A method for training an artificial neural network and / or testing the robustness of an artificial neural network.

Abstract

To provide a method, a device and a computer program that make it possible to train and / or verify the robustness of an artificial neural network.SOLUTION: The method includes: setting an input variable for the network which has a plurality of dimensions; determining upper and lower input variable limits for each dimension of an output side; determining lower and upper output variable limits for an output variable of the network with values from a value range restricted by the lower and upper input variable limits; and determining the lowest possible and highest possible values of a real-valued function with values from the value range restricted by the lower and upper output variable limits, where the input variable determines an output that proves the robustness of the network if the lowest possible and highest possible values lie within a preset interval of permissible values.SELECTED DRAWING: Figure 2

Description

[Technical Field]

[0001] Background technology Artificial neural networks are used for autonomous driving and other autonomous actions that enable machines to perform. Training and robustness testing of such artificial neural networks are essential for the reliability and certainty of such technical systems. [Background technology]

[0002] When integrating multiple systems, system integrators do not always have access to all aspects of the supplier's training and / or testing processes. These systems utilize pre-trained artificial neural networks, which in some cases undergo further training for specific tasks. [Prior art documents] [Non-patent literature]

[0003] [Non-Patent Document 1] Publication “Scaling provable adversarial defenses” by Eric Wong, Frank R. Schmidt, Jan Hendrik Metzen, and J. Zico Kolter (https: / / arxiv.org / abs / 1805.12514) [Non-Patent Document 2] Publication "On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models" by Sven Gowal, Krishnamurthy Dvijotham, Robert Stanforth, Rudy Bunel, Chongli Qin, Jonathan Uesato, Relja Arandjelovic, Timothy Mann, Pushmeet Kohli (https: / / arxiv.org / abs / 1810.12715) [Overview of the project] [Problems that the invention aims to solve]

[0004] Therefore, it is desirable to enable the training and / or testing of the robustness of such artificial neural networks.

[0005] The means to achieve this are disclosed, for example, in the publication "Scaling provable adversarial defenses" by Eric Wong, Frank R. Schmidt, Jan Hendrik Metzen, and J. Zico Kolter, which is available at https: / / arxiv.org / abs / 1805.12514.

[0006] The means for this purpose are disclosed, for example, in the publication "On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models" by Sven Gowal, Krishnamurthy Dvijotham, Robert Stanforth, Rudy Bunel, Chongli Qin, Jonathan Uesato, Relja Arandjelovic, Timothy Mann, and Pushmeet Kohli, which is available at https: / / arxiv.org / abs / 1810.12715. [Means for solving the problem]

[0007] Disclosure of the invention A robust artificial neural network can be trained or its robustness tested by the method, apparatus, and computer program described in the independent claim.

[0008] A method for training an artificial neural network and / or testing the robustness of an artificial neural network, wherein the artificial neural network is configured to determine an output variable, and the method is assumed to include setting up input variables for the network having multiple dimensions, the method including determining an upper input variable limit and a lower input variable limit for each dimension of the input variable or for each dimension on the output side of a linear layer of the artificial neural network without an activation function to which the input variable is mapped by the artificial neural network, wherein at the upper input variable limit, the disturbance variable model that can map the input variable to a disturbed input variable has the largest possible value in this dimension, and at the lower input variable limit, the disturbance variable model has the smallest possible value in this dimension. The method includes determining a lower output variable limit for the network's output variable by a value from a range of values ​​limited by the lower input variable limit and the upper input variable limit, and determining an upper output variable limit for the output variable by a value from a range of values ​​limited by the upper input variable limit and the lower input variable limit, for each dimension of the output variable. This method involves determining the smallest possible value of a real-valued function, in particular, from a range of values ​​limited by a lower limit and an upper limit on the output variable. This method also involves determining the largest possible value of a real-valued function, in particular, from a range of values ​​limited by a lower limit and an upper limit on the output variable, and determining an output that proves the robustness of the network if the smallest possible value and the largest possible value fall within a set interval of allowed values.

[0009] The input variable preferably represents a digital image, and more particularly, a video image, radar image, LiDAR image, ultrasound image, or infrared image.

[0010] The output variables preferably represent signals for drive control of a physical system, particularly a computer-controlled machine or information transmission system, preferably a robot, vehicle, household appliance, drive tool for manufacturing machinery, personal assistance system, access control system, monitoring system, or particularly a medical imaging system.

