Support method, recording medium, support program, and support system

Abstract

To provide a support method capable of performing parallel Bayesian optimization that takes into account heteroscedastic observation noise.SOLUTION: A support method is a method for supporting exploration of a value of an explanatory variable that maximizes or minimizes an expected value of an objective variable. The support method includes the steps of: outputting, from a first machine learning model ML1, a first predictive distribution f|D which is a predictive distribution of the expected value of the objective variable (step S1); outputting, from a second machine learning model ML2, a second predictive distribution g|D which is a predictive distribution of a variance of the objective variable (step S2); constructing a third predictive distribution MV|D that integrates the first predictive distribution f|D and the second predictive distribution g|D (step S3); and a recommended value acquisition process (step S4) of executing parallel Bayesian optimization on the basis of the third predictive distribution MV|D, at least one acquisition function AF, and an exploration range, and acquiring at least one recommended value of the explanatory variable that maximizes the acquisition function AF from within the exploration range.SELECTED DRAWING: Figure 3

Description

[Technical Field]

[0001] The present invention relates to an assistance method, a recording medium, an assistance program, and an assistance system. [Background technology]

[0002] A technique is known in which Bayesian optimization is performed on a data set in which explanatory variables and objective variables are associated, and the value of the explanatory variable that optimizes (minimizes or maximizes) the value of the objective variable is estimated (see, for example, Patent Document 1). Specifically, a machine learning model is made to learn the data set (machine learning), and a predictive distribution (posterior distribution) of the objective variable is output from the machine learning model. Then, based on the predictive distribution of the objective variable, an acquisition function, and a search range, values ​​(recommended values) of the explanatory variables that maximize the acquisition function are searched for within the search range as candidates for the optimal solution.

[0003] Parallel Bayesian optimization is a type of Bayesian optimization. Parallel Bayesian optimization uses the acquisition function used in general Bayesian optimization. Parallel Bayesian optimization allows multiple recommended values ​​to be searched for in a single process, allowing operators to more efficiently determine the values ​​of explanatory variables that optimize (minimize or maximize) the value of the objective variable.

[0004] Incidentally, the values ​​of the objective variable (observed values) obtained through an experiment may differ from experiment to experiment, even when the same explanatory variables (same experimental conditions) are used. In other words, the observed values ​​contain observation noise. Furthermore, the variance of the observation noise added to the objective variable value may differ depending on the value of the explanatory variables. Such observation noise is called heteroscedastic observation noise.

[0005] Heterogeneous Bayesian optimization has been proposed as a technique for performing Bayesian optimization that takes heterogeneous observation noise into account. Heterogeneous Bayesian optimization uses the predictive distribution of the expected value of the objective variable and the predictive distribution of the heterogeneous observation noise variance. Heterogeneous Bayesian optimization also uses an acquisition function that differs from that used in general Bayesian optimization. Specifically, the acquisition function expressed by the following equation (1) and the acquisition function expressed by the following equation (2) have been proposed as acquisition functions for use in heterogeneous Bayesian optimization.

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[0006] Equation (1) is defined based on the UCB (Upper Confidence Bound) acquisition function and the LCB (Lower Confidence Bound) acquisition function. In equation (1), the first term UCB represents the acquisition function for the predictive distribution of the expected value of the objective variable. The second term LCB represents the acquisition function for the predictive distribution of heteroscedastic observation noise variance.

[0007] Equation (2) is defined based on the EI (Expected Improvement) acquisition function. In equation (2), the first term, EI, represents the acquisition function for the predictive distribution of the expected value of the objective variable. The following equation (3), included in the second term, represents the expected value of the predictive distribution of heteroscedastic observation noise variance.

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[0008] The coefficient α included in Equation (1) and Equation (2) is a risk aversion coefficient. In heteroscedastic Bayesian optimization, the larger the value of the risk aversion coefficient α, the more actively the search for optimal solution candidates will avoid areas where observation noise is expected to be large. [Prior art documents] [Patent documents]

[0009] [Patent Document 1] Japanese Patent Application Publication No. 2023-174450 Summary of the Invention [Problem to be solved by the invention]

[0010] However, the acquisition function used in heteroscedastic Bayesian optimization cannot be used in parallel Bayesian optimization, which means that parallel Bayesian optimization cannot take heteroscedastic observation noise into account.

[0011] The present invention has been made in view of the above-mentioned problems, and an object of the present invention is to provide a support method, a recording medium, a support program, and a support system that are capable of performing parallel Bayesian optimization taking into account heteroscedastic observation noise. [Means for solving the problem]

[0012] According to one aspect of the present invention, a support method is a method for supporting a search for values ​​of explanatory variables that maximize or minimize an expected value of a dependent variable, the support method including the steps of: causing a first machine learning model capable of outputting a predictive distribution to output a first predictive distribution that is a predictive distribution of the expected value of the dependent variable; causing a second machine learning model capable of outputting a predictive distribution to output a second predictive distribution that is a predictive distribution of the variance of the dependent variable; constructing a third predictive distribution by integrating the first predictive distribution and the second predictive distribution; and performing parallel Bayesian optimization based on the third predictive distribution, at least one acquisition function, and a search range to obtain at least one recommended value of the explanatory variable within the search range that maximizes the acquisition function.

[0013] In one embodiment, a plurality of the recommended values ​​are acquired in the recommended value acquisition step.

[0014] In one embodiment, the first machine learning model and the second machine learning model each include a twice-differentiable kernel function, and the at least one acquisition function includes a Monte Carlo acquisition function.

[0015] In one embodiment, the at least one acquisition function includes Thompson sampling and a plurality of Monte Carlo acquisition functions having different properties, and the recommended value acquisition step acquires the plurality of recommended values ​​from the Thompson sampling and acquires the plurality of recommended values ​​from each of the plurality of Monte Carlo acquisition functions.

[0016] According to one aspect of the present invention, a recording medium is a computer-readable medium that stores an assistance program that defines the above-described assistance method.

[0017] According to one aspect of the present invention, an assistance program is a computer program executable by a computer, and defines the assistance method described above.

[0018] According to one aspect of the present invention, an assistance system is a system for assisting in a search for explanatory variable values ​​that maximize or minimize the expected value of a dependent variable. The assistance system includes a storage unit and a processing unit. The storage unit stores a first machine learning model capable of outputting a predictive distribution, a second machine learning model capable of outputting a predictive distribution, and at least one acquisition function. The processing unit causes the first machine learning model to output a first predictive distribution that is a predictive distribution of the expected value of the dependent variable, and causes the second machine learning model to output a second predictive distribution that is a predictive distribution of the variance of the dependent variable. The processing unit constructs a third predictive distribution by integrating the first predictive distribution and the second predictive distribution. The processing unit performs parallel Bayesian optimization based on the third predictive distribution, the acquisition function, and a search range, and obtains at least one recommended value of the explanatory variable within the search range that maximizes the acquisition function.

[0019] In one embodiment, the processing unit acquires a plurality of the recommended values.

[0020] In one embodiment, the first machine learning model and the second machine learning model each include a twice-differentiable kernel function, and the at least one acquisition function includes a Monte Carlo acquisition function.

