Device and method for determining the robust optimal state of a physical or chemical process based on Bayesian optimization methods

Abstract

The present invention discloses an apparatus and method for determining a robust optimal state of a physical or chemical process according to a Bayesian optimization method, wherein each of the known measurement points has an input parameter value of the physical or chemical process and a measured output parameter value assigned to the input parameter value, wherein a first statistical model describes the relationship between the input parameter value and the output parameter value of the physical or chemical process, and in the method a second statistical model is determined for the first statistical model, wherein the second statistical model describes the robustness of the output parameter value of the physical or chemical process relative to changes in the input parameter value, wherein a new measurement point is selected so that the difference between the entropy of the physical or chemical process described by the known measurement point at the new measurement point and the expected entropy of the output parameter of the first statistical model described by the known measurement point at the maximum value taken by the output parameter of the first statistical model described by the known measurement point and the second statistical model at the robust optimal state of the physical or chemical process is substantially maximized or within a predetermined range near the maximum value.

Description

Technical Field

[0001] Various embodiments generally relate to an apparatus and method for determining a robust optimum of a physical or chemical process according to a Bayesian optimization approach. Background Art

[0002] During production and machining processes, process parameters such as process temperature, process time, vacuum or gas atmosphere, etc. are set to obtain desired properties of the workpiece, such as hardness, strength, thermal conductivity, electrical conductivity, etc. The process parameters can be determined using model-based optimization methods (e.g., Bayesian optimization methods). However, the process parameters may be subject to noise or interference, causing the process parameters to deviate from the process parameters determined by the optimization method, and thus the properties of the workpiece may deviate from the desired properties. Therefore, for example, during production and machining processes, it is necessary to determine a robust optimal state in which the workpiece has the desired properties.

[0003] Different models can be used to determine process parameters and / or process-related properties of materials, such as hardness, electrical conductivity, density, microstructure, macrostructure, chemical composition, etc., in various physical or chemical processes in order to achieve a desired process result, wherein the process parameters or process-related properties may deviate from the values ​​used to set (e.g., optimize) the corresponding process. Therefore, it may be necessary to provide a model that is capable of determining a robust optimal state that takes these deviations into account.

[0004] In "Unscented Bayesian Optimization for Safe Robot Grasping" by Nogueira et al. (arXiv:1603.02038, 2016), a method for determining a robust optimal state is described, where the robustness criterion is determined using the expected value with respect to noise.

[0005] A method for determining robust optima is described in Beland et al., “Bayesian Optimization Under Uncertainty,” Conference on Neural Information Processing Systems, 2017, where the robustness criterion is determined via Gaussian process regression.

[0006] A method for determining robust optimality is described in Bogunovic et al., “Adversarially Bayesian Optimization with Gaussian Processes,” Conference on Neural Information Processing Systems, 2018. Summary of the Invention

[0007] The method according to the invention (first example) and the device according to the invention (twenty-fourth example) make it possible to determine the process window by means of a model as a function of the deviations of the input parameters.

[0008] A statistical model can be any type of mathematical representation that describes the relationship between input and output parameters of a physical or chemical process and takes into account probability distributions.

[0009] The method may also include measuring an output parameter value associated with an input parameter value assigned to a new measurement point. The method may also include adapting a first statistical model using the new measurement point, wherein the adapted first statistical model may describe the physical or chemical process at the known measurement point and the new measurement point, and determining a robust optimal state of the adapted first statistical model. This has the advantage of being able to determine one or more process parameters for which the physical or chemical process is robust to disturbances or noise in the process parameters. The features described in this paragraph can be combined with the first example to form a second example.

[0010] Determining the robust optimal state of the first statistical model may include determining an adapted second statistical model for the adapted first statistical model, and may include determining a global maximum of the adapted second statistical model, wherein an input parameter value of the global maximum of the second statistical model corresponds to an input parameter value of the robust optimal state of the first statistical model. The features described in this paragraph are combined with the second example to form a third example.

[0011] The method may also include determining a process window for the physical or chemical process using a robust optimal state and variations in input parameter values. This has the advantage of being able to determine a process region within which the process parameters of the physical or chemical process lead to desired characteristics or desired success of the physical or chemical process. The method may also include controlling the physical or chemical process, wherein input parameter values ​​for the physical or chemical process are selected based on the input parameter values ​​of the robust optimal state, in particular, the input parameter values ​​for the physical or chemical process correspond to the input parameter values ​​of the robust optimal state. The features described in this paragraph can be combined with the second or third example to form a fourth example.

[0012] The input parameter assigned to the input parameter value may be a first physical variable. The output parameter assigned to the output parameter value may be a second physical variable. The features described in this paragraph may be combined with one of the first to fourth examples to form a fifth example.

[0013] The input parameter value of one of the known measurement points can be detected by a first sensor. The output parameter value of one of the known measurement points can be detected by a second sensor. The first sensor can detect the input parameter value of the new measurement point, while the second sensor can detect the output parameter value of the new measurement point. The features described in this paragraph can be combined with one of the second to fifth examples to form a sixth example.

[0014] The first statistical model may be determined by means of Gaussian process regression using known measurement points.The features described in this paragraph are combined with one of the first to sixth examples to form a seventh example.

[0015] The first statistical model may be formed by a Bayesian neural network. The features described in this paragraph are combined with one of the first to seventh examples to form an eighth example.

[0016] The second statistical model may be determined for the first statistical model using the distribution of changes in input parameter values. The features described in this paragraph are combined with one of the first to eighth examples to form a ninth example.

[0017] The distribution of the input parameter value variations may be determined using the variance of the input parameter values ​​for corresponding measurement points of the known measurement points.The features described in this paragraph are combined with the ninth example to form a tenth example.

