Coating thickness real-time prediction control method integrating microarchitecture search and physical constraint

Abstract

The invention belongs to the technical field of nonlinear heat conduction control, and particularly discloses a coating thickness real-time prediction control method integrating microarchitecture search and physical constraint, which comprises the following steps: establishing a coating heat conduction physical model; utilizing a Newton Leibniz formula to convert a two-dimensional nonlinear Volterra type integral equation corresponding to the coating heat conduction physical model into an equivalent equation set; constructing a micro neural architecture search space; designing a double-layer optimization objective function; and performing online thickness prediction and feedback control. According to the method, differential neural architecture search and a physical information neural network are combined, the problem of nonlinear integral equation solving of thickness distribution in the heat conduction process of the coating material in the high-temperature environment is solved, and self-adaptive prediction and control of the coating thickness are achieved.

Description

Technical Field

[0001] The present invention relates to the technical field of nonlinear heat conduction control, and in particular to a real-time prediction and control method for coating thickness that integrates differentiable architecture search and physical constraints. Background Art

[0002] Traditional coating thickness control methods rely on numerical simulations based on physical models. Complex heat conduction integral equations must be established and solved using discretization methods such as finite element methods. This results in high computational costs and difficulty in handling strong nonlinear problems. In recent years, physical information neural networks (PINNs) have been used to solve integral equations by embedding differential equation constraints into neural network loss functions. However, the existing PINN framework uses a dense neural network (DNN) with a fixed structure, and the number of layers and neurons must be manually preset, leading to the following problems: (1) The network structure relies on empirical design and is difficult to adapt to the nonlinear heat conduction characteristics under different working conditions; (2) The complex network structure has a large number of parameters, low training efficiency, and is difficult to meet real-time control requirements; (3) The fixed-structure network is prone to falling into local optimality when processing multidimensional nonlinear integral equations, resulting in limited prediction accuracy.

[0003] The existing technologies have the following major drawbacks: (1) Traditional numerical methods have low computational efficiency when solving nonlinear heat conduction integral equations, making it difficult to achieve real-time control; (2) Fixed-structure PINN networks lack adaptability, requiring repeated network architecture design for different coating materials and thermal boundary conditions; (3) The selection of the number of neural network layers and neurons relies on trial-and-error methods, resulting in wasted computational resources and insufficient generalization performance; (4) Existing neural architecture search (NAS) methods are incompatible with physical constraints, and the discretization of the search space leads to low convergence efficiency. Therefore, a method that can improve the solution accuracy and generalization ability of complex nonlinear heat conduction models is urgently needed for real-time prediction and control of coating thickness. Summary of the Invention

[0004] In order to solve the problems existing in the prior art, the present invention provides a real-time prediction and control method for coating thickness that integrates differentiable architecture search and physical constraints. Through the adaptive neural network architecture search method, the number of layers and neuron distribution of the physical information neural network are automatically optimized to improve the efficiency of solving the nonlinear heat conduction equation; a continuous differentiable search space integrating physical constraints is constructed to realize gradient-driven network structure optimization; the computational cost of coating thickness prediction is reduced to meet the real-time control requirements in high-temperature environments; the solution accuracy and generalization ability of complex nonlinear heat conduction models are improved, and the problems mentioned in the above background technology are solved.

[0005] To achieve the above objectives, the present invention provides the following technical solution: a real-time prediction and control method for coating thickness that integrates microarchitecture search and physical constraints, comprising the following steps:

[0006] S1. Establish a physical model of coating heat conduction;

[0007] S2. Using the Newton-Leibniz formula, the two-dimensional nonlinear Volterra integral equation corresponding to the coating heat conduction physical model is transformed into an equivalent system of equations;

[0008] S3, building a differentiable neural architecture search space;

[0009] S4, double-layer optimization objective function design;

[0010] S5. Online thickness prediction and feedback control.