[0021] In one embodiment, the at least one acquisition function includes Thompson sampling and a plurality of Monte Carlo acquisition functions having different properties, and the processing unit obtains the plurality of recommended values ​​from the Thompson sampling and obtains the plurality of recommended values ​​from each of the plurality of Monte Carlo acquisition functions. [Effects of the Invention]

[0022] According to the support method, recording medium, support program, and support system of the present invention, parallel Bayesian optimization can be performed taking heteroscedastic observation noise into consideration. [Brief explanation of the drawings]

[0023] [Figure 1] 1 is a block diagram showing a configuration of a support system according to a first embodiment of the present invention. [Figure 2] FIG. 10 is a diagram illustrating an example of heteroscedastic observation noise. [Figure 3] 1 is a flowchart showing a support method according to the first embodiment of the present invention. [Figure 4] (a) is a diagram showing an example of observed values ​​of a dependent variable; (b) is a diagram showing examples of a first predictive distribution, a second predictive distribution, and a third predictive distribution; (c) is a diagram showing the first predictive distribution of (b); (d) is a diagram showing the second predictive distribution of (b); and (e) is a diagram showing the third predictive distribution of (b). [Figure 5] FIG. 10 is a diagram illustrating an example of the flow of a recommended value acquisition process. [Figure 6] 1 is a diagram showing the procedure of an experiment using the assistance method, recording medium, assistance program, and assistance system according to the first embodiment of the present invention. [Figure 7]FIG. 10 is a block diagram showing the configuration of a support system according to a second embodiment of the present invention. [Figure 8] FIG. 10 is a schematic diagram of a substrate processing system including a support system according to a third embodiment of the present invention. [Figure 9] FIG. 10 is a block diagram showing the configuration of a support system according to a third embodiment of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0024] Hereinafter, embodiments of the support method, recording medium, support program, and support system of the present invention will be described with reference to the drawings (FIGS. 1 to 9). However, the present invention is not limited to the following embodiments, and can be implemented in various forms without departing from the spirit of the present invention. Note that duplicated explanations may be omitted as appropriate. In addition, the same or equivalent parts in the drawings will be designated by the same reference symbols, and explanations will not be repeated.

[0025] [Embodiment 1] First, referring to FIG. 1, a recording medium 200, an assistance program SP, and an assistance system 100A according to this embodiment will be described. FIG. 1 is a block diagram showing the configuration of the assistance system 100A according to this embodiment. The assistance system 100A is a system that assists in the search for an explanatory variable value x (recommended value or candidate value) that optimizes (maximizes or minimizes) the expected value (predicted value) of a target variable. As shown in FIG. 1, the assistance system 100A according to this embodiment includes a terminal device 101A. The assistance program SP is installed in the terminal device 101A from the recording medium 200. The terminal device 101A executes the assistance program SP installed from the recording medium 200 to search for an explanatory variable value x that optimizes the expected value of the target variable. The terminal device 101A is an example of an "assistance device." The terminal device 101A may be, for example, a general-purpose computer or a dedicated computer.

[0026] Specifically, the recording medium 200 is a medium that can be read by a computer. A program (computer program) to be executed by a computer is non-temporarily recorded on the recording medium 200. The recording medium 200 records an assistance program SP. In other words, the recording medium 200 stores the assistance program SP. The assistance program SP is a computer program that can be executed by a computer.

[0027] The recording medium 200 may be, for example, a medium including a semiconductor memory such as an SD memory card or a USB (Universal Serial Bus) memory, or a medium including a magnetic disk such as a hard disk drive. Alternatively, the recording medium 200 may be an optical disk such as a CD (Compact Disk), a DVD (Digital Versatile Disk), or a Blu-ray disk, or may be a main memory device or an auxiliary memory device installed in another computer system.

[0028] The assistance program SP includes a first machine learning model ML1, a second machine learning model ML2, and a heterogeneous parallel Bayesian optimization program BP. The heterogeneous parallel Bayesian optimization program BP includes at least one acquisition function AF. In this embodiment, the assistance program SP further includes a preprocessing program FP.

[0029] The first machine learning model ML1 includes a model capable of outputting a predictive distribution. The first machine learning model ML1 learns (machine learns) a dataset (learning data) in which a dependent variable and an explanatory variable are associated, and outputs a predictive distribution of the expected value of the dependent variable. Hereinafter, the predictive distribution output from the first machine learning model ML1 may be referred to as a "first predictive distribution f|D." The first predictive distribution f|D is a conditional probability distribution.

[0030] The second machine learning model ML2 includes a model capable of outputting a predictive distribution. The second machine learning model ML2 learns (machine learns) a dataset (learning data) in which the variance of the dependent variable (variance of the observed values) is associated with the explanatory variables, and outputs a predictive distribution of the variance of the dependent variable. Hereinafter, the predictive distribution output from the second machine learning model ML2 may be referred to as the "second predictive distribution g|D." The second predictive distribution g|D is a conditional probability distribution.

[0031] The algorithm of the first machine learning model ML1 is not particularly limited as long as it includes an algorithm capable of outputting a predictive distribution. The first machine learning model ML1 may include, for example, a Gaussian process regression model. Similarly, the algorithm of the second machine learning model ML2 is not particularly limited as long as it includes a model capable of outputting a predictive distribution. The second machine learning model ML2 may include, for example, a Gaussian process regression model.

[0032] The preprocessing program FP includes a computer program that performs preprocessing on a dataset (training data). In this embodiment, the dataset (training data) preprocessed by the preprocessing program FP is input to a first machine learning model ML1. Similarly, the dataset (training data) preprocessed by the preprocessing program FP is input to a second machine learning model ML2.

[0033] The heterogeneous variance parallel Bayesian optimization program BP includes a computer program that constructs a third predictive distribution MV|D by integrating the predictive distribution (first predictive distribution f|D) output from the first machine learning model ML1 and the predictive distribution (second predictive distribution g|D) output from the second machine learning model ML2. The heterogeneous variance parallel Bayesian optimization program BP also includes a computer program that performs parallel Bayesian optimization based on the third predictive distribution MV|D, at least one acquisition function AF, and a search range, and outputs at least one explanatory variable value x (recommended value or candidate value) that maximizes the acquisition function AF. The third predictive distribution MV|D is a conditional probability distribution.

[0034] The terminal device 101A includes an operation unit 102, a display unit 103, an interface unit 104, a storage unit 105, and a processing unit .

[0035] The operation unit 102 includes a user interface device operated by the worker. The operation unit 102 inputs a signal according to the operation of the worker to the processing unit 106. The operation unit 102 may include, for example, a keyboard, a mouse, and a touch sensor. The touch sensor may be superimposed on the display surface of the display unit 103. A graphical user interface may be configured by superimposing the touch sensor on the display surface of the display unit 103. For example, the worker can operate the operation unit 102 to instruct the installation of the assistance program SP. The worker can also operate the operation unit 102 to instruct the execution of the assistance program SP.

[0036] The display unit 103 is controlled by the processing unit 106 to display various screens. For example, the display unit 103 may display the source code of the assistance program SP. For example, while the source code of the assistance program SP is displayed on the display unit 103, the operator can operate the operation unit 102 to set a dataset to be learned by the first machine learning model ML1. The display unit 103 includes a display device such as a liquid crystal display device or an organic electroluminescence (EL) display device.