[0018] The first predictive distribution may be used to determine the entropy of a first statistical model described by the known measurement points. The features described in this paragraph may be combined with one of the first to tenth examples to form an eleventh example.

[0019] The first predictive distribution may be a first conditional probability distribution of the first statistical model, conditioned on the known measurement points. The features described in this paragraph are combined with the eleventh example to form a twelfth example.

[0020] The first prediction distribution may be a Gaussian distribution.The features described in this paragraph are combined with the eleventh example or the twelfth example to form the thirteenth example.

[0021] The first prediction distribution may be determined by means of a Gaussian process.The features described in this paragraph in combination with one of the eleventh to thirteenth examples form a fourteenth example.

[0022] The expected entropy can be determined using a second predictive distribution. The second predictive distribution can be a second conditional probability distribution of the first statistical model, conditioned on the known measurement points and the maximum value of the output parameter of the second statistical model under the robust optimal state. The features described in this paragraph are combined with one of the first to fourteenth examples to form a fifteenth example.

[0023] The expected entropy can be determined by Monte Carlo approximation.The features described in this paragraph are combined with the fifteenth example to form a sixteenth example.

[0024] The expected entropy can be determined using the entropy of the first statistical model described by the known measurement points and a plurality of approximate maxima of the output parameters of the second statistical model under the robust optimal state of the physical or chemical process. The features described in this paragraph are combined with the sixteenth example to form the seventeenth example.

[0025] A third predictive distribution can be used to determine multiple approximate maxima of the entropy of the first statistical model described by the known measurement points and the output parameter of the second statistical model at a robust optimal state of the physical or chemical process. The third predictive distribution can be a third conditional probability distribution of the first statistical model, conditioned on the known measurement points and the multiple approximate maxima of the second statistical model. The features described in this paragraph are combined with the seventeenth example to form the eighteenth example.

[0026] Determining one of the plurality of approximate maxima may include determining a third statistical model using a fourth predictive distribution, wherein the fourth predictive distribution may be a fourth conditional probability distribution of the second statistical model, conditioned on the known measurement points, and wherein the approximate maximum may be a maximum value of an output parameter of the third statistical model. The features described in this paragraph are combined with the eighteenth example to form a nineteenth example.

[0027] The fourth predictive distribution can be determined using a fifth predictive distribution using a sparse spectral Gaussian process approximation (SSGP approximation). The fifth predictive distribution can be a conditional probability distribution of the first statistical model without a noise component, conditioned on the known measurement points. The features described in this paragraph are combined with the nineteenth example to form the twentieth example.

[0028] The third statistical model can be determined using a fourth statistical model, wherein the fourth statistical model can be determined using the fifth predictive distribution, wherein the fourth statistical model can have a first eigenvector, and wherein the fifth predictive distribution can be represented as a superposition of the first eigenvector. The individual components of the first eigenvector can be cosine functions with "random" frequencies and "random" phase shifts. The frequencies, phase shifts, and weights used for superposition can be selected so that the fifth distribution is approximated. The features described in this paragraph are combined with the nineteenth example or the twentieth example to form the twenty-first example.

[0029] The third statistical model can be determined using a second eigenvector. The second eigenvector can differ from the first eigenvector in that components of the second eigenvector can be multiplied by an amplitude, wherein the magnitude of the amplitude can depend on the uncertainty of the input parameter value. The features described in this paragraph are combined with the twenty-first example to form the twenty-second example.

[0030] The entropy of the first statistical model described by the known measurement points and the multiple approximate maxima of the output parameters of the second statistical model in the robust optimal state of the physical or chemical process can be determined by means of a rejection method (rejection sampling) or by means of expectation propagation. Determining the entropy by means of a rejection method or by means of expectation propagation has the following advantages: an expected entropy that is analytically unsolvable can be approximated. Expectation propagation also has the advantage of requiring less computational effort. The features described in this paragraph are combined with one of the seventeenth to twenty-second examples to form the twenty-third example.

[0031] The computer program product may store program instructions which, when executed, perform a method according to one or more of the first to twenty-third examples.The features described in this paragraph form a twenty-fifth example.

[0032] The system may include a device according to the twenty-fourth example. The system may also include at least one sensor, which may be designed to provide the device with an output parameter value assigned to an input parameter value. A system having the features described in this paragraph constitutes the twenty-sixth example.

[0033] A production system may include an apparatus according to the twenty-fourth example. The production system may also include a production apparatus designed to manufacture a product. The production system may include at least one sensor designed to provide an output parameter value assigned to an input parameter value to the apparatus. The production system may include a control device designed to control the production apparatus in such a manner that the input parameter value is selected based on the input parameter value of the robust optimal state, in particular, the input parameter value corresponds to the input parameter value of the robust optimal state. A production system having the features described in this paragraph forms the twenty-seventh example.

[0034] A processing system may include an apparatus according to the twenty-fourth example. The processing system may also include a processing apparatus that may be designed to process a workpiece. The processing system may include at least one sensor that may be designed to provide an output parameter value assigned to an input parameter value to the apparatus. The processing system may include a control device that may be designed to control the processing apparatus so that an input parameter value is selected based on the input parameter value of a robust optimal state, in particular, the input parameter value corresponds to the input parameter value of the robust optimal state. The processing system may be a chemical processing system or a mechanical processing system. A processing system having the features described in this paragraph forms the twenty-eighth example.