[0011] The output variable may be intended to classify sensor data, particularly for identifying objects within the sensor data or for semantic segmentation, or to provide regression of sensor data, particularly for identifying objects within the sensor data or for semantic segmentation, where the objects are preferably markings or objects on the road surface or objects representing markings or objects on the road surface, particularly objects representing road signs, pedestrians or vehicles.

[0012] It can be assumed that a lower bound on the input variables is determined for each dimension, and at this lower bound, the scalar product of the input variables and the unit vector of that dimension is as small as possible.

[0013] It is often assumed that for each dimension of the input variable, a lower limit of the input variable for that dimension is determined, and at this lower limit, the first scalar product of the input variable and the negative unit vector of that dimension is as large as possible. Furthermore, it is often assumed that for each dimension of the input variable, an upper limit of the input variable for that dimension is determined, and at this upper limit, the second scalar product of the input variable and the unit vector of that dimension is as large as possible.

[0014] It is often assumed that the output side of the linear layer of an artificial neural network without an activation function is defined by a matrix, and for each dimension of the output side, a first product is defined which can be determined by multiplying the negative transpose of the matrix by the unit vector of that dimension, and in this method, it is assumed that the lower bound of the input variables for this dimension is determined, and at this lower bound, the first scalar product of the input variables and the first product has the largest possible value, and for each dimension of the output side, a second product is defined which can be determined by multiplying the transpose of the matrix by the unit vector of that dimension, and in this method, it is assumed that the upper bound of the input variables for this dimension is determined, and at this upper bound, the second scalar product of the input variables and the second product has the largest possible value.

[0015] It is often assumed that artificial neural networks are intended to have a bias on the output side, in which case the lower and upper limits of the input variables are modified in relation to the bias.

[0016] It may be assumed that the linear layer of an artificial neural network without an activation function is the input layer of the artificial neural network, or that multiple linear layers without activation functions are arranged between the input side of the artificial neural network and the output side of the linear layer of the artificial neural network without an activation function.

[0017] It may be assumed that the input variables are determined or will be determined in relation to the measured signal, or that the input variables are selected or will be selected from a range of values.

[0018] When the smallest possible value or the largest possible value lies outside or on the boundary of an interval for which an acceptable value is set, it is often assumed that a value with the largest possible gap relative to this interval is determined from the interval between the smallest possible value and the largest possible value, and the network is trained to reduce this gap.

[0019] The apparatus includes a computing device configured to carry out this method.

[0020] The device may include at least one memory for input variables and an output device, where the input variables represent sensor data or sensor-measurable variables that can predict sensor data; the output device is configured to output a signal; and the computing device is configured to determine the signal in relation to an output variable on which an artificial neural network maps the input variables. For sensor data, for example, a drive control variable or a pre-processed sensor signal is output. For measurable, particularly physical, variables, for example, a sensor signal is output that replaces the sensor for other, particularly physical, variables.

[0021] The apparatus may include an input device configured to communicate with a sensor to detect sensor data, and at least one computing device configured to determine input variables in relation to the sensor data.

[0022] A computer program contains computer-readable instructions, and when these instructions are executed, the computer carries out this method.

[0023] Further advantageous embodiments will become apparent from the following description and drawings. [Brief explanation of the drawing]

[0024] [Figure 1] This diagram schematically shows a device for training artificial neural networks and / or testing the robustness of artificial neural networks. [Figure 2] This diagram shows a flowchart of a method for training an artificial neural network and / or testing the robustness of an artificial neural network. [Modes for carrying out the invention]

[0025] Figure 1 shows a schematic representation of the apparatus 100. The apparatus 100 includes at least one computing device, for example, at least one processor 102 and at least one memory 104.

[0026] In this example, at least one memory 104 contains a computer program that includes computer-readable instructions, which is used by at least one processor 102 to perform the method described below when the computer-readable instructions are executed.

[0027] At least one memory 104 in this example contains an artificial neural network. For example, the memory stores parameters that define the weights of the individual layers of the network, and / or hyperparameters that define the network's architecture and / or activation function.

[0028] In this example, the device 100 includes an input device 106 configured to receive sensor data from sensor 108. The input device 106 may be configured to select a value from a range of values ​​that reproduces a value representing the sensor data from sensor 108. It is assumed that at least one processor 102 is configured to select this value from this range. In this example, sensor 108 and the input device 106 are connected via a line 110 for communicating sensor data. In this example, the device has a data link 112 which connects at least one processor 102, at least one memory 104, and the input device 106 for data transmission.