[0037] The interface unit 104 exchanges information, data, or signals with the recording medium 200. Specifically, the interface unit 104 reads out the assistance program SP from the recording medium 200. The assistance program SP read out from the recording medium 200 is stored in the memory unit 105 by the processing unit 106. As a result, the assistance program SP is installed in the terminal device 101A.

[0038] For example, the interface unit 104 may be electrically connected to the recording medium 200 to input and output information, data, or signals to and from the recording medium 200. Specifically, the interface unit 104 may include a slot or a USB terminal. A card-shaped information carrier such as an SD memory card may be inserted into the slot. A USB memory may be inserted into the USB terminal, or the other end of a USB cable, one end of which is electrically connected to a hard disk drive, may be inserted into the USB terminal. Alternatively, the interface unit 104 may include an optical disc drive. The optical disc drive reads information (data) from a CD, DVD, or Blu-ray disc.

[0039] The storage unit 105 has a main storage device. The main storage device includes, for example, a semiconductor memory. The storage unit 105 may further include an auxiliary storage device. The auxiliary storage device includes, for example, at least one of a semiconductor memory and a hard disk drive. The storage unit 105 stores various computer programs and various data. Specifically, the storage unit 105 stores first learning data LD1 and second learning data LD2. The storage unit 105 also stores an assistance program SP installed from the recording medium 200. The assistance program SP is non-temporarily recorded in the storage unit 105. The storage unit 105 is a recording medium that non-temporarily records the assistance program SP.

[0040] The first training data LD1 is a data set used for training the first machine learning model ML1. Specifically, the first training data LD1 includes a data set that associates objective variables (observed values) with explanatory variables (experimental conditions).

[0041] For example, when using the assistance program SP to find an optimal solution for at least one of various parameters that define the operation of a substrate processing apparatus, the objective variable may be the particle count or an evaluation index value of the etching profile. In this case, the operator repeats an experiment in which the substrate processing apparatus processes multiple substrates multiple times while changing the parameter values ​​(experimental conditions) that are explanatory variables, and obtains the particle count or etching profile evaluation index value that is the objective variable for each substrate. Then, a data set (first learning data LD1) is created that associates the particle count or etching profile evaluation index value for each substrate with the corresponding parameter value (corresponding experimental condition).

[0042] The second training data LD2 is a dataset used for training the second machine learning model ML2. Specifically, the second training data LD2 includes a dataset that associates the variance of the dependent variable (variance of the observed values) with the explanatory variables (experimental conditions). Specifically, each value x of the explanatory variable is associated with one or more corresponding values ​​(observed values) of the dependent variable.

[0043] For example, when the objective variable is a particle count or an evaluation index value of an etching profile, even if a substrate processing apparatus processes multiple substrates using a certain parameter value, the particle count or etching profile evaluation index value will differ for each substrate. That is, the particle count or etching profile evaluation index value will be dispersed. Furthermore, the degree of variation in the particle count or etching profile evaluation index value will differ for each parameter value. That is, the particle count or etching profile evaluation index value will exhibit heteroscedasticity. In other words, the observed value of the objective variable (particle count or etching profile evaluation index value) contains heteroscedastic observation noise. The operator creates a data set (second learning data LD2) that associates each parameter value with the corresponding particle count dispersion or etching profile evaluation index value dispersion.

[0044] The processing unit 106 includes a processor. The processing unit 106 may include, for example, a central processing unit (CPU), a graphics processing unit (GPU), a neural network processing unit (NPU), or a quantum computer. Alternatively, the processing unit 106 may include a general-purpose computing device or a dedicated computing device. For example, the processing unit 106 may include a field-programmable gate array (FPGA) or an application-specific integrated circuit (ASIC).

[0045] The processing unit 106 performs various processes such as numerical calculations, information processing, and device control by executing computer programs stored in the storage unit 105 based on instructions from an operator input via the operation unit 102. For example, the processing unit 106 reads out an assistance program SP from the recording medium 200 via the interface unit 104 and stores the assistance program SP in the storage unit 105. The processing unit 106 also executes the assistance program SP.

[0046] Next, heteroscedastic observation noise will be described with reference to FIG. 2. FIG. 2 is a diagram showing an example of heteroscedastic observation noise. In FIG. 2, the horizontal axis represents the explanatory variable, and the vertical axis represents the response variable. As shown in FIG. 2, when the observation noise is heteroscedastic observation noise, the magnitude of the observation noise differs between a certain value x1 of the explanatory variable and another value x2 of the explanatory variable. Note that heteroscedastic Bayesian optimization searches for a value x of the explanatory variable that maximizes or minimizes the response variable (characteristic value) while avoiding areas with large observation noise.

[0047] Next, the assistance method, recording medium 200, assistance program SP, and assistance system 100A of this embodiment will be described with reference to Figures 1 to 3. Figure 3 is a flowchart showing the assistance method of this embodiment. Here, the assistance method refers to a method for assisting in the search for a value x (recommended value or candidate value) of an explanatory variable that optimizes (maximizes or minimizes) the expected value of a target variable. As shown in Figure 3, the assistance method of this embodiment includes steps S1 to S5.

[0048] The assistance method of this embodiment is executed, for example, by the terminal device 101A included in the assistance system 100A described with reference to Fig. 1. More specifically, the assistance method shown in Fig. 3 is executed by the terminal device 101A executing the assistance program SP read from the recording medium 200. In this case, the flowchart shown in Fig. 3 corresponds to the flow of processing executed by the processing unit 106 included in the terminal device 101A. In other words, Fig. 3 shows the processing executed by the processing unit 106 included in the assistance system 100A of this embodiment.

[0049] The process (assistance method) shown in Fig. 3 starts when the operator operates the operation unit 102 to instruct execution of the assistance program SP. When the process shown in Fig. 3 starts, the processing unit 106 causes the first machine learning model ML1 to learn (machine learn) the first training data LD1 and outputs a first predictive distribution f|D from the first machine learning model ML1 (step S1). The first predictive distribution f|D represents a predictive distribution (conditional probability distribution) of the expected value of the dependent variable.

[0050] Specifically, the operator operates the operation unit 102 to specify (set) the first training data LD1 as a data set to be trained by the first machine learning model ML1. Then, the operator operates the operation unit 102 to instruct training of the first training data LD1. The processing unit 106 executes the preprocessing program FP in response to the operator's instruction. As a result, preprocessing is performed on the first training data LD1. More specifically, each value x of the explanatory variable included in the first training data LD1 is normalized so that it falls within the range of 0 to 1. In addition, the objective variable is standardized. The processing unit 106 trains (machines learns) the preprocessed first training data LD1 in the first machine learning model ML1. As a result, a first predictive distribution f|D is output from the first machine learning model ML1 and stored in the storage unit 105.

[0051] Furthermore, the processing unit 106 causes the second machine learning model ML2 to learn (machine learn) the second training data LD2 and outputs a second predictive distribution g|D from the second machine learning model ML2 (step S2). The second predictive distribution g|D represents the predictive distribution (conditional probability distribution) of the variance of the dependent variable. In other words, the second predictive distribution g|D represents the predictive distribution of the observation noise.

