A three-dimensional temperature field reconstruction method based on transfer learning and physical constraints

Abstract

A three-dimensional temperature field reconstruction method based on transfer learning and physical constraints, involving three-dimensional temperature field reconstruction technology, solves the problem that the existing technology cannot realize the reconstruction of the temperature field under the condition of a very small number of known measuring points. The method includes adding physical constraints, pre-training and final training. The present invention obtains a source model through pre-training of enriched data of other arbitrary working conditions in similar scenarios and then migrates it to the target test measuring point data. The transfer learning method is used to capture the similarity of temperature distribution characteristics between different working conditions in similar scenarios, thereby realizing the reconstruction of the entire three-dimensional temperature field based on a very small number of measuring points. Compared with the existing data-driven reconstruction technology that needs to re-run the training process each time, it can greatly save time and improve the reconstruction efficiency.

Description

Technical Field

[0001] The invention relates to a three-dimensional temperature field reconstruction method based on transfer learning and physical constraints. Background Art

[0002] In many industrial fields such as aerospace, metallurgy, and microscale heat transfer, it is often necessary to obtain the overall temperature field of the observed object. However, in engineering practice, only partial measurement point data can be obtained, and there is no accurate boundary condition information for forward analytical calculation or numerical simulation. Therefore, this belongs to the category of heat conduction inverse problem, and the solution of the inverse problem itself is a recognized problem. Thanks to the development of machine learning, the temperature field reconstruction method based on data-driven has shown good results. However, most of the existing data-driven methods are suitable for scenarios with a large number of known measurement point data. When there are only a few known measurement points, the ill-posedness of the inverse problem itself will cause the existing temperature field that purely relies on data reconstruction to be almost incompatible with the actual situation. This problem is particularly obvious when facing complex three-dimensional scenes. It is undeniable that increasing the number of measurement points can help the reconstruction of the overall temperature field, but in engineering practice, a large number of measurement points will bring higher manpower and material costs, and the arrangement of measurement points is also limited in many scenarios, such as the small size of the research object, the interference of measurement points with actual flow, etc. Therefore, how to significantly reduce the demand for measurement point data while ensuring the accuracy of temperature field reconstruction is a technical problem that needs to be solved urgently. Summary of the invention

[0003] In order to solve the problem that the prior art cannot reconstruct the temperature field under the condition of a very small number of known measuring points, the present invention provides a three-dimensional temperature field reconstruction method based on transfer learning and physical constraints.

[0004] A three-dimensional temperature field reconstruction method based on transfer learning and physical constraints, the method includes adding physical constraints, pre-training and final training, the specific process is as follows:

[0005] Step 1: Add physical constraint PINN;

[0006] The loss function of supervised learning is set as:

[0007]

[0008] Where N t is the number of all known measuring points, is the true temperature, t i The temperature of the point predicted by the neural network;

[0009] Add the constraints of the physical control equation. The control equation of the temperature field in the solid domain is the Laplace equation in the Cartesian coordinate system:

[0010]

[0011] In the formula, x i ,y i and z i are the coordinates of the i-th measuring point in the x, y and z directions respectively;

[0012] Assume the function according to the form of the heat conduction equation:

[0013]

[0014] The loss function after adding physical constraints is:

[0015] MSE = MSE t +MSE f (4)

[0016]

[0017] In the formula, MSE t is the temperature constraint of the known measurement point, MSE f is the constraint on the residual of the control equation; is the i-th measuring point The true value of the temperature at i is the predicted temperature value of the i-th measuring point; N f is the number of all measurement points in the target domain, is the jth measuring point at a temperature of The value of the heat conduction equation function when ;

[0018] By continuously reducing the total loss function value, the prediction results of PINN can be constrained to meet the temperature and heat conduction equations of the measuring point;

[0019] Step 2: Pre-training stage;

[0020] According to the loss function after adding physical constraints in step 1, the PINN source model is obtained by pre-training with data obtained by numerical simulation of another arbitrary working condition in a similar scenario, and its loss function is:

[0021] MSE 1 =MSE t1 +MSE f1 (7)

[0022]

[0023] In the formula, all variables have the same meaning as in formulas (4)-(6). is the true value of the temperature at the mth measuring point, is the predicted temperature value of the mth measuring point; N t1 is the number of all measurement points in the target domain, is the nth measuring point at a temperature of The value of the heat conduction equation function when N f1 is the number of points used to calculate the function value of the heat conduction equation. The subscript 1 means that all points and the corresponding temperature data are the temperature fields under the source condition. t1 is the temperature constraint of the known measurement point under the source condition, MSE f1 is the constraint on the residual of the control equation under the source condition; the subscript 1 represents that all points and the corresponding temperature data are the temperature field under the source condition; the subscript 1 represents that all points and the corresponding temperature data are the temperature field under the source condition;

[0024] Step 3: Final training phase;

[0025] The PINN source model obtained in the pre-training phase of step 2 is applied to the target working condition, and the training is adjusted through the known measurement points of the target working condition. The loss function is:

[0026] MSE 2 =MSE t2 +MSE f2 (10)

[0027]

[0028] In the formula, the subscript 2 represents that all measuring points and the corresponding temperatures are the temperature fields under the target working conditions;

[0029] After the training is completed, the final temperature field reconstruction neural network for the target working condition is obtained.

[0030] Beneficial effects of the present invention: The present invention obtains a source model by pre-training with rich data of numerical simulation of other arbitrary working conditions in similar scenarios, and then migrates it to the target test point data. The transfer learning method is used to capture the similarity of temperature distribution characteristics between different working conditions in similar scenarios, thereby realizing the reconstruction of the entire three-dimensional temperature field based on a very small number of measurement points. Compared with the existing temperature reconstruction technology, it can well solve the problem of extremely low reconstruction accuracy due to a very small number of measurement points. The temperature reconstruction results of the flat air film case described in the present invention under different methods are shown in Figure 1. Figure 4 As shown (the temperature distribution of three lines on the outer surface is selected as a comparison), it can be seen from the comparison that the three-dimensional temperature field reconstruction method based on transfer learning and physical constraints (TL-PINN) proposed in the present invention has significantly higher reconstruction accuracy when the number of known measurement points is extremely small.

[0031] In addition, the source neural network model obtained in the pre-training stage can be reused. Each time facing a new test condition, it is only necessary to input the new test measurement point data into the source model for fine-tuning training to obtain the final model. The fine-tuning training only takes a very short time. Compared with the existing data-driven reconstruction technology, which needs to re-run the training process every time, it can greatly save time and improve the reconstruction efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 It is a principle block diagram of the overall method of a three-dimensional temperature field reconstruction method based on transfer learning and physical constraints described in the present invention;

[0033] Figure 2 This is an algorithm verification case, namely: a schematic diagram of the reconstruction of the three-dimensional temperature field of a flat plate with an air film;

[0034] Figure 3 Schematic diagram of the fully connected neural network structure used in the present invention;

[0035] Figure 4 Prediction results of three models when 45 measurement points are randomly selected from the entire domain: (a) conventional BPNN; (b) PINN; (c) TL-PINN. DETAILED DESCRIPTION

[0036] Combination Figures 1 to 4 This embodiment describes a three-dimensional temperature field reconstruction method based on transfer learning and physical constraints, which is implemented by the following steps:

[0037] 1. Add physical constraints (PINN). Conventional fully connected BP neural network directly takes the point coordinates as input and the temperature value of the point as output. It establishes the mapping from coordinates to temperature through training with known measurement point data, which is a typical supervised learning. Its loss function is

[0038]

[0039] Where N t is the number of all known measuring points, is the true temperature, t i The temperature of the point predicted by the neural network. On this basis, PINN needs to add constraints of the physical control equation, assuming that it is a steady-state calculation, the thermal conductivity of the material is set to a constant, and there is no internal heat source. Therefore, the control equation of the solid domain temperature field in the Cartesian coordinate system is the Laplace equation

[0040]

[0041] In the formula, x i ,y i and z i are the coordinates of the i-th measuring point in the x, y and z directions respectively.