[0029] In this example, for training purposes, or for inference by the artificial neural network on these values, the input variable x ∈ R of the artificial neural network is used. nIt is assumed that the input variable x is determined. It may be assumed that the input variable x is determined from training data or test data which includes multiple sensor data measured in advance by the sensor 108. It may be assumed that the training data or test data includes a value for the input variable x that reproduces the sensor data measurable by the sensor 108. It may be assumed that the training data or test data includes a value for the input variable x, and that using this value, the generative artificial neural network can generate sensor data measurable by the sensor 108. The sensor data, input variable x, training data or test data may be stored in at least one memory 104.

[0030] The sensor data and / or input variable x may represent a digital image, in particular a video image, radar image, LiDAR image, ultrasound image, or infrared image.

[0031] The artificial neural network may be configured to classify sensor data, in particular, to identify objects within the sensor data or for semantic segmentation.

[0032] The artificial neural network may be configured to provide regressions on sensor data, particularly for identifying objects within the sensor data or for semantic segmentation.

[0033] An artificial neural network is configured to output an output variable y. The output variable y represents a signal for driving and controlling a physical system.

[0034] In this example, the device 100 includes an output device 114, which is configured to output signals for driving and controlling a physical system in relation to an output variable y. In this example, the output device 114 is connected to at least one processor 102 via a data link 112.

[0035] The output device 114 is configured, for example, to drive and control a computer-controlled machine 116 in relation to an output variable y. In this example, the output device 114 and the computer-controlled machine 116 are connected via a control line 118 for signal transmission. The computer-controlled machine 116 is, for example, a robot, a vehicle, a household appliance, a power tool, a manufacturing machine, a personal assistance system, an access control system, a monitoring system, or in particular, a medical imaging system.

[0036] Additionally or alternatively, it may be assumed that the output device 114 is configured to drive and control the information transmission system in relation to the output variable y.

[0037] For autonomous driving or other applications, the output variable y may be intended to classify the sensor data, particularly to identify objects within the sensor data or for semantic segmentation.

[0038] For autonomous driving or other applications, the output variable y may be intended to provide a regression of sensor data, particularly for identifying objects within the sensor data or for semantic segmentation.

[0039] In this context, the object preferably represents a marking or object on the road surface or a marking or object on the road surface. The object may represent, for example, a road sign, a pedestrian, or a vehicle.

[0040] In this example, the artificial neural network includes the first layer, which will be referred to as the input layer below. In this example, the artificial neural network includes the last layer, which will be referred to as the output layer below.

[0041] In this example, at least one hidden layer is located within the network between the input layer and the output layer. The input layer, in this example, is a layer without an activation function. The output layer, in this example, is a layer with an activation function. Hidden layers within the network may be assumed to be formed with or without an activation function. A layer without an activation function will also be referred to as an affine layer below.

[0042] The weights of stratum j are, in this example, matrix A j It is stored in [location].

[0043] Matrix A for the input layer maps the input variable x of the input layer to the output side of the input layer. In this example, the input layer has multiple n dimensions.

[0044] The weights of multiple adjacent affine layers can be considered as being merged into a single matrix A. In this case, the output side is the output side of the last layer of the merged layers. In the example of two adjacent affine layers A1 and A2, these affine layers A1 and A2 are merged into matrix A = A2·A1.

[0045] A method for training an artificial neural network and / or testing its robustness is described below. The steps of this method are schematically shown in Figure 2.

[0046] Artificial neural networks have an output variable y ∈ R k It is configured to determine this.

[0047] In the following, two cases are distinguished. Case 1: y=f w (x) Case 2: y=f A,b,w (x)

[0048] In Case 1, the linear layer of the artificial neural network without an activation function is the input layer of the artificial neural network.

[0049] In Case 2, a plurality of linear layers without an activation function are arranged between the input side of the artificial neural network and the output side of the linear layer of the artificial neural network without an activation function. In Case 2, at least the input layer is affine. This means that the artificial neural network

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[0050] In both cases, a disturbance variable model T(x) ⊂ R n is assumed, whereby the input variable x is a perturbed input variable

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[0051] For the disturbance variable model T(x), in this example, the function

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[0052] For the case of \(1 \lt p \lt \infty\),

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[0053] When \(1\leq i\leq N\),

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[0054] L p For the disturbance variable model with the norm, the upper limit input variable limit is

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[0055] In contrast, according to this method, it is possible to provide a solution without determining the maximization part \(T1(x)\).