[0052] Specifically, the operator operates the operation unit 102 to specify (set) the second training data LD2 as a data set to be trained by the second machine learning model ML2. Then, the operator operates the operation unit 102 to instruct training of the second training data LD2. The processing unit 106 executes the preprocessing program FP in response to the operator's instruction. As a result, preprocessing is performed on the second training data LD2. More specifically, each value x of the explanatory variable included in the second training data LD2 is normalized so that it falls within the range of 0 to 1. In addition, the objective variable is standardized. The processing unit 106 trains (machines trains) the preprocessed second training data LD2 in the second machine learning model ML2. As a result, the second predictive distribution g|D is output from the second machine learning model ML2 and stored in the storage unit 105.

[0053] The order of execution of the first learning process (step S1) in which the first machine learning model ML1 learns the first training data LD1 and the second learning process (step S2) in which the second machine learning model ML2 learns the second training data LD2 can be reversed. Alternatively, the first learning process (step S1) and the second learning process (step S2) may be executed in parallel. In this case, the operator operates the operation unit 102 to instruct learning of the first training data LD1 and the second training data LD2.

[0054] After acquiring the first predictive distribution f|D and the second predictive distribution g|D, the processing unit 106 integrates the first predictive distribution f|D and the second predictive distribution g|D to construct a third predictive distribution MV|D (step S3). Specifically, when the first predictive distribution f|D is expressed by the following formula (4) and the second predictive distribution g|D is expressed by the following formula (5), the third predictive distribution MV|D is expressed by the following single formula (6). Note that formulas (3) and (4) express the predictive distribution (conditional probability distribution) output from the Gaussian process regression model.

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[0055] Specifically, heteroscedastic Bayesian optimization, which is Bayesian optimization that takes heteroscedastic observation noise into account, searches for the value x of the explanatory variable that maximizes or minimizes the objective function MV(x) expressed by the following equation (7). MV(x)=f(x)-αg(x) (7)

[0056] In equation (7), f(x) represents the objective variable. g(x) represents heteroscedastic observation noise. Furthermore, coefficient α represents the risk aversion coefficient. When f(x) is expressed using a Gaussian process regression model, it is expressed by the following equation (8). When g(x) is expressed using a Gaussian process regression model, it is expressed by the following equation (9). In equations (8) and (9), "m f " and "m g " indicates the mean function. "k f " and "k g " indicates the kernel function.

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[0057] The first predictive distribution f|D expressed by the above formula (4) indicates the predictive distribution output from the Gaussian process regression model f expressed by formula (8). The second predictive distribution g|D expressed by the above formula (5) indicates the predictive distribution output from the Gaussian process regression model g expressed by formula (9). The third predictive distribution MV|D expressed by the above formula (6) is constructed by integrating formulas (4) and (5) based on the objective function MV(x), using a linear transformation of the multivariate normal distribution and the properties of the sum of multivariate normal distributions when independence holds.

[0058] After constructing the third predictive distribution MV|D, the processing unit 106 executes a recommended value acquisition step (step S4) to acquire at least one recommended value (or candidate value) of the explanatory variable. Specifically, the processing unit 106 executes parallel Bayesian optimization based on the third predictive distribution MV|D, at least one acquisition function AF, and the search range, and acquires at least one value x (recommended value or candidate value) of the explanatory variable that maximizes the acquisition function AF from within the search range.

[0059] Before executing parallel Bayesian optimization, the operator operates the operation unit 102 to set (input) the type of acquisition function AF to be used for parallel Bayesian optimization in the source code of the assistance program SP. Specifically, the operator selects at least one of the acquisition functions AF included in the assistance program SP and sets the selected acquisition function AF in the source code of the assistance program SP. The operator also operates the operation unit 102 to set (input) a search range value and the number of explanatory variable values ​​x to be presented from one acquisition function AF in the source code of the assistance program SP. The number of explanatory variable values ​​x to be presented from one acquisition function AF indicates an integer equal to or greater than 1. The number of explanatory variable values ​​x to be presented from one acquisition function AF may be preset as a fixed value in the assistance program SP.

[0060] When 1 is set as the number of explanatory variable values ​​x to be presented from the acquisition function AF, the processing unit 106 acquires one recommended value from one acquisition function AF. When an integer equal to or greater than 2 is set as the number of explanatory variable values ​​x to be presented from the acquisition function AF, the processing unit 106 acquires two or more recommended values ​​(multiple recommended values) from one acquisition function AF. For example, when the number of explanatory variable values ​​x to be presented from one acquisition function AF is set to three, six recommended values ​​(or candidate values) are acquired by performing parallel Bayesian optimization using two acquisition functions AF.

[0061] Finally, the processing unit 106 causes the display unit 103 to display the recommended values ​​(or candidate values) of the explanatory variables obtained by performing parallel Bayesian optimization (step S5), and ends the processing (assistance method) shown in FIG.

[0062] Next, the first predictive distribution f|D, the second predictive distribution g|D, and the third predictive distribution MV|D will be described with reference to FIGS. 4(a) to 4(e). FIG. 4(a) is a diagram showing an example of observed values ​​of a dependent variable. The observed value data shown in FIG. 4(a) is one-dimensional data. FIG. 4(b) is a diagram showing an example of the first predictive distribution f|D, the second predictive distribution g|D, and the third predictive distribution MV|D. FIG. 4(c) is a diagram showing the first predictive distribution f|D of FIG. 4(b). FIG. 4(d) is a diagram showing the second predictive distribution g|D of FIG. 4(b). FIG. 4(e) is a diagram showing the third predictive distribution MV|D of FIG. 4(b). In FIGS. 4(a) to 4(e), the horizontal axis represents the explanatory variable, and the vertical axis represents the dependent variable.

[0063] The first predictive distribution f|D, the second predictive distribution g|D, and the third predictive distribution MV|D in Figures 4(b) to 4(e) show predictive distributions for the observed values ​​(one-dimensional data) in Figure 4(a). More specifically, Figures 4(b) to 4(e) show the expected values ​​and standard deviation ranges of the first predictive distribution f|D, the second predictive distribution g|D, and the third predictive distribution MV|D.

[0064] As described above with reference to FIGS. 1 to 3 and 4(a) to 4(e), according to this embodiment, Bayesian optimization can be performed based on one predictive distribution (third predictive distribution MV|D). Therefore, unlike heteroscedastic Bayesian optimization, an acquisition function used in general Bayesian optimization can be used. Therefore, parallel Bayesian optimization can be performed. Furthermore, the third predictive distribution MV|D is a predictive distribution that integrates the first predictive distribution f|D and the second predictive distribution g|D based on the objective function MV(x) that formulates heteroscedastic Bayesian optimization. Therefore, by using the third predictive distribution MV|D, Bayesian optimization that takes heteroscedastic observation noise into account can be performed. Therefore, according to this embodiment, parallel Bayesian optimization that takes heteroscedastic observation noise into account can be performed.

