202213257 Auslandsfassung 1 Description Method for providing a physically explainable fault infor- mation of a bearing by a fault detection model The present disclosure relates to a fault detection apparatus and a computer-implemented method for providing physically explainable fault information of a bearing built in a machine by a fault detection model. Sensors are omnipresent in all kinds of heavy machinery, mo- tors, and similar equipment. One especially important appli- cation field of sensors is detecting faults of bearings in rotating machinery such as motors, turbines, and pumps etc. Faults in bearings are among the most common causes of mal- function for rotating equipment. This malfunction can be de- tected in vibration patterns. To obtain the required infor- mation, sensors such as position transducers, velocity sen- sors, accelerometers and spectral emitted energy sensors can be installed either directly on the bearings or mounted on these machines. This allows to obtain measurements that can be used for vibration analysis by extracting vibration fre- quencies and amplitudes. If the bearings are subject to dif- ferent damages, geometrical imperfections or malfunction, the sensor values typically represent suspicious patterns and anomalies. While there exist multiple tools and methods de- rived from physical theory that would allow, in principle, to obtain this information from the sensor measurements, this is still a very challenging task. However, there are multiple problems related to detecting faults in bearings from sensor data that still need to be solved. Since time series of sensor data are only indirect measurements of real physical mechanisms and have a very com- plex data structure, machine learning model trained to detect faults in bearings need to be complex themselves and are, therefore, often not human-understandable, often called black-box algorithms. This means that it is not possible to 202213257 Auslandsfassung 2 understand how the model behaves, why the model predicts faults of the bearing and consequently whether the results of the algorithm are in line with physical reasoning. This leads to obstacles for data scientists / model developers to build a better and more robust model and to obstacles for domain experts, like Engineers or operation personal to understand and trust the results of the model. It also leads to problems with the detection of root-causes for faults in the bearing, e.g., the bearing or crankshaft faults and consequently to a lack of acceptance from user side. Signal processing approaches for detecting faults on bearings are standard in application and well-grounded in theory. Identified faults in a bearing may occur at different parts of the bearing. In a bearing, faults can occur either in the bearings outer race, in the inner race, in the cage or at balls of a ball bearing. Different approaches and formulas are necessary to describe the physical relationship between measured sensor data and the individual type of fault. Since these formulas often describe the damage frequency for the different kinds of damage types of the bearing based on the attributes and the rotation speed of the bearing, the differ- ent phenomena can be physically explained. To perform fault diagnosis of these different damages on acceleration data, it is important to know these physical attributes of the bearing as well as the rotation speed the bearing was running at when the data was recorded. Since some of the fault effects are amplitude modulated in the vibration spectrum and overlaid by resonance effects, different pre-processing steps are applied to the recorded raw data to reveal the specific fault fre- quencies. However, the signal processing approaches often suffer from a couple of problems. E.g., a resulting vibration spectrum highly depends on the mounting position of a speed sensors and a possible load of the machine. Further, noisy sensor signals with multiple confounding factors impede the signal- processing. 202213257 Auslandsfassung 3 Also, machine learning approaches have been considered to perform fault detection in bearings. In order to perform the training of the machine learning algorithm, it is necessary to have access to a sufficient amount of ideally labeled training data containing realistic vibration signals measured during actual operation of the machine of interest with healthy and defective parts. Nevertheless, such algorithms are inherently very complex black box models. This means that it is entirely unclear based on which logic such methods form their decision. Therefore, it is the object of the present application to provide an apparatus and method for fault detection of roll- ing objects by machine learning methods which outputs physi- cally interpretable information, which even indicate root- causes for a detected fault. A further specific object is to improve the interpretability of bearing fault detection with machine learning algorithms trained on either vibration or electric current data of rotating machines. This object is solved by the features of the independent claims. The dependent claims contain further developments of the invention. A first aspect concerns a computer-implemented method for providing physically explainable fault information of a bear- ing built in a machine by a fault detection model, comprising the steps: - obtaining sensor data measured at the bearing as input data relating to an input data domain and a fault detection model which is already trained on sensor data related to said input data domain to output a predicted failure value of the bear- ing by processing the obtained sensor data, - mapping the measured sensor data from the input data domain to a selected data domain and resulting in an augmented fault detection model which outputs augmented predicted failure value related to the selected data domain instead of the in- 202213257 Auslandsfassung 4 put data domain, wherein the selected data domain has a phys- ical meaning to the bearing’s fault, - performing a feature attribution on the augmented fault de- tection model for the obtained sensor data, quantifying an importance of at least one individual feature of the input data to the augmented failure value related to the selected data domain, and - displaying the individual feature and the respective quan- tified importance in the selected data domain at a user in- terface. The method is based on a “conventional” fault detection model relating input