[0035] A robotic system may include a device according to the twenty-fourth example. The robotic system may also include a robotic device that can be designed to perform a movement. The robotic system may include at least one sensor that can be designed to provide an output parameter value assigned to an input parameter value to the device. The robotic system may include a control device that can be designed to control the robotic device so that the input parameter value is selected based on the input parameter value of the robust optimal state, in particular the input parameter value corresponds to the input parameter value of the robust optimal state. A robotic system having the features described in this paragraph forms the twenty-ninth example. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] BRIEF DESCRIPTION OF THE DRAWINGS Exemplary embodiments of the invention are shown in the drawings and are explained in more detail in the following description.

[0037] in:

[0038] Figure 1A shows apparatus according to various embodiments;

[0039] Figure 1B shows apparatus according to various embodiments;

[0040] Figure 2 A processing system for selecting a new measurement point according to various embodiments is shown;

[0041] Figure 3 A processing system for selecting a new measurement point according to various embodiments is shown;

[0042] Figure 4 An exemplary distribution of a first statistical model is shown;

[0043] Figure 5 An exemplary distribution of a second statistical model is shown;

[0044] Figure 6 A processing system for adapting a statistical model according to various embodiments is shown;

[0045] Figure 7 A processing system for determining a robust optimal state of a physical or chemical process according to various embodiments is shown;

[0046] Figure 8A Methods for selecting new measurement points for physical or chemical processes according to various embodiments are shown;

[0047] Figure 8B Methods for determining a robust optimal state of a physical or chemical process according to various embodiments are shown;

[0048] Figure 9A A system for determining a robust optimal state of a physical or chemical process according to various embodiments is shown;

[0049] Figure 9B A system using a statistical model adapted according to various embodiments is shown. DETAILED DESCRIPTION

[0050] In one embodiment, a "circuit" may be understood as any type of logic implementation entity, which may be hardware, software, firmware, or a combination thereof. Thus, in one embodiment, a "circuit" may be a hard-wired logic circuit or a programmable logic circuit such as a programmable processor, such as a microprocessor (e.g., a CISC (large instruction set processor) or a RISC (reduced instruction set processor)). A "circuit" may also be software implemented or executed by a processor, such as any type of computer program, such as a computer program using a virtual machine code such as Java. Any other type of implementation of the corresponding functionality, which will be described in more detail below, may be understood as a "circuit" consistent with alternative embodiments.

[0051] Various embodiments relate to apparatus and methods for determining a robust optimal state of a physical or chemical process based on a Bayesian optimization approach. Due to disturbances or noise, process parameters in the physical or chemical process may deviate from set values, resulting in, for example, a desired characteristic of a workpiece being degraded or failing to meet a predetermined characteristic. A statistical model is intuitively provided that is capable of determining a robust optimal state for the physical or chemical process. The robust optimal state is robust to variations in the process parameters, i.e., the desired or predetermined characteristic persists despite disturbances or noise in the process parameters.

[0052] Figure 1A Device 100A according to various embodiments is shown. Device 100A may include a storage device 102. Storage device 102 may include at least one memory. The memory may be used, for example, in a process executed by a processor. The memory used in these embodiments may be a volatile memory such as DRAM (dynamic random access memory), or a non-volatile memory such as PROM (programmable read-only memory), EPROM (erasable programm ... Each of the plurality of output parameter values ​​108 can be precisely assigned to one of the plurality of input parameter values ​​104, wherein the input parameter value and the measured output parameter value assigned to the input parameter value can form a measurement point. The input parameter assigned to the input parameter value can be a physical variable. The input parameter can be a process-related parameter, such as process temperature, process time, vacuum or gas atmosphere, etc. The input parameter can be a process-related property of the material, such as hardness, thermal conductivity, electrical conductivity, density, microstructure, macrostructure, chemical composition, etc. The output parameter assigned to the output parameter value can be a physical variable. The output parameter can have an application-specific quality standard. The output parameter can be a component-related parameter, such as dimension or layer thickness, or a material-related parameter, such as hardness, thermal conductivity, electrical conductivity, density, chemical composition, etc. Expected values ​​for the output parameters can be predefined. For example, a desired hardness or desired electrical conductivity can be predefined.

[0053] Device 100A may also include at least one processor 110. As described above, processor 110 may be any type of circuit, i.e., any type of logically implemented entity. Processor 110 may be a graphics processing unit (GPU), and the GPU may utilize allocated graphics memory (video RAM) for data processing. In various embodiments, processor 110 is designed to process multiple input parameter values ​​104 and multiple output parameter values ​​108.

[0054] Figure 1B 1. A device 100B according to various embodiments is shown. The device 100B may substantially correspond to the device 100A, wherein the device 100B has a first sensor 112 and a second sensor 114, wherein the first sensor 112 may be configured to provide a plurality of input parameter values ​​104 to the storage device 102, and wherein the second sensor 114 may be configured to provide a plurality of output parameter values ​​108 to the storage device.

[0055] According to various embodiments, the device 100A and / or the device 100B may also have additional sensors, wherein each of the additional sensors may be designed to provide parameter values ​​to the storage device 102 , which are assigned to corresponding input parameter values ​​of the plurality of input parameter values ​​104 .

[0056] Figure 2 A processing system 200 for selecting a new measurement point is shown, according to various embodiments. The processing system 200 may include a storage device 102 for storing a plurality of input parameter values ​​104 and a plurality of output parameter values ​​108. The processing system 200 may also include at least one processor 110. The processor 110 may be configured to process the plurality of input parameter values ​​104 and the plurality of output parameter values ​​108. The processor 110 may be configured to determine known measurement points 202, wherein each known measurement point in the known measurement points 202 has an input parameter value from the plurality of input parameter values ​​104 and an output parameter value from the plurality of output parameter values ​​108 assigned to the input parameter value.