[0011] Preferably, in step S1, a two-dimensional nonlinear Volterra integral equation is established for the heat conduction process of the coating in a high temperature environment to describe the relationship between the coating thickness distribution μ(x, y) and the heat flux density and material properties. The specific equation is:

[0012]

[0013] where (x,y)∈Ω=[0,1] 2 is the two-dimensional spatial coordinate of the substrate surface, which represents the normalized spatial coordinate; μ(x, y) is the real-time coating thickness at position (x, y); μ(s, t) is the real-time coating thickness at position (s, t); f(x, y) is the intensity distribution function of the external deposition source, and its specific form is:

[0014] Preferably, in step S2, the Newton-Leibniz formula is used to transform the two-dimensional nonlinear Volterra integral equation corresponding to the coating heat conduction physical model into an equivalent system of equations, which is expressed as follows:

[0015]

[0016] Where, Γ c is the control equation, F2(x,y) is the integral term The corresponding primitive function of the second-order integral operator, Γ t is the transformation equation, F2(y,t)'t is the partial derivative of F2(y,t) with respect to t, and F1(s,t) is the integral term The corresponding primitive function of the first-order integral operator, Γ d Define the solution constraints for the boundary conditions.

[0017] Preferably, in step S3, the number of layers of the feedforward neural network DNN used to solve the two-dimensional nonlinear Volterra integral equation corresponding to the coating heat conduction physical model is set to M, the number of hidden neurons in each layer is N, and a differentiable neural architecture search space is constructed through layer selection and neuron selection.

[0018] Preferably, the layer selection, i.e., converting the discrete layer number selection into a differentiable probability distribution through the retain / skip operation of Softmax relaxation, specifically includes the following:

[0019] Define the hidden layer output as y = ρ1·X + ρ2·σ(w·X+b), where X = {(x,y)|(x,y)∈Ω}, w is the weight of the hidden layer, b is the bias, σ is the activation function of the hidden layer, ρ1 is the weight of the skipped hidden layer, and ρ2 is the weight of the retained hidden layer;

[0020] The search for the DNN layer is to find the highest weight among ρ1 and ρ2; if ρ1<ρ2, the hidden layer is retained; if ρ1≥ρ2, the hidden layer is skipped; when ρ1=ρ2, skipping will reduce the number of hidden layers.

[0021] Preferably, the neuron selection is to realize the continuous expression of the number of neurons by introducing a binary mask matrix and Gumbel-Softmax sampling; specifically, the following are included:

[0022] Based on the layer selection, the hidden layer output is defined as y = ρ1·X + ρ2·σ(w·X+b)·[g×mask] T , where g is the weight of the number of neurons, mask is a predefined binary mask matrix; the possible number of neurons in the hidden layer of the neural network is defined as N i (i=1,…,N n )≤N, to achieve the selection of the number of neurons, the number of neurons in the M hidden layers is first filled to N, and the extra neurons are disabled by multiplying the filled neurons by a tensor mask to simulate different numbers of neurons; the output of the hidden layer is:

[0023]

[0024] At this time, g=[g1,g2,…,g N ], g i (i=1,2,…,N) is the weight of each neuron number, mask i (i=1,2,…,N) is the mask corresponding to each number of neurons, and its shape is 1*N, that is, if the number of neurons is N i (i=1,…,N), then the corresponding mask mask i (i=1,2,…,N) Top N i elements are 1, and the rest (NN i ) elements are 0;

[0025] The search for DNN neurons is to find g i (i=1,2,…,N n) with the highest weight.

[0026] Preferably, in step S4, the following is specifically included:

[0027] S41, inner layer physical optimization: fix the architecture parameters γ, including layer weights ρ = [ρ1, ρ2] and neuron number weights g, and update the network parameters θ, including weights w and biases b, through the Adam optimizer;

[0028] Prioritize satisfying the residual Loss of the heat conduction equation c and deposition rate constraint Loss F , to ensure that the solution conforms to physical laws; the residual error of the heat conduction equation Loss c Defined as:

[0029]

[0030] Where, is the number of sample points, Represents the set of all sample points as the input of the neural network; l(·) is a specific metric function, usually the L2 norm or its variant;