signals of sensor data measured, e.g., over time, to a predicted failure value, e.g., whether a fault is present (value 1) or not (value 0). A feature attribution ap- plied to the predicted failure value would provide data points in the input data domain, e.g., in the time domain. The data feature comprises at least one but mostly several adjacent sensor data points. These data features in the input domain would not provide information about the underlying specific fault. The quantified data features in the input da- ta domain provide neither a hint to the root-cause for the predicted failure value nor is it interpretable in a way that is in line with accepted physical theory. In contrast to that, the feature attribution performed on the augmented fault detection model, which is related to the selected data domain, provides features in the selected data domain instead of features in the input data domain. The augmented predicted failure value resulting from the augmented fault detection model is equivalent to, especially even the same as, the pre- dicted failure value resulting from the obtained fault detec- tion model. The selected data domain has a physical meaning to the fault of the bearing and is therefore interpretable, e.g., comparable with typical fault frequencies of the con- sidered bearing. In an embodiment of the method, the domain mapping consists of multiple concatenated domain mappings. 202213257 Auslandsfassung 5 Such concatenated domain mappings can model/reflect different domain transitions and provide therefore more flexibility with respect to faults being physically explainable in anoth- er domain as the input data domain, in which the fault detec- tion model was trained on. In an embodiment of the method, the at least one domain map- ping is performed by applying an invertible, bijective trans- formation function onto the measured input data. This ensures a unique and unambiguous mapping between the output of the fault detection model in the input data domain (i.e., the predicted failure value) and the output of the augmented fault model (i.e., the augmented failure value) in the selected data domain. The predicted failure value and the augmented predicted failure value are equivalent, i.e., they have the same value. In an embodiment of the method, the feature attribution is performed by any model agnostic feature attribution method applicable to a type of machine learning model which is used for the fault detection model. This ensures that the data features which are most relevant to the output of the augmented fault detection model can be analysed independently on the augmented learning model, and only dependent on the “original” fault detection model. In an embodiment of the method, the fault detection model is a deep neural network, especially am Autoencoder, a Convolu- tional Neural Network or a Deep Belief Network. Deep neural networks can learn extreme complicated patterns, they are flexible and able to cope with high-dimensional com- plex sensor input data. 202213257 Auslandsfassung 6 In an embodiment of the method, the sensor data is vibration data or electric current data measured at or near the bear- ing. Vibration data are especially indicative to fault indications of bearings, as defects in the bearing produce cyclic inter- ruptions due to imperfections of e.g., balls. The machine with the defective bearing often requires more electrical current than in un-defective state. In an embodiment of the method, the sensor data measured at the bearing are measured in the time domain and the mapping is performed into a frequency domain. Sensors mainly measure a physical parameter of the machine over time and are therefore in most cases available in time domain. As faults of bearings result in cyclic appearing dis- turbances in sensor data measured over time, the frequency domain of the signal envelope is the most likely domain where a physical interpretation seems possible. In an embodiment of the method, an alarm is automatically output to the user interface, if the quantified importance of the augmented fault detection model is detected at a prede- fined frequency, which is related to a root cause of the bearing. The alarm attracts the attention of an operation personnel in such cases indicating a fault of the bearing with high proba- bility. This allows to take fast measures at the machine, e.g., change settings, stop the machine, or schedule mainte- nance work. In an embodiment of the method, the output quantified im- portance is displayed in a color-code with respect to quanti- ty. 202213257 Auslandsfassung 7 This facilitates to recognize critical or high relevant val- ues, e.g., in a continuous spectrum of importance values. The color-code may indicate different values of quantity in dif- ferent colors of a predefined color scale or in different in- tensity of one color. In an embodiment of the method, the fault detection model of a bearing is trained on a signal envelope of the measured sensor data in a time domain and analyzed for specific fault frequencies in the frequency domain. The signal envelope highlights the time distance between two peaks in the sensor data measured in the time domain. Charac- teristic faults of the bearing depend on parameters like the diameter of the balls or a pitch diameter of the bearing, and produce disturbances in different intervals of time, which relate directly to frequencies depending on the value of above parameters. In an embodiment of the method, the domain mapping is per- formed by a Fourier transformation function. Fourier Transformation or a Fast Fourier Transformation func- tion are well known and require few processing capacities. In an embodiment of the method, the machine is a rotating ma- chine, especially a motor, a turbine, a pump, or a press. A second aspect concerns a fault detection apparatus for providing physically explainable fault information of a bear- ing built in a machine by a fault detection model, comprising at least one processor con-figured to perform the steps: - obtaining sensor data measured at the bearing as input data relating to an input data domain and a fault detection model which is trained on sensor data related to said input data domain to output a predicted failure value of the bearing by processing the obtained sensor data, 202213257 Auslandsfassung 8 - mapping the measured sensor data from the input data domain to a selected data domain resulting in an augmented fault de- tection