[0056] This method includes step 202.

[0057] In step 202, the input variable \(x\in R\) having a plurality of \(n\) dimensions nThe following is set up. In step 202, in this example, the disturbance variable model T(x) is set up. In step 202, in this example, the real-valued function l:R is set up in particular. k →R is set. In step 202, in this example, the interval I⊂R of values ​​that results in a authenticated artificial neural network is set.

[0058] It may be assumed that the input variable x is determined in relation to the measured signal, or that the input variable x is selected from a range of values.

[0059] Next, step 204 is executed.

[0060] Step 204 includes substeps, which in Case 1 are performed for each dimension 1 ≤ i ≤ n of the input variable x. In Case 2, these substeps are performed for each dimension 1 ≤ i ≤ N on the output side of the linear layer of the artificial neural network without an activation function, to which the input variable x is mapped by the artificial neural network.

[0061] Part 1: Upper limit of input variable u i decision In this example, the upper limit of the input variable u i This is determined, and this upper limit of the input variable u i In this case, the disturbance variable model T(x)⊂R n However, it has the largest possible value in dimension i.

[0062] Part 2: Lower limit of input variable l i decision In this example, the lower limit of the input variable l i This is determined, and this lower limit of the input variable l i In this case, the disturbance variable model T(x)⊂R n However, it has the smallest possible value in dimension i.

[0063] Case 1: In this method, for each dimension of the input variable x (1 ≤ i ≤ n), the lower bound of this dimension i is defined as the input variable limit l. i It is often assumed that the lower limit of the input variable l will be determined, and this lower limit of the input variable l i In this case, the input variable x and the negative unit vector of dimension i -e i The first scalar product with <-e i x〉 has the largest possible value. Preferably, these limits are determined for the output variable l ≤ y ≤ u, where l,u ∈ R k In this example, the lower bound of the input variable is defined as follows: l i =-opt(-e i ;x) This will be decided.

[0064] In this method, for each dimension of the input variable x, 1 ≤ i ≤ n, the upper limit of the input variable u of this dimension i is defined. i It is often assumed that the upper limit of the input variable u will be determined, and this upper limit of the input variable u i In this case, the input variable x and the unit vector e of dimension i are i The second scalar product with <e i The value of x> is as large as possible. Preferably, these limits are such that, in this example, the output variable y = f w These are determined for (x). Preferably, these limits are determined for the output variable l ≤ y ≤ u, where l,u ∈ R. k In this example, the upper limit of the input variable is defined as follows: u i =opt(e i ;x) This will be decided.

[0065] It can be assumed that the artificial neural network has a bias b applied to the input variable x, and in this method, the lower limit of the input variable l i and upper limit of input variable u i It is assumed that this will be corrected in relation to bias b.

[0066] Case 2: In this example, the output of a linear layer in an artificial neural network without an activation function is defined by matrix A of an affine layer, or is integrated by matrix A into sequentially consecutive affine layers starting from the first layer of the artificial neural network.

[0067] For each output dimension 1 ≤ i ≤ N defined by this matrix A, a first product is defined, which is the negative transition -A of matrix A. T And this unit vector e of dimension i i It can be determined by multiplication with .

[0068] In this method, for each dimension i, the lower limit of the input variables l is defined. i It is conceivable that this lower limit of input variables l will be determined. i In this case, the first scalar product of the first product with input variable x <-A T e i The value of x is as large as possible.

[0069] For each output dimension 1 ≤ i ≤ N defined by this matrix A, a second product is defined, which is the transition A of matrix A. T And this unit vector e of dimension i i It can be determined by multiplication with .

[0070] In this method, the upper limit of the input variables u of this dimension i is i It is conceivable that this upper limit of input variables u will be determined, and this upper limit of input variables u i In this case, the second scalar product of the input variable x and the second product <A T e i The value of x is as large as possible.

[0071] These limitations, in this example, apply to the output variable y=f A,b,w These are determined for (x). Preferably, these limits are determined for the output variable l≦y≦u, where l,u∈R k That is the case.