[0031] Deposition rate constraint Loss F Defined as:

[0032] Loss F =w t L t +w d L d

[0033] Among them, w t =1 and w d =30 is the loss function Loss F The transformation equation Γ t and its solution constraint Γ d The corresponding weight;

[0034]

[0035] is the number of sample points of the transformation equation in the computational domain;

[0036]

[0037] is the number of sample points of the solution constraint in the computational domain;

[0038] S42, outer layer structure optimization: fix the network parameters θ, including weight w and bias b, update the architecture parameters γ through the Proximal Gradient method, and minimize the predicted thickness Mean square error MSE with the measured value μ μ And the first-order and second-order integral operator primitive functions that minimize the integral term and the true value F m Mean square error (MSE) F ;

[0039] MSE μ The definition formula is as follows:

[0040]

[0041] MSE F The definition formula is as follows:

[0042]

[0043] Preferably, in step S5, a real-time control closed loop is designed based on the trained optimal neural network, the temperature field data collected in real time is input into the trained neural network, and a predicted value of the coating thickness distribution is output.

[0044] The beneficial effects of the present invention are:

[0045] 1) Adaptive network structure: Automatically determines the optimal number of layers and neuron distribution through gradient optimization, reducing the number of parameters by 86% compared to traditional fixed networks and increasing training speed by 3.9 times;

[0046] 2) Physical constraint embedding: Incorporating the heat conduction integral equation into the loss function as a hard constraint improves prediction accuracy by two orders of magnitude compared to purely data-driven methods;

[0047] 3) Real-time control capability: Online prediction delay is less than 50ms, meeting the real-time adjustment requirements of the plasma spraying process;

[0048] 4) Strong nonlinear processing: for μ 2 The nonlinear integral term of the term is optimized, and the adaptive extraction of high-order features is achieved through a mask mechanism, reducing the error fluctuation by 72% under complex working conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 Schematic diagram of the search framework of the real-time prediction control method for coating thickness that integrates microarchitecture search and physical constraints of the present invention;

[0050] Figure 2 This is a schematic diagram comparing the mean square error (MSE) of the search framework of the real-time prediction control method for coating thickness that integrates differentiable architecture search and physical constraints with the traditional fixed network structure. DETAILED DESCRIPTION

[0051] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0052] The present invention provides a technical solution: a real-time prediction and control method for coating thickness that integrates microarchitecture search and physical constraints, comprising the following steps:

[0053] S1. Establish a physical model of coating heat conduction;

[0054] In step S1, a two-dimensional nonlinear Volterra integral equation is established for the heat conduction process of the coating in a high-temperature environment to describe the relationship between the coating thickness distribution μ(x, y) and the heat flux density and material properties. The specific equation is:

[0055]

[0056] where (x,y)∈Ω=[0,1] 2 is the two-dimensional spatial coordinate of the substrate surface (unit: m), which represents the normalized spatial coordinate; μ(x, y) is the real-time coating thickness at position (x, y) (unit: μm); μ(s, t) is the real-time coating thickness at position (s, t) (unit: μm), which characterizes the autocatalytic effect of the deposited area on the current deposition rate; f(x, y) is the external deposition source intensity distribution function (including the plasma flow field control term), and its specific form is:

[0057] S2. Using the Newton-Leibniz formula, the two-dimensional nonlinear Volterra integral equation corresponding to the coating heat conduction physical model is transformed into an equivalent system of equations;

[0058] In step S2, the Newton-Leibniz formula is used to transform the two-dimensional nonlinear Volterra integral equation corresponding to the coating heat conduction physical model into an equivalent system of equations, which is expressed as follows:

[0059]

[0060] Where, Γ c is the control equation, F2(x,y) is the integral term The corresponding primitive function of the second-order integral operator, Γ t is the transformation equation, F2(y,t)' t is the partial derivative of F2(y,t) with respect to t, and F1(s,t) is the integral term The corresponding primitive function of the first-order integral operator, Γ d Define the solution constraints for the boundary conditions.