[0042] Therefore, according to the heat conduction equation, we assume that the function

[0043]

[0044] Then the loss function of PINN is

[0045] MSE = MSE t +MSE f (4)

[0046]

[0047] In the formula, MSE t Represents the temperature constraint of a known measurement point, MSE f Represents the constraint of the residual of the control equation. By continuously reducing the total loss function value, the prediction results of PINN can be constrained to satisfy the temperature of the measuring point and the heat conduction equation as much as possible.

[0048] 2. Pre-training stage. As we all know, the solution of inverse problems is very difficult, but the forward calculation of known boundary conditions is very easy. At the same time, the temperature field distribution obtained under different working conditions in similar scenarios is often similar, and the difference lies only in the numerical value and a small part of the boundary area, that is, the internal laws and characteristics of the temperature distribution are similar, which meets the requirements of transfer learning. Therefore, when the test point data is small, the data obtained by numerical simulation of another arbitrary working condition in a similar scenario is first pre-trained to obtain the source PINN model, and its loss function is

[0049] MSE 1 =MSE t1 +MSE f1 (7)

[0050]

[0051] In the formula, the meanings of all variables are the same as those in formulas (4)-(6), and the subscript 1 represents that all points and corresponding temperature data are the temperature fields under the source condition;

[0052] 3. Final training stage: The source model obtained in the pre-training stage is applied to the target working condition and fine-tuned through known measurement points. The loss function is:

[0053] MSE 2 =MSE t2 +MSE f2 (10)

[0054]

[0055] Wherein, the subscript 2 represents that all measuring points and the corresponding temperatures are the temperature fields under the target working condition; after the training is completed, the final temperature field reconstruction neural network for the target working condition (i.e., the working condition to which the test measuring point belongs) can be obtained.

[0056] Specific implementation method 2: Combination Figures 2 to 4 This embodiment is an example of a three-dimensional temperature field reconstruction method based on transfer learning and physical constraints described in the first embodiment: Taking a flat air film scene as an example, the model is as follows: Figure 2 As shown in the figure, some test points in the entire solid domain are known, and the goal is to reconstruct the entire three-dimensional temperature field. The process is as follows:

[0057] 1. Add physical constraints (PINN);

[0058] The coordinates of each point in the solid domain (x i ,y i , z i ) is the neural network input, and the three-dimensional heat conduction equation is added to the loss function. After preliminary training and testing, the structure of the neural network is determined to be 3-8*5-1, that is, the input layer is 3 neurons, representing the coordinates (x i ,y i , z i ), the hidden layer is 5 layers, each layer has 8 neurons, the output layer has 1 neuron, that is, the temperature value of the point, and the layers are fully connected. The neural network structure is as follows Figure 3 shown.

[0059] 2. Pre-training stage;

[0060] Commercial software (fluid simulation software ANSYS Fluent 19.0) is used to perform numerical simulation of any working condition for the model, obtain the temperature distribution in the air film scene, extract the coordinates of each node in the solid domain and the corresponding temperature, and use them as data to train the established physical constraint neural network (PINN) to obtain the source model. Note that the working condition is arbitrarily given by humans, not an experimental condition, and the data obtained from the numerical simulation is also rich.

[0061] 3. Final training phase;

[0062] The source neural network trained with a large amount of other arbitrary working condition data in the pre-training stage is transferred to the target test condition. The coordinates of a very small number of test measurement points and the corresponding temperatures are used as the input and output of the neural network to fine-tune the source model. After convergence, the final neural network for reconstructing the three-dimensional temperature field of the test condition is obtained.