[0065] Next, an acquisition function (acquisition function AF) used for the third predictive distribution MV|D will be described. As already described, an acquisition function used in general Bayesian optimization can be used for the third predictive distribution MV|D. For example, the assistance program SP may include, as the acquisition function AF, at least one of a Probability of Improvement (PI) acquisition function, an Expected Improvement (EI) acquisition function, a Log EI acquisition function, a Lower Confidence Bound (LCB) acquisition function, an Upper Confidence Bound (UCB) acquisition function, Thompson Sampling (TS), a Joint Entropy Search (JES), a Predictive Entropy Search (PES), a Max-value Entropy Search (MES), and a Knowledge Gradient.

[0066] Furthermore, in this embodiment, the first machine learning model ML1 includes a Gaussian process regression model. The kernel function of the Gaussian process regression model included in the first machine learning model ML1 may be a twice-differentiable kernel function. Similarly, the second machine learning model ML2 includes a Gaussian process regression model. The kernel function of the Gaussian process regression model included in the second machine learning model ML2 may be a twice-differentiable kernel function. The twice-differentiable kernel function is, for example, an RBF kernel or a Matern-5 / 2 kernel.

[0067] According to this embodiment, since the kernel functions of the Gaussian process regression models included in each of the first machine learning model ML1 and the second machine learning model ML2 are twice-differentiable kernel functions, a Monte Carlo acquisition function can be used as the acquisition function (acquisition function AF) used for the third predictive distribution MV|D. Therefore, the assistance program SP may include the Monte Carlo acquisition function as the acquisition function AF. Alternatively, the assistance program SP may include a computer program that acquires the Monte Carlo acquisition function from the acquisition function AF. Specifically, the Monte Carlo acquisition function is an acquisition function that allows Monte Carlo approximation of the gradient of the acquisition function. For example, acquisition functions such as the PI acquisition function, the EI acquisition function, the LCB acquisition function, the UCB acquisition function, and the entropy search can be regarded as Monte Carlo acquisition functions by formula transformation. The processing unit 106 may execute the assistance program SP and acquire the Monte Carlo acquisition function by Monte Carlo approximation of the gradient of the acquisition function AF specified by the operator.

[0068] Next, the recommended value acquisition step (step S4 in FIG. 3) will be described with reference to FIG. 5. FIG. 5 is a diagram showing an example of the flow of the recommended value acquisition step (step S4 in FIG. 3). As shown in FIG. 5, the recommended value acquisition step (step S4 in FIG. 3) may include steps S41 to S44. Step S41 represents a step of accepting settings (input) of multiple Monte Carlo acquisition functions with different properties. Step S42 represents a step of accepting settings (input) of the number of recommended values ​​(or candidate values) to be presented from one acquisition function AF. Step S43 represents a step of accepting settings (input) of values ​​in the search range. Step S44 represents a step of executing parallel Bayesian optimization.

[0069] When the processing unit 106 starts the process shown in FIG. 5, it accepts the setting (input) of multiple Monte Carlo acquisition functions having different properties (step S41). Specifically, an input field (line) for accepting the setting (input) of multiple Monte Carlo acquisition functions is provided in the source code of the assistance program SP. The operator operates the operation unit 102 to input information indicating multiple Monte Carlo acquisition functions having different properties into the source code of the assistance program SP. The multiple Monte Carlo acquisition functions having different properties include, for example, a PI acquisition function, an EI acquisition function, an LCB acquisition function, a UCB acquisition function, and an acquisition function such as entropy search. Note that the operator may input information indicating multiple acquisition functions AF having different properties, and the processing unit 106 may acquire multiple Monte Carlo acquisition functions corresponding to the set multiple acquisition functions AF by calculation.

[0070] Next, the processing unit 106 accepts the setting (input) of the number of recommended values ​​(or candidate values) to be presented from one acquisition function AF (step S42). Specifically, an input field (line) for accepting the setting (input) of the number of recommended values ​​(or candidate values) to be presented from one acquisition function AF is provided in the source code of the assistance program SP. The input field (line) for accepting the setting (input) of the number of recommended values ​​(or candidate values) accepts an integer value of 1 or greater. The operator operates the operation unit 102 to input the number of recommended values ​​(or candidate values) to be presented from one acquisition function AF into the source code of the assistance program SP.

[0071] Next, the processing unit 106 accepts the setting (input) of the value of the search range (step S43). Specifically, an input field (line) for accepting the setting (input) of the value of the search range is provided in the source code of the assistance program SP. The operator operates the operation unit 102 to input the value of the search range into the source code of the assistance program SP.

[0072] Next, the processing unit 106 executes parallel Bayesian optimization (step S44), which ends the process shown in FIG.

[0073] The processing unit 106 may accept a setting (input) of the number of recommended values ​​(or candidate values) for each of the acquisition functions AF used in the parallel Bayesian optimization. Specifically, the source code of the assistance program SP may be provided with an input field (line) for accepting a setting of the number of recommended values ​​(or candidate values) for each acquisition function AF used in the parallel Bayesian optimization.

[0074] 5, the processes of steps S41 to S43 are executed after the third predictive distribution MV|D is constructed (after step S3 in FIG. 3 is executed), but some or all of the processes of steps S41 to S43 may be executed before the third predictive distribution MV|D is constructed (before step S3 in FIG. 3 is executed). Specifically, the processes of steps S41 to S43 may be executed before parallel Bayesian optimization (step S44) is executed. For example, the processes of steps S41 to S43 may be executed before step S1 in FIG. 3 is executed.

[0075] In the example shown in FIG. 5, the processes of steps S41 to S43 are executed in the order of step S41, step S42, and step S43, but the execution order of step S41, step S42, and step S43 can be reversed.

[0076] In this embodiment, in step S4 (recommended value acquisition step) of FIG. 3, the processing unit 106 executes parallel Bayesian optimization based on Thompson sampling and parallel Bayesian optimization based on each of a plurality of Monte Carlo acquisition functions set by the operator.

[0077] According to this embodiment, multiple recommended values ​​(or candidate values) can be obtained from Thompson sampling, and multiple recommended values ​​(or candidate values) can be obtained from each of multiple Monte Carlo acquisition functions with different properties. Therefore, the operator can more efficiently determine the value x of the explanatory variable that optimizes (maximizes or minimizes) the objective variable.

[0078] An upper limit may be set for the number of recommended values ​​(or candidate values) presented from one acquisition function AF. Specifically, an upper limit may be set in advance in the value of the input field for setting the number of recommended values ​​(or candidate values). The upper limit may be, for example, 5. By setting an upper limit for the number of recommended values ​​(or candidate values), bias is less likely to occur among the multiple recommended values ​​(or candidate values) presented from the multiple acquisition functions AF. Therefore, the operator can more efficiently determine the value x of the explanatory variable that optimizes (maximizes or minimizes) the objective variable.

[0079] Next, the procedure of an experiment using the assistance method, recording medium 200, assistance program SP, and assistance system 100A of this embodiment will be described with reference to Fig. 6. Fig. 6 is a diagram showing the procedure of an experiment using the assistance method, recording medium 200, assistance program SP, and assistance system 100A of this embodiment.

[0080] The experimental procedure shown in Fig. 6 is started after executing the support method shown in Fig. 3. As shown in Fig. 6, the operator performs multiple experiments using multiple recommended values ​​(currently recommended values) obtained by executing the support method shown in Fig. 3 (step S11), and obtains multiple experimental results (multiple values ​​of the objective variable) (step S12). That is, the operator performs multiple experiments and obtains multiple observed values. Then, the operator determines whether any of the experimental results (values ​​of the objective variable) is an optimal value (maximum or minimum value) (step S13).