[0061] S3, building a differentiable neural architecture search space;

[0062] In step S3, the number of layers of the feedforward neural network DNN used to solve the two-dimensional nonlinear Volterra integral equation corresponding to the coating heat conduction physical model is assumed to be M, the number of hidden neurons in each layer is N, and a differentiable neural architecture search space is constructed through layer selection and neuron selection.

[0063] The layer selection, i.e., converting the discrete layer number selection into a differentiable probability distribution through the retain / skip operation of Softmax relaxation, specifically includes the following:

[0064] The output of the hidden layer (∈M) is defined as y = ρ1·X + ρ2·σ(w·X+b), where X = {(x,y)|(x,y)∈Ω}, w is the hidden layer weight, b is the bias, σ is the hidden layer activation function, and ρ1 and ρ2 are layer selection weights. ρ1 is the weight for skipping hidden layers, and ρ2 is the weight for retaining hidden layers. The DNN layer search is to find the highest weight between ρ1 and ρ2. If ρ1 < ρ2, the hidden layer is retained; if ρ1 ≥ ρ2, the hidden layer is skipped. When ρ1 = ρ2, skipping reduces the number of hidden layers, streamlining the neural network structure and improving the efficiency of neural network training.

[0065] The neuron selection, i.e., the continuous expression of the number of neurons is achieved by introducing a binary mask matrix and Gumbel-Softmax sampling, specifically includes the following:

[0066] Based on the layer selection, the hidden layer output is defined as y = ρ1·X + ρ2·σ(w·X+b)·[g×mask] T , where g is the weight of the number of neurons, mask is a predefined binary mask matrix; the possible number of neurons in the hidden layer of the neural network is defined as N i (i=1,…,N n )≤N, to achieve the selection of the number of neurons, first fill the number of neurons in the M hidden layers to N, and then simulate different numbers of neurons by multiplying the filled neurons by a tensor mask and disabling the extra neurons. The output of the hidden layer is:

[0067]

[0068] At this time, g=[g1,g2,…,g N ], g i (i=1,2,…,N) is the weight of each neuron number, mask i(i=1,2,…,N) is the mask corresponding to each number of neurons, and its shape is 1*N, that is, if the number of neurons is N i (i=1,…,N), then the corresponding mask mask i (i=1,2,…,N) Top N i elements are 1, and the rest (NN i ) elements are 0. The search of DNN neurons is to find g i (i=1,2,…,N n ). If there are neurons with the same number of weights, the one with the smaller number of neurons is selected.

[0069] S4, double-layer optimization objective function design;

[0070] In step S4, the specific steps include:

[0071] S41, inner layer physical optimization: fix the architecture parameters γ (including layer weight ρ and neuron number weight g), update the network parameters θ (including weight w and bias b) through the Adam optimizer, and give priority to satisfying the residual of the heat conduction equation (Loss c ) and deposition rate constraint (Loss F ), ensuring that the solution complies with physical laws; the residual of the heat conduction equation (Loss c ) is defined as:

[0072]

[0073] Where, is the number of sample points, Represents the set of all sample points as the input of the neural network. l(·) is a specific metric function, usually the L2 norm or its variant.

[0074] Deposition rate constraint (Loss F ) is defined as:

[0075] Loss F =w t L t +w d L d

[0076] Among them, w t =1 and w d =30 is the loss function Loss F The transformation equation Γ t and its solution constraint Γ d The corresponding weight.

[0077]

[0078] is the number of sample points of the transformation equation in the computational domain;

[0079]

[0080] is the number of sample points of the solution constraint in the computational domain;

[0081] S42, outer layer structure optimization: fix the network parameters θ (including weight w and bias b), update the architecture parameters γ (including layer selection weight and neuron number weight) through the ProximalGradient method, and minimize the predicted thickness The mean square error (MSE) of the measured value μ μ ) and minimize the integral term first-order and second-order integral operator primitive functions and the true value F m The mean square error (MSE) F ),

[0082] MSE μ The definition formula is as follows:

[0083]

[0084] MSE F The definition formula is as follows:

[0085]

[0086] S5. Online thickness prediction and feedback control.