[0072] It can be assumed that the artificial neural network has a bias b applied to the output side. In this case, this method has a lower limit of input variables l. i and upper limit of input variable u i It is assumed that this is modified in relation to bias b. If the artificial neural network has two consecutive affine layers, for example, matrix A = A2·A1 and bias b = A2b1 + b2.

[0073] Next, step 206 is executed.

[0074] In step 206, for each dimension of the output variable y (1 ≤ j ≤ k), a lower limit of the output variable is set for the output variable y.

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[0075] In this example, the upper limit of the output variable u y and lower limit of the output variable l y The upper limit of input variables u1, ..., u n and lower limit input variable limits l1, ..., l n It is determined by the value within the range of values ​​limited by [the specified criteria].

[0076] In the following examples, the upper limit of the output variable u in Case 1 is shown. y and lower limit of the output variable l y This explains three methods for determining [the answer].

[0077] Method 1: Element x of input variable x i Each artificial neural network f w (x) is assumed to include an activation layer φ:R→R, and element x i One upper limit input variable for each limit x i ≦u iThis is stipulated.

[0078] In this example,

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[0079] Method 2: Artificial neural network f for input variable x w It is assumed that (x) contains the scalar product <c,x> + β with the bias β, and element x i One upper limit input variable for each limit x i ≦u i and one lower limit of the input variable l i ≤x i This is stipulated.

[0080] In this example, vector c + and c - This was decided, and here, [Number] and [Number] is. Subsequently, L = 〈c - , u〉 + 〈c + , l〉 + β and U = 〈c - , l〉 + 〈c + , u〉 + β are determined. As a result of the scalar product, L i ≤ 〈c, x〉 ≤ U i is determined, where the values for l i ≤ x i ≤ u i are used.

[0081] The upper output variable limit u y and the lower output variable limit l y are determined by the mapping of the result of the scalar product 〈c, x〉 + β between the remaining part of the artificial neural network f w (x) and the bias.

[0082] Method 3: It is assumed that the artificial neural network f w (x) = A·x + b for the input variable x includes a linear layer A with a bias b.

[0083] In this example, for each element y j of the output variable y = A·x + b, the scalar product with column j is determined from the matrix A, and here, for each scalar product, the calculation is performed as described in Method 2.

[0084] If no bias is set, this calculation is performed with bias b = 0 in this example.

[0085] In Case 2, this calculation is performed as described above for Case 1, where the input variable x is replaced by the output side defined by the matrix A.

[0086] Subsequently, step 208 is executed.

[0087] In step 208, in particular, a function l: R k →R of as small a value as possible

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[0088] Subsequently, step 210 is executed.

[0089] In step 210, in particular, a function l: R k →R of as large a value as possible

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[0090] Subsequently, step 212 is executed.

[0091] In step 212, when the as small a value as possible y - and the as large a value as possible y + are within the set interval I ⊂ R of acceptable values, an output for proving the robustness of the network is determined. <000054o> Optionally, training may be assumed. In step 214, it is checked whether the as small a value as possible y - or the as large a value as possible y + is outside the set interval I ⊂ R of acceptable values or on its boundary.

[0093] The smallest possible value y - or the largest possible value y + However, if the interval I⊂R is outside the set acceptable range or is on its boundary, step 216 is executed. Otherwise, in this example, training is terminated.

[0094] Optionally, it may be assumed that input variables are detected after training and mapped to output variables, in which case signals or sensor data for driving and controlling a physical system are specifically determined and / or output in relation to each output variable.

[0095] In step 216, the smallest possible value y - and the largest possible value y + The value is determined from the interval between and , and this value is the largest possible interval.

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[0096] Next, step 218 is executed.

[0097] In step 218, the network is trained to reduce this interval.

[0098] Next, step 202 is executed.