[0081] If the operator determines that none of the experimental results are optimal (No in step S13), he or she updates the first learning data LD1 and the second learning data LD2 using the multiple values ​​x of the explanatory variable used in the current experiment (the current recommended value) and the multiple experimental results of the current experiment (multiple values ​​of the objective variable) (step S14), and again executes the support method described with reference to Fig. 3. Then, the operator repeats updating the first learning data LD1 and the second learning data LD2 (step S14), the support method described with reference to Fig. 3, and the experiment (steps S11 and S12) until he or she determines that the experimental results are optimal.

[0082] If the operator determines that any of the experimental results is an optimal value (Yes in step S13), the operator ends the experiment. For example, the operator may end the experiment if the operator determines that any of the experimental results has converged to a value within the target range.

[0083] The first embodiment of the present invention has been described above with reference to FIGS. 1 to 6. According to the first embodiment, it is possible to perform Bayesian optimization that takes heteroscedastic observation noise into consideration. Furthermore, according to the first embodiment, it is possible to perform parallel Bayesian optimization and obtain multiple recommended values ​​(or candidate values) of explanatory variables in a single process. Therefore, compared to the case where one recommended value (or candidate value) is obtained per process, the operator can more efficiently determine the value x of the explanatory variable that optimizes (minimizes or maximizes) the value of the objective variable.

[0084] In the embodiment described with reference to FIGS. 1 to 6, the assistance system 100A (terminal device 101A) acquires the assistance program SP from the recording medium 200. However, the assistance system 100A (terminal device 101A) may acquire the assistance program SP from another computer system. For example, the terminal device 101A may be communicatively connected to the other computer system via a cable and acquire the assistance program SP from the other computer system. Alternatively, the terminal device 101A may be communicatively connected to the other computer system via a line network such as the Internet and acquire the assistance program SP from the other computer system. The other computer system may be a general-purpose computer or a dedicated computer. The other computer system may be a server.

[0085] 1 to 6, the assistance program SP is installed in the terminal device 101A, but the assistance program SP does not have to be installed in the terminal device 101A. The terminal device 101A may execute the assistance program SP stored in the recording medium 200.

[0086] [Embodiment 2] Next, a second embodiment of the present invention will be described with reference to Figures 3 and 7. However, only the differences from the first embodiment will be described, and the same matters as in the first embodiment will not be described. The second embodiment differs from the first embodiment in that the server 300 outputs at least one recommended value (candidate value) of the explanatory variable based on the assistance program SP.

[0087] 7 is a block diagram showing the configuration of a support system 100B of embodiment 2. As shown in FIG.

[0088] In the second embodiment, the terminal device 101B includes an operation unit 102, a display unit 103, a storage unit 105, a communication unit 107, and a processing unit .

[0089] The communication unit 107 is connected to a network and performs communication with the server 300. The network includes, for example, the Internet, a local area network (LAN), a public telephone network, and a short-range wireless network. The communication unit 107 includes a communication device. The communication unit 107 is, for example, a network interface controller.

[0090] The communication unit 107 is controlled by the processing unit 106 and exchanges information, data, or signals with the server 300. For example, the communication unit 107 transmits to the server 300 the first learning data LD1, the second learning data LD2, information for setting the type of acquisition function AF to be used in the parallel Bayesian optimization, information for setting the number of explanatory variable values ​​x (recommended values ​​or candidate values) to be presented from one acquisition function AF, and information for setting the value of the search range.

[0091] The server 300 includes a communication unit 301 , a storage unit 302 , and a processing unit 303 .

[0092] The communication unit 301 is connected to a network and communicates with the terminal device 101B. The communication unit 301 includes a communication device. The communication unit 301 is, for example, a network interface controller. The communication unit 301 is controlled by the processing unit 303 to exchange information, data, or signals with the terminal device 101B. For example, the communication unit 301 receives from the terminal device 101B first learning data LD1, second learning data LD2, information for setting the type of acquisition function AF to be used in parallel Bayesian optimization, information for setting the number of explanatory variable values ​​x (recommended values ​​or candidate values) to be presented from one acquisition function AF, and information for setting the value of a search range.

[0093] The storage unit 302 has a main storage device and an auxiliary storage device. The main storage device includes, for example, a semiconductor memory. The auxiliary storage device includes, for example, a hard disk drive. The storage unit 302 stores the assistance program SP. The storage unit 302 also stores the first learning data LD1 and the second learning data LD2 received from the terminal device 101B, information for setting the type of acquisition function AF to be used for parallel Bayesian optimization, information for setting the number of explanatory variable values ​​x (recommended values ​​or candidate values) to be presented from one acquisition function AF, and information for setting the value of the search range.

[0094] The processing unit 303 includes a processor. The processing unit 303 may include, for example, a CPU, a GPU, an NPU, or a quantum computer. Alternatively, the processing unit 303 may include a general-purpose computing device or a dedicated computing device. For example, the processing unit 303 may include an FPGA or an ASIC. The processing unit 303 executes the assistance program SP based on an instruction from the terminal device 101B.

[0095] Next, the support method, support program SP, and support system 100B of the second embodiment will be described with reference to Fig. 7 and Fig. 3. The processing unit 303 starts the process shown in Fig. 3 based on a command from the terminal device 101B.

[0096] 3 starts, the processing unit 303 causes the first machine learning model ML1 to learn (machine learn) the first training data LD1 received from the terminal device 101B, and causes the first machine learning model ML1 to output a predictive distribution of the expected value of the dependent variable (first predictive distribution f|D) (step S1). The processing unit 303 also causes the second machine learning model ML2 to learn (machine learn) the second training data LD2 received from the terminal device 101B, and causes the second machine learning model ML2 to output a predictive distribution of the variance of the dependent variable (second predictive distribution g|D) (step S2).

[0097] After obtaining the first predictive distribution f|D and the second predictive distribution g|D, the processing unit 303 integrates the first predictive distribution f|D and the second predictive distribution g|D to construct a third predictive distribution MV|D (step S3).

[0098] After constructing the third predictive distribution MV|D, the processing unit 303 executes parallel Bayesian optimization based on the third predictive distribution MV|D, at least one acquisition function AF, and the search range, and acquires at least one explanatory variable value x (recommended value or candidate value) that maximizes the acquisition function AF from within the search range (step S4). Then, the processing unit 303 transmits information indicating the at least one recommended value (or candidate value) to the terminal device 101B, and causes the display unit 103 to display the recommended value (or candidate value) of the explanatory variable (step S5). As a result, the process (support method) shown in FIG. 3 ends.