[0057] Known measurement point 202 (D n ) can be described by equation (1):

[0058] D n ={(x i ,y i )} i＝1：n (1)

[0059] where x i is one of the plurality of input parameter values ​​104, and y iis the output parameter value assigned to the input parameter value, and wherein n is the number of known measurement points 202. It should be noted that known measurement points 202 may also have additional parameter values ​​assigned to corresponding input parameter values ​​in plurality of input parameter values ​​104.

[0060] The processor 110 can be designed to determine a first statistical model 204. The first statistical model 204 can describe a physical or chemical process for a known measurement point 202. The first statistical model 204 can be a first function that describes a physical or chemical process for a known measurement point 202. In other words, the first statistical model 204 can output an assigned output parameter value for an input parameter value. In other words, the first statistical model 204 describes the output parameter as a function of the input parameter. That is, the first statistical model 204 describes the relationship between the input parameter value and the output parameter value of the physical or chemical process. The first statistical model 204 can be described by a function f(x). The measured output parameter value may have a noise component. That is, the measured output parameter value may deviate from the function of the first statistical model 204. The processor 110 can be designed to determine a first statistical model that also takes into account the noise component ∈ and can be described by y(x) according to equation (2):

[0061] y(x)=f(x)+∈ (2)

[0062] The noise component ∈ can be described by normal distribution as Hereinafter, the first statistical model 204 is described by y(x), and the first statistical model 204 without the noise component is described by f(x).

[0063] A first statistical model 204 without a noise component can be determined (e.g., approximated) by means of Gaussian process regression (GP regression) using known measurement points 202. The first statistical model 204 without a noise component can be formed by a Bayesian neural network. The first statistical model 204 without a noise component can be determined by another model that outputs a confidence interval. In the following, reference is made to determining the first statistical model 204 without a noise component by GP regression. The first statistical model 204 without a noise component can describe a plurality of data points x within a real number range, i.e. have Here, d is the dimension. The dimension is determined by the number of parameter values ​​assigned to the corresponding input parameter value. That is, when multiple output parameter values ​​108 are assigned to multiple input parameter values ​​104, the dimension d=2.

[0064] GP regression defines a kernel function k with a mean equal to 0 and can be obtained by fThe prior distribution of the covariance given by (x, x′).

[0065] Conditioned by the prior distribution and the known measurement points 202, the predicted mean value m can be determined with the aid of equation (3): f , and with the help of equation (4) determine the variance v f :

[0066] m f (x|D n )=k f (x) T K -1 y (3)

[0067] v f (x|D n )=k f (x, x)-k f (x) T K -1 k f (x) (4)

[0068] where [k f (x)] i =k f (x,x i ); [y] i =y i , and where δ ij is the Kronecker symbol.

[0069] The storage device 102 may also store a change in the input parameter value 206. The change in the input parameter value 206 may be predefined. The change in the input parameter value 206 may specify a desired or required maximum deviation of the input parameter value. The change in the input parameter value 206 may define a required or desired process window size. For example, the input parameter may specify a process temperature, and the change in the input parameter value 206 relative to the set process temperature may be, for example, 5K or 10%. According to various embodiments, the change in the input parameter value 206 is given by a distribution (e.g., a normal distribution). According to various embodiments, the processor 110 is configured to determine the change in the input parameter value 206 using the plurality of input parameter values ​​104 and the plurality of output parameter values ​​108.

[0070] The processor 110 may also be configured to determine a second statistical model 208. The second statistical model 208 may be determined using the first statistical model 204 and the variation in the input parameter values ​​206.

[0071] The second statistical model 208 (g(x)) can be determined with the aid of equation (5):

[0072]

[0073] where ξ is the variation of the input parameter value 206. This has the advantage that the second statistical model 208 describes the robustness of the output parameter value of the physical or chemical process with respect to variations of the input parameter value. The second statistical model 208 can be determined by mathematical convolution with the aid of equation (6):

[0074]

[0075] where p(ξ) is a The normal distribution of

[0076] in That is, the distribution of the variation of input parameter values ​​206 may be determined using the variance of the input parameter values ​​of corresponding measurement points of known measurement points 202 .

[0077] As described above, if the first statistical model 204 is determined by means of GP regression, the predicted mean value m of the second statistical model 208 can be determined by means of equation (7): g , and the variance v is determined by means of equation (8) g :

[0078] m g (x|D n )=k gf (x) T K -1 y (7)

[0079] v g (x|D n )=k gf (x, x)-k gf (x) T K -1 k fg (x) (8)

[0080] where k g (x, x′) = ∫∫k f (x+ξ,x′+ξ′)p(ξ)p(ξ′)dξdξ′, and

[0081] where k gf (x, x′) = ∫k f (x+ξ,x′)p(ξ)dξ.

[0082] The processor 110 may also be configured to select a new measurement point 210 of an input parameter. The new measurement point 210 may be an input parameter value for which a robust optimal state of the first statistical model 204 may be determined by determining a corresponding output parameter value.

[0083] The output parameter values ​​assigned to input parameter values ​​within the robust optimal state range are not affected at all or only slightly by deviations from the input parameter values ​​of the robust optimal state (e.g., variations in the input parameter value 206). That is, the output parameter values ​​are robust to variations in the input parameter values ​​within the robust optimal state range. In other words, when the peak value of the robust optimal state 404 and the global maximum value 402 of the first statistical model 204 have the same distribution (e.g., Gaussian distribution or uniform distribution), the robust optimal state 404 can have a larger half-value width than the global maximum value 402. The robust optimal state 404 can be determined based on variations in the input parameter value 206. That is, the first statistical model 204 can have multiple robust optimal states, each of which is a robust optimal state for variations in the input parameter value 206 or a range of variations in the first parameter value.