Claims

1. A method for training an artificial neural network and / or testing the robustness of an artificial neural network, wherein the artificial neural network is configured to determine an output variable, The method includes setting input variables for the network having multiple dimensions (202), The method includes determining an upper limit (204) and a lower limit (204) of the input variable for each dimension of the input variable, or for each dimension of the output side of the linear layer of the artificial neural network without an activation function to which the input variable is mapped by the artificial neural network, wherein in the upper limit, the disturbance variable model that can map the input variable to the disturbed input variable has the largest possible value in the dimension, and in the lower limit, the disturbance variable model has the smallest possible value in the dimension. The method includes determining a lower limit of the output variable for each dimension of the output variable by a value from a range of values ​​limited by the lower limit of the input variable and the upper limit of the input variable (206), and determining an upper limit of the output variable for the output variable by a value from a range of values ​​limited by the upper limit of the input variable and the lower limit of the input variable (206), The method includes determining the smallest possible value of a real-valued function, in particular, from a range of values ​​limited by the lower limit of the output variable and the upper limit of the output variable (208), The method includes determining the largest possible value of the function in particular, which is a real-valued function, by a value from the range of values ​​limited by the lower limit of the output variable and the upper limit of the output variable (210), When the smallest possible value and the largest possible value fall within the interval set for the allowable values, an output is determined that proves the robustness of the network (212). A method characterized by the following features.

2. The method according to claim 1, wherein the input variable represents a digital image, and in particular represents a video image, radar image, LiDAR image, ultrasound image, or infrared image.

3. The method according to claim 1, wherein the output variable represents a signal for drive control of a physical system, in particular a computer-controlled machine or information transmission system, preferably a robot, a vehicle, a household appliance, a drive tool for a manufacturing machine, a personal assistance system, an access control system, a monitoring system, or in particular a medical imaging system.

4. The method according to claim 1, wherein the output variable classifies the sensor data, in particular for identifying objects within the sensor data or for semantic segmentation, or the output variable provides a regression of the sensor data, in particular for identifying objects within the sensor data or for semantic segmentation, wherein the objects are preferably markings or objects on the road surface or objects representing markings or objects on the road surface, in particular objects representing road signs, pedestrians or vehicles.

5. For each dimension of the input variable, the lower limit of the input variable for that dimension is determined, and at the lower limit of the input variable, the first scalar product of the input variable and the negative unit vector of that dimension has the largest possible value. The method according to claim 1, wherein for each dimension of the input variable, the upper limit of the input variable for that dimension is determined, and at the upper limit of the input variable, the second scalar product of the input variable and the unit vector of that dimension has the largest possible value.

6. The output side of the linear layer of the artificial neural network without an activation function is defined by a matrix, For each dimension of the output side, a first product is defined that can be determined by multiplying the negative transpose of the matrix by the unit vector of that dimension. The above method assumes determining the lower limit of the input variable for the dimension, where the first scalar product of the input variable and the first product has the largest possible value. For each dimension of the output side, a second product is defined that can be determined by multiplying the transpose of the matrix by the unit vector of that dimension. The method according to claim 1, wherein the method assumes determining the upper limit of the input variable for the dimension, and in the upper limit of the input variable, the second scalar product of the input variable and the second product has the largest possible value.

7. The method according to claim 5 or 6, wherein the artificial neural network is biased with respect to the input variables or the output side, and the lower limit and upper limit of the input variables are modified in relation to the bias.

8. The method according to claim 6, wherein the linear layer of the artificial neural network without an activation function is the input layer of the artificial neural network, or a plurality of linear layers without an activation function are arranged between the input side of the artificial neural network and the output side of the linear layer of the artificial neural network without an activation function.

9. The method according to claim 1, wherein the input variable is determined or determined in relation to a measured signal, or the input variable is selected or selected from a range of values.

10. If the smallest possible value or the largest possible value is outside or on the boundary of the set interval of the allowable values ​​(214), then a value that is as large as possible relative to the interval is determined from the interval between the smallest possible value and the largest possible value (216). The method according to claim 1, wherein the network is trained to reduce the interval (218).

11. An apparatus (100) comprising at least one computing device (102) configured to carry out the method described in claim 1.

12. The device (100) includes at least one memory (104) for the input variables and an output device (114), The input variable represents sensor data, or the input variable represents a variable measurable by the sensor that can predict the sensor data. The apparatus (100) according to claim 11, wherein the output device (114) is configured to output a signal, and the computing device (102) is configured to determine the signal in relation to the output variable on which the artificial neural network maps the input variable.

13. The apparatus (100) includes an input device (106), which is configured to communicate with a sensor (108) to detect sensor data. The apparatus (100) according to claim 11, wherein the at least one computing device (102) is configured to determine the input variable (x) in relation to the sensor data.

14. A computer program that includes a computer-readable instruction, characterized in that when the computer-readable instruction is executed, it causes the computer to perform the method described in claim 1.

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