[0099] Specifically, the processing unit 303 transmits screen information indicating the source code of the assistance program SP to the terminal device 101B, and causes the display unit 103 of the terminal device 101B to display a screen indicating the source code of the assistance program SP. As described with reference to FIG. 5, the operator operates the operation unit 102 to input, onto the screen indicating the source code of the assistance program SP, information indicating at least one acquisition function AF to be used in parallel Bayesian optimization, information indicating the number of recommended values ​​(or candidate values) to be presented from one acquisition function AF, and information indicating a value of the search range. As a result, the information input onto the screen indicating the source code of the assistance program SP by the operator operating the operation unit 102 is transmitted from the terminal device 101B to the server 300. The processing unit 303 executes parallel Bayesian optimization based on the information acquired from the terminal device 101B and the third predictive distribution MV|D.

[0100] The second embodiment of the present invention has been described above with reference to Fig. 3 and Fig. 7. According to the second embodiment, similar to the first embodiment, parallel Bayesian optimization can be performed taking heteroscedastic observation noise into consideration.

[0101] [Embodiment 3] Next, a third embodiment of the present invention will be described with reference to Figures 8 and 9. However, only differences from the first and second embodiments will be described, and a description of the same aspects as the first and second embodiments will be omitted. The third embodiment differs from the first and second embodiments in that the control device 10 included in the substrate processing system 1000 also functions as the support system 100C.

[0102] 8 is a schematic diagram of a substrate processing system 1000 including a support system 100C of this embodiment. In detail, FIG. 8 is a schematic plan view of a substrate processing apparatus 400 included in the substrate processing system 1000.

[0103] As shown in Fig. 8, the substrate processing system 1000 includes a substrate processing apparatus 400 and a control device 10. The substrate processing apparatus 400 processes a substrate W. The control device 10 controls the substrate processing apparatus 400. In this embodiment, the substrate W is a disk-shaped semiconductor wafer. The substrate processing apparatus 400 is a single-wafer processing apparatus, and processes the substrates W one by one.

[0104] In detail, the substrate processing apparatus 400 includes a plurality of substrate processing units 2, a fluid cabinet 401, a plurality of fluid boxes 402, a plurality of load ports LP, an indexer robot IR, and a center robot CR.

[0105] A cassette CA is placed on each of the load ports LP. The cassette CA accommodates a stack of substrates W. The cassette CA is, for example, a FOUP (Front Opening Unified Pod), a SMIF (Standard Mechanical Interface) pod, or an OC (Open Cassette).

[0106] The indexer robot IR transports substrates W between the cassette CA and the center robot CR. The center robot CR transports substrates W between the indexer robot IR and a plurality of substrate processing units 2. Note that a placement stage (path) on which the substrate W is temporarily placed may be provided between the indexer robot IR and the center robot CR, and the device may be configured so that the substrate W is transferred indirectly between the indexer robot IR and the center robot CR via the placement stage.

[0107] The substrate processing units 2 form a plurality of towers TW (four towers TW in FIG. 8). The towers TW are arranged to surround the center robot CR in a plan view. Each tower TW includes a plurality of substrate processing units 2 (three substrate processing units 2 in FIG. 8) stacked one above the other.

[0108] The fluid cabinet 401 contains a fluid. Specifically, the fluid cabinet 401 contains a processing liquid. Alternatively, the fluid cabinet 401 may contain a processing liquid and a gas.

[0109] The processing liquid is not particularly limited as long as it is a liquid that comes into contact with the substrate W. The processing liquid may include, for example, dilute hydrofluoric acid (DHF), hydrofluoric acid (HF), hydrofluoric nitric acid (a mixture of hydrofluoric acid and nitric acid (HNO3)), buffered hydrofluoric acid (BHF), ammonium fluoride, HFEG (a mixture of hydrofluoric acid and ethylene glycol), phosphoric acid (H3PO4), sulfuric acid, acetic acid, nitric acid, hydrochloric acid, ammonia water, hydrogen peroxide water, organic acids (e.g., citric acid, oxalic acid), organic alkalis (e.g., tetramethylammonium hydroxide (TMAH)), sulfuric acid / hydrogen peroxide water mixture (SPM), ammonia / hydrogen peroxide water mixture (SC1), hydrochloric acid / hydrogen peroxide water mixture (SC2), isopropyl alcohol (IPA), surfactants, corrosion inhibitors, pure water (e.g., deionized water), carbonated water, electrolytic ionized water, hydrogen water, ozone water, or hydrochloric acid water at a dilute concentration (e.g., about 0.001% by weight to about 0.01% by weight). The gas may include, for example, an inert gas. The inert gas is, for example, nitrogen gas.

[0110] Each fluid box 402 corresponds to one of the multiple towers TW. The fluid in the fluid cabinet 401 is supplied to all of the substrate processing units 2 included in the corresponding tower TW via one of the fluid boxes 402.

[0111] Each of the substrate processing units 2 processes one substrate W at a time. Specifically, each of the substrate processing units 2 processes the substrate W by supplying a processing liquid to the substrate W. For example, each of the substrate processing units 2 performs a cleaning process or an etching process on the substrate W.

[0112] The control device 10 controls the operation of each part of the substrate processing apparatus 400. For example, the control device 10 controls the substrate processing unit 2, the fluid cabinet 401, the fluid box 402, the load port LP, the indexer robot IR, and the center robot CR. The control device 10 includes a control unit 11 and a memory unit 12.

[0113] The control unit 11 controls the operation of each unit of the substrate processing apparatus 400 based on various information stored in the storage unit 12. The control unit 11 includes a processor. The control unit 11 may include, for example, a CPU, a GPU, an NPU, or a quantum computer. Alternatively, the control unit 11 may include a general-purpose computing device or a dedicated computing device. For example, the control unit 11 may include an FPGA or an ASIC.

[0114] The memory unit 12 stores various types of information for controlling the operation of the substrate processing apparatus 400. For example, the memory unit 12 stores various types of data and various computer programs. The various types of data include recipe data. The recipe data indicates a recipe that defines the processing content, processing conditions, and processing procedure for the substrate W. In the recipe, various setting values ​​(recipe parameter values) are set as processing conditions.

[0115] The storage unit 12 includes a main storage device. The main storage device includes, for example, a semiconductor memory. The storage unit 12 may further include an auxiliary storage device. The auxiliary storage device includes, for example, at least one of a semiconductor memory and a hard disk drive.

[0116] 9 is a block diagram showing the configuration of a support system 100C of this embodiment. In detail, FIG. 9 is a block diagram showing the configuration of a control device 10.

[0117] As shown in Figures 8 and 9, the control device 10 also functions as an assistance system 100C. More specifically, as shown in Figure 9, an assistance program SP is installed in the control device 10 from a recording medium 200. Like the terminal device 101A described with reference to Figures 1 to 6, the control device 10 executes the assistance program SP installed from the recording medium 200 and searches for values ​​(recommended values ​​or candidate values) of explanatory variables that optimize the expected value of the objective variable. The control device 10 is an example of an "assistance device."

[0118] 9, the control device 10 further includes an operation unit 13, a display unit 14, and an interface unit 15. The configurations of the operation unit 13, the display unit 14, and the interface unit 15 are similar to those of the operation unit 102, the display unit 103, and the interface unit 104 described with reference to FIG. 1, and therefore, description thereof will be omitted.