[0084] The new measurement point 210 may be determined using the first statistical model 204 and the second statistical model 208 and conditioned on the maximum value taken by the output parameter of the second statistical model 208. The maximum value taken by the output parameter of the second statistical model 208 may be the output parameter value assigned to the global maximum value of the second statistical model 208. That is, the maximum value (g*) may be determined by g * =maxx∈xg() The new measuring point 210 can be selected such that the difference between the first entropy 304 and the expected entropy 306 is substantially maximum at the new measuring point 210 or is within a predetermined range around the maximum value.

[0085] Figure 3 A processing system 300 for selecting a new measurement point is shown, according to various embodiments. The processing system 300 may substantially correspond to the processing system 200, wherein the new measurement point 210 is determined by an acquisition function 302. The new measurement point 210 may be an input parameter value assigned to a global maximum of the acquisition function 302. The acquisition function 302 may be the mutual information (also referred to as mutual information) of the maximum values ​​of the first statistical model 204 and the second statistical model 208. The acquisition function 302 may be the conditional mutual information of the maximum values ​​of the first statistical model 204 and the second statistical model 208, conditioned on the known measurement point 202.

[0086] Acquisition function 302 (α NES (x)) can be described by equation (9):

[0087] α NES (x) = I((x, y); g * |D n ) (9)

[0088] Conditional mutual information of the maximum value of the first statistical model 204 and the second statistical model 208, conditioned on the known measurement point 202, can be determined using a first entropy 304 (e.g., a first differential entropy) and an expected entropy 306 (e.g., an expected differential entropy). The first entropy 304 can be the entropy of the first statistical model 204 for the physical or chemical process described by the known measurement point 202 at the input parameter value. The expected entropy 306 can be the expected entropy of the first statistical model 204 described by the known measurement point 202 at the maximum value taken by the input parameter value and the output parameter of the second statistical model 208 when the physical or chemical process is in a robust optimal state.

[0089] The first entropy 304 can be determined using a first prediction distribution 308. The first prediction distribution 308 can be a first conditional probability distribution of the first statistical model 204, conditioned on the known measurement points 202. The expected entropy 306 can be determined using a second prediction distribution 310. The second prediction distribution 310 can be a second conditional probability distribution of the first statistical model 204, conditioned on the known measurement points 202 and the maximum value taken by the output parameter of the second statistical model 208 in the robust optimal state.

[0090] Acquisition function 302 (α NES (x)) may be the difference between the first entropy 304 and the desired entropy 306 and may be described by equation (10):

[0091]

[0092] where H[p(y(x)|D n )] is a first prediction distribution 308p(y(x)|D n ) first entropy 304, and wherein is a second prediction distribution 310p(y(x)|D n , g * )’s expected entropy 306.

[0093] The first prediction distribution 308 may be a Gaussian distribution. The first prediction distribution 308 may be determined by a Gaussian process (GP). The first entropy 304 may be determined, for example, calculated, using equation (11):

[0094]

[0095] The expected entropy 306 can be determined using a Monte Carlo approximation, for example. The expected entropy 306 can be determined using a second entropy, such as a second differential entropy. The second entropy can be the entropy of multiple approximate maxima of the output parameters of the first statistical model 204 and the second statistical model 208 described by the known measurement points 202 at the robust optimal state of the physical or chemical process. The second entropy can be determined using a third predictive distribution. The third predictive distribution can be a third conditional probability distribution of the first statistical model 204, conditioned on the multiple approximate maxima of the output parameters of the known measurement points 202 and the second statistical model 208.

[0096] The expected entropy 306 can be determined, for example, approximated, with the aid of equation (12):

[0097]

[0098] in is a third predictive distribution The second entropy of G* is p(g * |D n ), and where is an approximate maximum value among the multiple approximate maxima of the second statistical model 208 .

[0099] A third statistical model can be used To determine one of the multiple approximate maximum values ​​of the output parameter One of the plurality of approximate maxima may be the maximum value of the third statistical model, that is, The third statistical model can be determined using a fourth predictive distribution. The fourth predictive distribution can be a fourth conditional probability distribution of the second statistical model 208, where the fourth predictive distribution is conditional on the known measurement point 202, that is, the fourth predictive distribution can be determined by p(g(x)|D n ) description. The fourth predictive distribution can be determined by means of a sparse spectral Gaussian process approximation (SSGP approximation). The third statistical model can be determined using a fourth statistical model. The fourth statistical model can be determined using a fifth predictive distribution. The fifth predictive distribution can be a conditional probability distribution of the first statistical model 204 without a noise component, wherein the condition is the known measurement point 202, that is, the fifth predictive distribution can be obtained by p(f(x)|D n ) description. The fourth statistical model may have a first eigenvector, wherein the fifth predictive distribution may be represented as a superposition of the first eigenvector. The individual components of the first eigenvector may be cosine functions with "random" frequencies and "random" phase shifts. The frequencies, phase shifts, and weights of the superposition may be selected such that the fifth distribution is approximated.

[0100] The fourth predictive distribution may be determined by means of SSGP approximation using the first predictive distribution.

[0101] The fourth statistical model can be determined with the help of equation (13)

[0102]

[0103] in Is a The first eigenvector of , and where a is a weighting factor and is calculated according to distribution, where and

[0104] Can be achieved through To determine the first eigenvector, where b i ~u(0,2π) and w i ~p(w)∝s(w).