[0063] like Figure 4As shown in Figure 2, in order to reflect the advantages of this method (TL-PINN) over the existing neural network methods (conventional BPNN, PINN), Figure 2 45 test points were randomly selected in the flat plate model area, and three straight lines l were extracted at different positions on the upper surface (z = 1) 1 :y=20;l 2 :y=50;l 3 : y=70, examine the change of temperature prediction value T of three neural network methods with the horizontal coordinate x. Figure 4 The solid line in the middle represents the true value of temperature, the dashed line represents the temperature prediction value of the neural network, and the dotted line in (c) represents the temperature value of the pre-training condition of this method. It can be seen that the solid lines in (a) and (b) deviate greatly from the dashed lines, while those in (c) are very close, which fully demonstrates that the existing conventional BPNN and PINN cannot accurately reconstruct the three-dimensional temperature field under a small amount of measurement point data, while the method of the present invention (TL-PINN) has a stronger temperature field reconstruction capability. In addition, since the method of the present invention only needs to use the target measurement points for fine-tuning training, the source model obtained by pre-training can be used all the time, so the required training time is much less than that of the existing methods. Therefore, from the perspective of temperature field reconstruction accuracy and neural network training time, the method of the present invention has greater advantages and practical value than the existing methods.

[0064] The technical features of the above-described embodiments may be arbitrarily combined. To make the description concise, not all possible combinations of the technical features in the above-described embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0065] The above-mentioned embodiments only express several implementation methods of the present invention, and the descriptions thereof are relatively specific and detailed, but they cannot be understood as limiting the scope of the invention patent. It should be pointed out that, for ordinary technicians in this field, several variations and improvements can be made without departing from the concept of the present invention, and these all belong to the protection scope of the present invention. Therefore, the protection scope of the patent of the present invention shall be subject to the attached claims.

Claims

1. A three-dimensional temperature field reconstruction method based on transfer learning and physical constraints, characterized by: The method includes adding physical constraints, pre-training and final training. The specific process is as follows: Step 1: Add physical constraint PINN; Set the loss function of supervised learning to: Where N t is the number of all known measuring points, t(x i ,y i , z i ) is the true temperature, t i The temperature of the point predicted by the neural network; Add the constraints of the physical control equation. The control equation of the temperature field in the solid domain is the Laplace equation in the Cartesian coordinate system: In the formula, x i ,y i and z i are the coordinates of the i-th measuring point in the x, y and z directions respectively; Assume the function according to the form of the heat conduction equation: Then the loss function after adding physical constraints is: MSE'=MSE t +MSE f (4) In the formula, MSE t is the temperature constraint of the known measurement point, MSE f is the constraint on the residual of the control equation; is the i-th measuring point The true value of the temperature at t i is the predicted temperature value of the i-th measuring point; N f is the number of all measurement points in the target domain, is the jth measuring point at a temperature of The value of the heat conduction equation function when ; By continuously reducing the total loss function value, the prediction results of PINN can be constrained to meet the temperature and heat conduction equations of the measuring point; Step 2: Pre-training stage; According to the loss function after adding physical constraints in step 1, the PINN source model is obtained by pre-training with data obtained by numerical simulation of another arbitrary working condition in a similar scenario, and its loss function is: MSE1=MSE t1 +MSE f1 (7) In the formula, is the true value of the temperature at the mth measuring point, is the predicted temperature value of the mth measuring point; N t1 is the number of all measurement points in the target domain, is the nth measuring point at a temperature of The value of the heat conduction equation function when N f1 is the number of points used to calculate the function value of the heat conduction equation. The subscript 1 means that all points and the corresponding temperature data are the temperature fields under the source condition. t1 is the temperature constraint of the known measurement point under the source condition, MSE f1 is the constraint of the residual of the control equation under the source condition; the subscript 1 means that all points and the corresponding temperature data are the temperature field under the source condition; Step 3: Final training phase; The PINN source model obtained in the pre-training phase of step 2 is applied to the target working condition, and the training is adjusted through the known measurement points of the target working condition. The loss function is: MSE2=MSE t2 +MSE f2 (10) In the formula, the subscript 2 represents that all points and the corresponding temperature data are the temperature fields under the target working conditions; After the training is completed, the final temperature field reconstruction neural network for the target working condition is obtained.

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