[0087] In step S5, a real-time control closed loop is designed based on the trained optimal neural network. Real-time temperature field data is input into the trained neural network, which outputs a predicted coating thickness distribution value, μ(x, y). Based on the predicted thickness distribution, the plasma flow rate (with a control range of ±15%) and deposition angle (±0.5°) are dynamically adjusted to achieve a coating thickness uniformity error of ≤3.2μm (a 67% reduction compared to open-loop control).

[0088] The present invention provides a real-time prediction control method for coating thickness that integrates microarchitecture search and physical constraints. Figure 1As shown in Table 1, in order to compare with the neural architecture searched by the present invention, five reference neural network architectures were manually designed from the search space based on experience: (a) a 10-layer fully connected network, a neural network with the most parameters in the search space (10 layers, 100 neurons per layer); (b) a 2-layer wide network, a shallow (2 hidden layers) neural network with the most neurons per layer (100 neurons per layer); (c) a 10-layer narrow network, a neural network with the most hidden layers (10 layers) but the least neurons per layer (10 neurons per layer); (d) an increasing structure network, a neural network with an increasing number of neurons; (e) a decreasing structure network, a neural network with a decreasing number of neurons, as shown in Table 1.

[0089] Table 1 Neural architectures searched by the present invention and five reference neural network architectures

[0090] name Architecture The present invention [2,80,40,60,60,40,1] 10-layer fully connected network [2,100*10,1] 2-layer wide network [2,100*2,1] 10-layer narrow network [2,20*10,1] Incremental structure network [2,20,40,60,80,100,100,100,100,100,100,1] Decreasing structure network [2,100,80,60,40,20,20,20,20,20,20,1]

[0091] Table 1 details these reference architectures as sequences, where the first and last elements of the sequence represent the input and output, and the remaining elements represent the number of neurons in each layer. All five reference architectures, as well as the proposed search architecture, were trained from scratch for 8,000 epochs (one epoch is defined as the completion of training on all training data) under the same experimental conditions using 16,000 training points and 1,600 solution constraint points. All experiments were repeated five times, and the average value was calculated.

[0092] The present invention is a real-time prediction control method for coating thickness that integrates microarchitecture search and physical constraints. The search framework is compared with the mean square error (MSE) of the traditional fixed network structure. Figure 2 As shown. Figure 2 As can be seen from the latest experimental data, the method of the present invention has a more significant advantage over the existing technology: under the condition of 16,000 training samples, the mean square error of the search architecture based on the present invention is 4.24×10 -8 , compared with the 10-layer fully connected network (9.69×10 -7 ), 2-layer wide network (1.78×10 -6 ) and a 10-layer narrow network (7.09×10 -6 ) were increased by 22.8 times, 42.0 times and 167.3 times respectively; and significantly outperformed the incremental structure network (3.02×10 -6 ) and decreasing structure network (2.40×10 -7 ), with accuracy increased by 71.2 times and 5.7 times respectively. In terms of computational efficiency, when the NVIDIA RTX 4070Ti platform achieves the same accuracy, the training time of this invention is only 1.2 hours, which is 74.5% shorter than the traditional fixed structure network (4.7 hours), and the parameter size is reduced to 1.2×104 , which is 86.2% lower than that of a 10-layer fully connected network (8.7×10), achieving a simultaneous breakthrough in accuracy and efficiency.

[0093] The present invention utilizes a continuous relaxation of the search space, increasing search efficiency by 15.7 times and reducing memory usage by 89%. Using a physical-structural dual-stage optimization approach, training time is reduced by 64% and convergence speed is increased by 3.9 times. Through Gumbel-Softmax mask sampling, the neuron activation rate reaches 32.7%, increasing energy efficiency by 3.1 times. The present invention also utilizes a real-time closed-loop control design, achieving a thickness uniformity error of ≤3.2μm and a single inference latency of ≤0.8ms.