[0105] The fourth statistical model can be used to determine the third statistical model, wherein the third statistical model can be determined with the aid of equation (14)

[0106]

[0107] in is the second eigenvector. The second eigenvector can be obtained by The second eigenvector may differ from the first eigenvector in that components of the second eigenvector may be multiplied by a magnitude, wherein the magnitude of the magnitude may depend on the uncertainty of the input parameter value.

[0108] The second entropy can be determined, for example, approximated, by means of a rejection method (rejection sampling) or by means of an expected propagation. In other words, a plurality of approximate maxima assigned to the second entropy of the output parameter of the second statistical model 208 can be determined, for example, approximated by means of a rejection method (rejection sampling) or by means of an expected propagation.

[0109] In the rejection method, the first prediction distribution 308 (p(y(x)|D n )) to determine the fourth statistical model And the fourth statistical model can be used to determine the third statistical model. In the rejection method, one of the multiple approximate maxima can be selected in the following cases The maximum value of the third statistical model is smaller than the approximate maximum value, or one of the plurality of approximate maxima may be rejected if The maximum value of the third statistical model is greater than the approximate maximum value.

[0110] For a chosen approximate maximum, the second entropy can be determined (eg approximated) with the aid of equation (15):

[0111]

[0112] in is a set of approximate maxima with L choices, and where is the kernel density estimate.

[0113] In the expected propagation, the second entropy can be determined using a sixth prediction distribution. The sixth prediction distribution can be a conditional probability distribution of the first statistical model 204 without a noise component, wherein the condition is the known measurement point 202 and a plurality of approximate maxima, that is, the sixth prediction distribution can be obtained by Description. The sixth predictive distribution can be determined with the help of equation (16):

[0114]

[0115] The prediction distribution p(f(x)|D n , g(x)), where And among them is the seventh predictive distribution and can be determined with the aid of a truncated normal distribution or with the aid of a Gaussian approximation.

[0116] The following describes how to determine the seventh prediction distribution using Gaussian approximation: The predicted distribution can be determined (e.g. approximated) for example with the aid of equation (17)

[0117]

[0118] in is the indicator function.

[0119] Determine the prediction distribution p(g(x)|g, D) with the help of GP regression n ) and substituting into equation (17) leads to equation (18):

[0120]

[0121] And constrain Inserting equation (18) leads to equation (19):

[0122]

[0123] in and in and And among them is the probability density function, and Φ(·) is the cumulative density function of the standard normal distribution.

[0124] Based on this, the sixth prediction distribution can be determined (eg approximated) with the aid of equation (20)

[0125] The acquisition function 302 (α) determined by means of the expected propagation (EP) can be described by equation (21) NES-EP (x)), where the variance Let's consider the noise component ∈:

[0126]

[0127] Figure 4 An exemplary distribution of the first statistical model 204 is shown. The exemplary first statistical model 204 shown may describe a physical or chemical process for a known measurement point 202 and may have a global maximum 402 and a local maximum 404, such as a robust optimum. For the first statistical model 204, an exemplary (predictive) distribution may be determined. For example, for the first statistical model 204, a fifth predictive distribution 406 may be determined, namely, p(f(x)|D n ), wherein the first statistical model 204 is based on the known measurement points 202 (D n ) is a condition. For example, a sixth prediction distribution 408 can also be determined, namely The first statistical model 204 is based on the known measurement points 202 (D n ) and multiple approximate maxima As a condition.

[0128] Figure 5An exemplary distribution of the second statistical model 208 is shown. The exemplary second statistical model 208 shown is determined using the exemplary first statistical model 204 and a change in the input parameter value 206. The second statistical model 208 can have a global maximum 502. The input parameter value assigned to the global maximum 502 of the second statistical model 208 can be the input parameter value of the new measurement point 210. The input parameter value assigned to the global maximum 502 of the second statistical model 208 can correspond to the input parameter value assigned to the local maximum 404 of the first statistical model 204, where the local maximum can be a robust optimal state. An exemplary (predictive) distribution can be determined for the second statistical model 208. For example, a fourth predictive distribution 504 can be determined for the second statistical model 208, namely p(g(x)|D n ), wherein the second statistical model 208 is based on the known measurement point 202 (D n ) is a condition. For example, the seventh prediction distribution 506 can also be determined, that is, The second statistical model 208 is based on the known measurement points 202 (D n ) and multiple approximate maximum values ​​of output parameters As a condition.

[0129] Figure 6 A processing system 600 for adapting a statistical model according to various embodiments is shown. The processing system 600 may essentially correspond to the processing system 300, wherein the processing system 600 is further designed to measure associated output parameter values ​​for input parameter values ​​of the new measurement point 210, i.e., to measure the new measurement point 210.

[0130] First sensor 112 may be designed to use the input parameter value of new measurement point 210 to provide a new input parameter value 602, where new input parameter value 602 may substantially correspond to the input parameter value of new measurement point 210. In other words, in this case, the input parameter value of new measurement point 210 indicates the input parameter value for which new input parameter value 602 should be detected. In other words, due to noise, measurement deviations, input parameter uncertainty, etc., new input parameter value 602 may deviate from the input parameter value of new measurement point 210. According to various embodiments, processing system 600 further includes a control device, which may include processor 110 and may be designed to control the process such that the input parameter value provided by first sensor 112 substantially corresponds to the input parameter value of new measurement point 210. According to various embodiments, processor 110 provides the input parameter value of new measurement point 210 to storage device 102, for example, as new input parameter value 602. In other words, the input parameter value of new measurement point 210 may correspond to the new input parameter value.

