a1) is obtained by solving the optimization problem (8) At this time, MSE PN Included in Iteration result from the previous step (b1) is obtained by solving the optimization problem (9) called (c1) k = k + 1; Determine whether the loop stop condition is met: If yes, end the loop and return If not, return to step (a1) and execute parameter Θ U and θ G Iterate the transfer process to solve the optimization problems (8) and (9); (2) Initialization: Set k = 0 and use Initialize the network NetU-RP; 2.1) In addition to x and t, the output of CPINN And the measured values are input into NetU-RP in an either-or manner through a cyclic delay of time according to whether the hardware sensors are available; 2.2) Execute the following loop when k < M: (a2) Train NetU-RP using formula (4); 2.3) Determine whether the loop stop condition is met: If yes, end the loop and return the candidate differential term obtained by automatic differentiation If not, return to step (a2). 2.4) Use formula (10) to combine candidate differential terms Calculate the AIC value in; 2.5) Select the combination of differential terms with the minimum AIC as the final model for approximating formula (1).

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