[0094] It should be noted that, in this document, the terms "comprises," "includes," or any other variations thereof are intended to encompass non-exclusive inclusion, such that a process, method, article, or apparatus comprising a series of elements includes not only those elements but also other elements not explicitly listed, or elements inherent to such process, method, article, or apparatus. In the absence of further limitations, an element defined by the phrase "comprising a ..." does not exclude the presence of other identical elements in the process, method, article, or apparatus comprising the element.

[0095] The terms used in the embodiments of the present invention are only for the purpose of describing specific embodiments and are not intended to limit the present invention. The singular forms "a", "an", "the" and "the" used in the embodiments of the present invention and the appended claims are also intended to include plural forms unless the context clearly indicates otherwise.

[0096] It should be understood that the term "and / or" as used herein is merely a description of the relationship between associated objects, indicating that three possible relationships exist. For example, "A and / or B" can represent: A exists alone, A and B exist simultaneously, or B exists alone. Furthermore, the character " / " in this document generally indicates that the associated objects are in an "or" relationship.

[0097] The word "if," as used herein, may be interpreted as "at the time of" or "when" or "in response to determining" or "in response to detecting," depending on the context. Similarly, the phrases "if it is determined" or "if (stated condition or event) is detected" may be interpreted as "when it is determined" or "in response to the determination" or "when detecting (stated condition or event)" or "in response to detecting (stated condition or event)," depending on the context.

[0098] The references to "first" and "second" in the embodiments merely distinguish similar objects and do not represent a specific ordering of the objects. It is understood that the specific order or precedence of "first" and "second" can be interchanged where appropriate. It should be understood that the objects distinguished by "first" and "second" can be interchanged where appropriate, so that the embodiments described herein can be implemented in an order other than that illustrated or described herein.

[0099] Although the present invention has been described in detail with reference to the aforementioned embodiments, it is still possible for those skilled in the art to modify the technical solutions described in the aforementioned embodiments, or to make equivalent substitutions for some of the technical features therein. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A real-time prediction and control method for coating thickness that integrates differentiable architecture search and physical constraints, characterized in that: The steps include: S1. Establish a physical model of coating heat conduction; S2. Using the Newton-Leibniz formula, the two-dimensional nonlinear Volterra integral equation corresponding to the coating heat conduction physical model is transformed into an equivalent system of equations; S3, building a differentiable neural architecture search space; S4, double-layer optimization objective function design; S5. Online thickness prediction and feedback control.

2. The method for real-time prediction and control of coating thickness by integrating differentiable architecture search and physical constraints according to claim 1 is characterized in that: In step S1, a two-dimensional nonlinear Volterra integral equation is established for the heat conduction process of the coating in a high-temperature environment to describe the relationship between the coating thickness distribution μ(x, y) and the heat flux density and material properties. The specific equation is: where (x,y)∈Ω=[0,1] 2 is the two-dimensional spatial coordinate of the substrate surface, which represents the normalized spatial coordinate; μ(x, y) is the real-time coating thickness at position (x, y); μ(s, t) is the real-time coating thickness at position (s, t); f(x, y) is the intensity distribution function of the external deposition source, and its specific form is:

3. The method for real-time prediction and control of coating thickness by integrating differentiable architecture search and physical constraints according to claim 1 is characterized in that: In step S2, the Newton-Leibniz formula is used to transform the two-dimensional nonlinear Volterra integral equation corresponding to the coating heat conduction physical model into an equivalent system of equations, which is expressed as follows: Where, Γ c is the control equation, F2(x,y) is the integral term The corresponding primitive function of the second-order integral operator, Γ t is the transformation equation, F2(y,t)' t is the partial derivative of F2(y,t) with respect to t, and F1(s,t) is the integral term The corresponding primitive function of the first-order integral operator, Γ d Define the solution constraints for the boundary conditions.

4. The method for real-time prediction and control of coating thickness by integrating differentiable architecture search and physical constraints according to claim 1 is characterized in that: In step S3, the number of layers of the feedforward neural network DNN used to solve the two-dimensional nonlinear Volterra integral equation corresponding to the coating heat conduction physical model is assumed to be M, the number of hidden neurons in each layer is N, and a differentiable neural architecture search space is constructed through layer selection and neuron selection.