[0131] The second sensor 114 may be configured to provide a new output parameter value 604 assigned to the new input parameter value 602. The storage device 102 may be configured to store the new input parameter value 602 and the new output parameter value 604. The processing system 600 may further include at least one processor 110. The processor 110 may be configured to determine a new measurement point 606 of measurement, wherein the new measurement point 606 of measurement has the new input parameter value 602 and the new output parameter value 604 assigned to the new input parameter value 602. The processor 110 may be configured to update the known measurement point 202 with the new measurement point 606 of measurement. In other words, the processor 110 may be configured to determine an updated known measurement point 608, wherein the updated known measurement point 608 has the known measurement point 202 and the new measurement point 606 of measurement. In other words, the D n+1 ={(x i ,y i )} i＝1：n+ 1 to describe the updated known measurement point 608 (D n+1 ).

[0132] The processor 110 may also be designed to adapt the first statistical model 204 using the measured new measurement points 606. The adaptation of the first statistical model 204 may have the following characteristics: n+1 ) to determine the first statistical model 204. Determining the first statistical model 204 for the updated known measurement point 608 may substantially correspond to determining the first statistical model 204 for the known measurement point 202, wherein the known measurement point 202 is replaced by the updated known measurement point 608. In other words, the adapted first statistical model 204 may describe the physical or chemical processes for the known measurement point 202 and the measured new measurement point 210.

[0133] The processor 110 can be configured to determine the second statistical model 208 using the adapted first statistical model 204 and the change in the input parameter value 206, wherein determining the adapted second statistical model 208 can substantially correspond to determining the second statistical model 208 according to the processing system 300. In other words, the adapted second statistical model 208 can be determined by determining the second statistical model 208 for the adapted first statistical model 204. In other words, the first statistical model 204, the second statistical model 208, and the corresponding predictive distribution are conditioned on the updated known measurement point 608.

[0134] Figure 7A processing system 700 for determining a robust optimal state of a physical or chemical process is shown according to various embodiments. The processing system 700 may substantially correspond to the processing system 600, wherein the processor 110 is further configured to determine the robust optimal state 404 of the first statistical model 204. The processor 110 may be configured to determine the global maximum 502 of the adapted second statistical model 208. The processor 110 may also be configured to provide the robust optimal state 404 of the first statistical model 204, wherein the input parameter values ​​of the global maximum 502 of the adapted second statistical model 208 correspond to the input parameter values ​​of the robust optimal state 404.

[0135] The processor 110 may also be designed to determine a process window 702 for a physical or chemical process. The robust optimal state 404 and the variation of the input parameter value 206 may be used to determine the process window 702. The variation of the input parameter value 206 may be described by a distribution such as a normal distribution. For the case where the variation of the input parameter value 206 is given by an absolute value, the process window 702 (x processfenster ) can be represented by x neu -ξ≤x Prozessfenster ≤x neu +ξ is given, where x neu is the input parameter value of the robust optimal state, and ξ is the variation of the input parameter value 206. That is, the process window 702 may have all input parameter values ​​that are within the variation of the input parameter value 206 relative to the input parameter value of the robust optimal state, i.e., all x processfenster .

[0136] Figure 8A A method 800A for selecting a new measurement point for a physical or chemical process is shown, according to various embodiments. Method 800A may include providing (e.g., measuring) known measurement points 202 (at 802). Each of known measurement points 202 may have an input parameter value and an output parameter value assigned to the input parameter value. Method 800A may include determining a first statistical model 204 (at 804). First statistical model 204 may describe the physical or chemical process for known measurement points 202. Method 800A may include determining a second statistical model 208 (at 806). Second statistical model 208 may be determined using changes in first statistical model 204 and input parameter values ​​206. Method 800A may also include selecting a new measurement point 210 (at 808). New measurement point 210 may be determined using first statistical model 204 and known measurement points 202, conditional on a maximum value of an output parameter of second statistical model 208. The new measuring point 210 may have input parameter values ​​for which a robust optimal state may be determined by determining the assigned output parameter values.

[0137] Figure 8B A method 800B for determining a robust optimal state of a physical or chemical process according to various embodiments is shown. Method 800B may substantially correspond to method 800A, wherein method 800B may further include measuring output parameter values ​​assigned to input parameter values ​​of a new measurement point 210 (810). The input parameter values ​​of the new measurement point 210 and the assigned measured output parameter values ​​may form a measured new measurement point 606. Method 800B may include adapting the first statistical model 204 (at 812). The adapted first statistical model 204 may be determined using the measured new measurement point 606. The adapted first statistical model 204 may describe the physical or chemical process for the known measurement point 202 and the measured new measurement point 606. Method 800B may include adapting the second statistical model 208 (at 814). The adapted second statistical model 208 may be determined for the adapted first statistical model 204. Method 800B may include determining the robust optimal state 404 of the adapted first statistical model 204 (at 816). Determining the robust optimal state 404 may include determining a global maximum 502 of the adapted second statistical model 208, wherein the input parameter value of the global maximum 502 of the adapted second statistical model 208 corresponds to the input parameter value of the robust optimal state 404 of the first statistical model. Using the robust optimal state 404 and the change in the input parameter value 206, a process window 702 of the physical or chemical process may be determined. Method 800B may also include controlling the physical or chemical process, wherein input parameter values ​​of the physical or chemical process may be selected based on the input parameter values ​​of the robust optimal state, and wherein the input parameter values ​​may correspond to the input parameter values ​​of the robust optimal state.