[0119] 1, the storage unit 12 stores the first learning data LD1, the second learning data LD2, and the assistance program SP read from the recording medium 200 by the interface unit 15. In the substrate processing system 1000, the explanatory variables may include, for example, at least one of various setting values ​​(recipe parameter values) defined by a recipe and various setting values ​​(apparatus parameter values) set for the substrate processing apparatus 400. When the substrate processing unit 2 performs a cleaning process, the objective variable may be, for example, the number of particles. When the substrate processing unit 2 performs an etching process, the objective variable may be, for example, an evaluation index value of an etching profile.

[0120] The control unit 11 executes the assistance program SP in the same manner as the processing unit 106 described with reference to FIGS. 1 to 6, and outputs at least one value x (recommended value or candidate value) of the explanatory variable that optimizes the expected value of the objective variable.

[0121] The third embodiment of the present invention has been described above with reference to Fig. 8 and Fig. 9. According to the third embodiment, similarly to the first and second embodiments, parallel Bayesian optimization can be performed taking heteroscedastic observation noise into consideration.

[0122] In the third embodiment, the substrate processing apparatus 400 performs a cleaning process or an etching process on the substrate W, but the substrate processing apparatus 400 is not particularly limited as long as it is an apparatus for processing a substrate. For example, the substrate processing apparatus 400 may be a coating apparatus, a developing apparatus, an exposure apparatus, a baking apparatus, or a film forming apparatus.

[0123] Furthermore, in the third embodiment, the substrate processing apparatus 400 is of a single wafer type, but the substrate processing apparatus 400 may be of a batch type.

[0124] In addition, in the third embodiment, the substrate processing apparatus 400 processes a disk-shaped semiconductor wafer, but the substrate to be processed is not limited to a semiconductor wafer. The substrate to be processed may be a glass substrate for a photomask, a glass substrate for a liquid crystal display, a glass substrate for a plasma display, a substrate for an FED (Field Emission Display), a substrate for an optical disk, a substrate for a magnetic disk, or a substrate for a magneto-optical disk. The shape of the substrate to be processed is also not limited to a disk shape.

[0125] The embodiments of the present invention have been described above with reference to the drawings (FIGS. 1 to 9). However, the present invention is not limited to the above embodiments and can be implemented in various forms without departing from the spirit of the present invention. Furthermore, the components disclosed in the above embodiments can be modified as appropriate. For example, some of the components shown in one embodiment may be added to the components of another embodiment, or some of the components shown in one embodiment may be deleted from the embodiment.

[0126] The drawings mainly show each component in a schematic manner to facilitate understanding of the invention, and the thickness, length, number, spacing, etc. of each component shown in the drawings may differ from the actual ones due to the convenience of creating the drawings. Furthermore, the configuration of each component shown in the above embodiment is merely an example and is not particularly limited, and it goes without saying that various modifications are possible within a range that does not substantially deviate from the effects of the present invention.

[0127] 1 to 9, the assistance program SP includes the preprocessing program FP, but the assistance program SP does not have to include the preprocessing program FP. For example, a preprocessing program created by an operator may be stored in the storage unit 105.

[0128] In addition, in the embodiment described with reference to FIGS. 1 to 9, parallel Bayesian optimization based on Thompson sampling and parallel Bayesian optimization based on each of a plurality of Monte Carlo acquisition functions are performed, but parallel Bayesian optimization based on one type of acquisition function AF may also be performed.

[0129] Furthermore, in the embodiment described with reference to Figures 1 to 9, multiple recommended values ​​(or candidate values) are obtained for each processing, but it is also possible to obtain one recommended value (or candidate value) for each processing.

[0130] In the embodiment described with reference to FIGS. 1 to 9, the operator sets multiple Monte Carlo acquisition functions with different properties. However, the multiple Monte Carlo acquisition functions used in parallel Bayesian optimization may be preset as fixed values ​​in the assistance program SP. [Industrial Applicability]

[0131] The present invention is useful for methods and systems that perform Bayesian optimization for predictive distributions that contain heteroscedastic observation noise. [Explanation of symbols]

[0132] 10: Control device 11: Control section 12: Storage section 100A: Support System 100B: Support System 100C: Support System 101A: Terminal device 101B: Terminal device 105: Storage section 106: Processing section 200: Recording medium 302: Storage section 303: Processing section AF: Acquisition Function BP: Heteroscedastic parallel Bayesian optimization program LD1: First training data LD2: Second training data ML1: First machine learning model ML2: Second machine learning model S1: Step S2: Step S3: Step S4: Step SP: Support Program

Claims

1. A method for assisting in searching for values ​​of explanatory variables that maximize or minimize an expected value of a target variable, comprising: outputting a first predictive distribution, which is a predictive distribution of an expected value of the dependent variable, from a first machine learning model capable of outputting a predictive distribution; outputting a second predictive distribution, which is a predictive distribution of the variance of the dependent variable, from a second machine learning model capable of outputting a predictive distribution; constructing a third predictive distribution by integrating the first predictive distribution and the second predictive distribution; a recommended value acquisition step of performing parallel Bayesian optimization based on the third predictive distribution, at least one acquisition function, and a search range to acquire at least one recommended value of the explanatory variable that maximizes the acquisition function from the search range; and how to help.

2. The support method according to claim 1 , wherein a plurality of the recommended values ​​are acquired in the recommended value acquisition step.

3. the first machine learning model and the second machine learning model each include a twice-differentiable kernel function; 3. The method of claim 1, wherein the at least one acquisition function comprises a Monte Carlo acquisition function.

4. the at least one acquisition function includes Thompson sampling and a plurality of distinct Monte Carlo acquisition functions; The support method according to claim 3 , wherein in the recommended value obtaining step, a plurality of the recommended values ​​are obtained from the Thompson sampling, and a plurality of the recommended values ​​are obtained from each of the plurality of Monte Carlo acquisition functions.

5. A computer-readable recording medium having a support program that defines the support method according to claim 1 or 2 recorded thereon.

6. A support program that can be executed by a computer and that defines the support method according to claim 1 or 2.

7. A support system for supporting a search for values ​​of explanatory variables that maximize or minimize an expected value of a target variable, a storage unit that stores a first machine learning model capable of outputting a predictive distribution, a second machine learning model capable of outputting a predictive distribution, and at least one acquisition function; a processing unit that causes the first machine learning model to output a first predictive distribution, which is a predictive distribution of an expected value of the dependent variable, and causes the second machine learning model to output a second predictive distribution, which is a predictive distribution of a variance of the dependent variable; Equipped with The processing unit constructing a third predictive distribution by integrating the first predictive distribution and the second predictive distribution; and performing parallel Bayesian optimization based on the third predictive distribution, the acquisition function, and a search range to obtain at least one recommended value of the explanatory variable from within the search range that maximizes the acquisition function.

8. The assistance system according to claim 7 , wherein the processing unit acquires a plurality of the recommended values.

9. the first machine learning model and the second machine learning model each include a twice-differentiable kernel function; 9. The assistance system of claim 7 or claim 8, wherein the at least one acquisition function comprises a Monte Carlo acquisition function.

10. the at least one acquisition function includes Thompson sampling and a plurality of distinct Monte Carlo acquisition functions; The assistance system according to claim 9 , wherein the processing unit obtains the plurality of recommended values ​​from the Thompson sampling and obtains the plurality of recommended values ​​from each of the plurality of Monte Carlo acquisition functions.

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