5. The method for real-time prediction and control of coating thickness by integrating differentiable architecture search and physical constraints according to claim 4 is characterized in that: The layer selection, i.e., converting the discrete layer number selection into a differentiable probability distribution through the retain / skip operation of Softmax relaxation, specifically includes the following: Define the hidden layer output as y = ρ1·X + ρ2·σ(w·X+b), where X = {(x,y)|(x,y)∈Ω}, w is the weight of the hidden layer, b is the bias, σ is the activation function of the hidden layer, ρ1 is the weight of the skipped hidden layer, and ρ2 is the weight of the retained hidden layer; The search for the DNN layer is to find the highest weight among ρ1 and ρ2; if ρ1<ρ2, the hidden layer is retained; if ρ1≥ρ2, the hidden layer is skipped; when ρ1=ρ2, skipping will reduce the number of hidden layers.

6. The method for real-time prediction and control of coating thickness by integrating differentiable architecture search and physical constraints according to claim 4, characterized in that: The neuron selection, i.e., the continuous expression of the number of neurons is achieved by introducing a binary mask matrix and Gumbel-Softmax sampling, specifically includes the following: Based on the layer selection, the hidden layer output is defined as y = ρ1·X + ρ2·σ(w·X+b)·[g×mask] T , where g is the weight of the number of neurons, mask is a predefined binary mask matrix; the possible number of neurons in the hidden layer of the neural network is defined as N i (i=1,…,N n )≤N, to achieve the selection of the number of neurons, the number of neurons in the M hidden layers is first filled to N, and the extra neurons are disabled by multiplying the filled neurons by a tensor mask to simulate different numbers of neurons; the output of the hidden layer is: At this time, g=[g1,g2,…,g N ], g i (i=1,2,…,N) is the weight of each neuron number, mask i (i=1,2,…,N) is the mask corresponding to each number of neurons, and its shape is 1*N, that is, if the number of neurons is N i (i=1,…,N), then the corresponding mask mask i (i=1,2,…,N) Top N i elements are 1, and the rest (NN i ) elements are 0; The search for DNN neurons is to find g i (i=1,2,…,N n ) with the highest weight.

7. The method for real-time prediction and control of coating thickness by integrating differentiable architecture search and physical constraints according to claim 1, characterized in that: In step S4, the specific steps include: S41, inner layer physical optimization: fix the architecture parameters γ, including the layer weights ρ = [ρ1, ρ2] and the neuron number weight g, and update the network parameters θ, including the weights w and biases b, through the Adam optimizer; Prioritize satisfying the residual Loss of the heat conduction equation c and deposition rate constraint Loss F , to ensure that the solution conforms to physical laws; the residual error of the heat conduction equation Loss c Defined as: Where, is the number of sample points, Represents the set of all sample points as the input of the neural network; l(·) is a specific metric function, usually the L2 norm or its variant; Deposition rate constraint Loss F Defined as: Loss F =w t L t +w d L d Among them, w t =1 and w d =30 is the loss function Loss F The transformation equation Γ t and its solution constraint Γ d The corresponding weight; is the number of sample points of the transformation equation in the computational domain; is the number of sample points of the solution constraint in the computational domain; S42, outer layer structure optimization: fix the network parameters θ, including weight w and bias b, update the architecture parameters γ through the Proximal Gradient method, and minimize the predicted thickness Mean square error MSE with the measured value μ μ And the first-order and second-order integral operator primitive functions that minimize the integral term and the true value F m Mean square error (MSE) F ; MSE μ The definition formula is as follows: MSE F The definition formula is as follows:

8. The method for real-time prediction and control of coating thickness by integrating differentiable architecture search and physical constraints according to claim 1, characterized in that: In step S5, based on the trained optimal neural network, a real-time control closed loop is designed, the real-time collected temperature field data is input into the trained neural network, and the coating thickness distribution prediction value μ(x, y) is output.

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