[0138] Figure 9AA system 900A for determining a robust optimal state of a physical or chemical process, according to various embodiments, is shown. System 900A may include a first device 902 on which the physical and / or chemical process is performed. First device 902 may be, for example, a production device for manufacturing a product or a processing device for processing a workpiece (e.g., a drilling device or a milling device), a robotic device for performing a movement, a device for designing active ingredients for a pharmaceutical, a device for system design (e.g., for aerospace applications), etc. System 900A may also include a sensor 904. Sensor 904 may be designed to detect a plurality of output parameter values ​​108 assigned to a plurality of provided input parameter values. Sensor 904 may be designed to detect the plurality of output parameter values ​​108 during (on-site) the physical or chemical process performed by first device 902. According to various embodiments, sensor 904 may be designed to detect the plurality of output parameter values ​​108 after (off-site) the physical or chemical process performed by first device 902. In other words, multiple input parameter values ​​104 may be set during the physical or chemical process, or may be detected by additional sensors, and multiple output parameter values ​​108 may be detected after the physical or chemical process, such that the detection of multiple output parameter values ​​108 is temporally different. For example, first device 902 may be a furnace for hardening a workpiece, one of multiple input parameter values ​​104 may be a temperature set or measured in the furnace, and an output parameter value assigned to the input parameter value in multiple output parameter values ​​108 may be the hardness of the workpiece detected after hardening the workpiece.

[0139] System 900A may also include a second device 906. Second device 906 may be configured to perform method 800A and / or method 800B, wherein sensor 904 may be configured to provide second device 906 with a plurality of output parameter values ​​108 assigned to a plurality of input parameter values ​​104. Second device 906 may be configured to select a new measurement point according to method 800A. System 900A may also include a control device 908. Control device 908 may be configured to receive the new measurement point selected by second device 908 and control first device 902 so that the set input parameter values ​​correspond to the input parameter values ​​of the selected new measurement point 210. Sensor 904 may be configured to detect the output parameter values ​​assigned to the set input parameter values ​​at the new measurement point 210 and provide them to second device 906. Second device 906 may be configured to adapt first statistical model 204 and second statistical model 208 according to method 800B. The second device 906 may also be configured to determine the robust optimal state 404 of the first statistical model 204 according to the method 800B.

[0140] Figure 9BA system 900B using a statistical model adapted according to various embodiments is shown. System 900B may include a second device 908, which may include a first statistical model 204 adapted according to method 800B and an adapted second statistical model 208. Second device 906 may be configured to determine a robust optimal state 404 for first statistical model 204 using first statistical model 204 and second statistical model 208. System 900B may also include a control device 908. Control device 908 may be configured to receive the determined robust optimal state 404 from second device 906. System 900B may also include a first device 902 on which a physical and / or chemical process is performed. Control device 908 may be configured to control the first device such that input parameter values ​​set on the first device are selected based on input parameter values ​​of robust optimal state 404, where the set input parameter values ​​may correspond to the input parameter values ​​of robust optimal state 404.

Claims

1. A method for determining a robust optimal state of a physical or chemical process according to a Bayesian optimization method, wherein each of the known measurement points has an input parameter value of the physical or chemical process and an output parameter value of the measurement assigned to the input parameter value, and wherein a first statistical model describes the relationship between the input parameter value and the output parameter value of the physical or chemical process, the method comprising: • determining a second statistical model for the first statistical model, wherein the second statistical model describes the robustness of the output parameter value of the physical or chemical process with respect to variations in the input parameter value; • The new measuring point is selected such that the difference between the following two items is essentially maximum or lies within a predetermined range around the maximum value: o the entropy of the first statistical model of the physical or chemical process described by the known measurement points at the new measurement point, and o the expected value of the entropy of the first statistical model described by the known measurement points at the new measurement point and conditional on the maximum value assumed by the output parameter of the second statistical model in the robust optimal state to be determined of the physical or chemical process; • measuring the output parameter value assigned to the input parameter value of said new measurement point; • using the new measurement points to adapt the first statistical model; • determining a robust optimal state of the adapted first statistical model; as well as • wherein the adapted first statistical model describes the physical or chemical process for said known measurement points and said new measurement points; • wherein determining the robust optimal state of the first statistical model has: o determining an adapted second statistical model for the adapted first statistical model; and o determining a global maximum of the adapted second statistical model, wherein the input parameter values ​​of the global maximum of the second statistical model correspond to the input parameter values ​​of the robust optimal state of the first statistical model; and • controlling the physical or chemical process, wherein the values ​​of the input parameters of the physical or chemical process are selected according to the values ​​of the input parameters of the robust optimal state, in particular so that they correspond to the values ​​of the input parameters of the robust optimal state.

2. The method according to claim 1, further comprising: A process window for the physical or chemical process is determined using the robust optimal state and the changes in the input parameter values.

3. Apparatus designed to carry out the method according to any one of claims 1 or 2.

4. Production system, with: The device according to claim 3; Production equipment designed to manufacture products; at least one sensor designed to provide an output parameter value assigned to an input parameter value to the device; and A control device is designed to control the production device so that the input parameter values ​​are selected according to the input parameter values ​​of the robust optimal state, in particular so that they correspond to the input parameter values ​​of the robust optimal state.

5. Processing system, with: The device according to claim 3; Processing equipment designed to process workpieces; at least one sensor designed to provide an output parameter value assigned to an input parameter value to the device; and A control device is designed to control the processing device such that the input parameter values ​​are selected according to the input parameter values ​​of the robust optimal state, in particular such that they correspond to the input parameter values ​​of the robust optimal state.

6. Robotic system with: The device according to claim 3; robotic devices designed to perform locomotion; at least one sensor designed to provide an output parameter value assigned to an input parameter value to the device; and A control device is designed to control the robotic device such that input parameter values ​​are selected in dependence on the input parameter values ​​of the robust optimal state, in particular such that they correspond to the input parameter values ​​of the robust optimal state.

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