model which outputs augmented predicted failure value related to the selected data domain, wherein the selected da- ta domain has a physical meaning to the fault of the bearing, - performing a feature attribution on the augmented fault de- tection model for the obtained sensor data, quantifying an importance of at least one individual feature of the input data to the augmented failure value related to the selected data domain, and - displaying the individual feature and the respective quan- tified importance in the selected data domain at a user in- terface. A third aspect concerns a computer program product directly loadable into the internal memory of a digital computer, com- prising software code portions for performing the steps as described before, when said product is run on said digital computer. The invention will be explained in more detail by reference to accompanying figures. Similar objects will be marked by the same reference signs. Figure 1 illustrates an embodiment of the inventive comput- er-implemented method by a flow diagram. Figure 2 schematically illustrates in more detail an inter- action of the various processing steps. Figure 2A schematically illustrates a domain mapping and re- sulting augmented fault detection model. 202213257 Auslandsfassung 9 Figure 2B schematically illustrates characteristics of a transformation function applied for the domain map- ping. Figure 3A schematically illustrates an output of a fault de- tection model in the input domain. Figure 3B schematically illustrates an output of an embodi- ment of the augmented fault detection model in the selected domain. Figure 4 schematically illustrates an output of feature at- tribution of the fault detection model in the input domain compared with the output of feature attribu- tion of the augmented fault detection model in the selected domain. Figure 5 illustrates an embodiment of the inventive fault detection apparatus as a block diagram. It is noted that in the following detailed description of em- bodiments, the accompanying drawings are only schematic, and the illustrated elements are not necessarily shown to scale. Rather, the drawings are intended to illustrate functions and the co-operation of components. Here, it is to be understood that any connection or coupling of functional units, devices, components or other physical or functional elements could al- so be implemented by a direct connection or an indirect con- nection coupling element, e.g., via one or more intermediate elements. A connection or a coupling of entities or compo- nents can for example be implemented by a wire-based, a wire- less connection and/or a combination of a wire-based and a wireless connection. Functional units can be implemented by dedicated hardware, e.g., processor, firmware or by software, and/or by a combination of dedicated hardware and firmware and software. It is further noted that each functional unit described for an apparatus can perform a functional step of the related method and vice versa. 202213257 Auslandsfassung 10 First, a description of standard approaches toward detecting bearing faults is provided, that are standard in application and well-grounded in theory. However, they often suffer from a range of problems. Examples are (too) noisy raw signals of sensor data, multiple confounding factors. These standard ap- proaches suffer of being inflexible and not able to cope with high-dimensional complex sensor data. As this approach is based on a physical derivation of the bearing behaviour dur- ing fault conditions it is described in detail below. Localized faults in a bearing may, especially a rolling ele- ment bearing, occur at different parts of the bearing, either in the bearings outer race, in the inner race, in the cage or at a rolling element. Depending on the fault, different ap- proaches are necessary. For the formulars below, we will use the following notation: d is the bearings ball diameter, ^^ is pitch diameter, ^^ _^ is the shaft speed, ^^ is the number of rolling elements and ^^ is the bearing contact angle. An inner ring damage is caused by irregularities on the inner ring of a bearing. When the rolling elements strike this fault, a shock is introduced that emits high frequency reso- nance. The envelope spectrum shows this fault at the BFPI frequencies: An outer ring damage is caused by irregularities on the outer ring of a bearing. The envelope spectrum shows this fault at the BFPI frequencies: Wear or deformation will cause the cage to move from its cen- tric position. Unbalance forces occur which lead to shock pulses. 202213257 Auslandsfassung 11 Damaged rolling elements periodically touch the bearings in- ner ring and outer ring and generate shock signals. Since the rolling elements rotate around themselves and at the same time experience a relative movement through the cage, side- bands are to be expected. The envelope spectrum shows peaks at: Since these formulas describe the damage frequency for the different kinds of bearing damage types based on the bearing physical attributes and the rotation speed, the different phenomena can be physically explained. To perform fault diag- nosis of these different bearing damages on acceleration da- ta, it is important to know these physical attributes of the bearing as well as the rotation speed the bearing was running at when the data was recorded. Since some of the fault effects are amplitude modulated in the vibration spectrum and overlaid by resonance effects, different pre-processing steps are applied to the recorded raw data to reveal the specific fault frequencies. It should be noted that a resulting vibration spectrum highly depends on the mounting position of a speed sensors and possible load of the machine. A signal processing-based approach applies a bandpass filter to the acceleration signal that contains the running noise of the bearing. Afterwards the envelope signal of the prefil- tered signal is computed and is transferred to the frequency domain. By calculating the envelope, the amplitude-modulated damage fault signal can be de-modulated, and the resulting envelope 202213257 Auslandsfassung 12 spectrum shows the different bearing faults in form of a peak at the characteristic frequencies BPFI, BPFO, FTF and BSF as defined above. A common way of obtaining the envelope spectrum is to calcu- late the analytical signal and then transfer it to the fre- quency domain. Since strong background noise such as impulse electromagnetic noise and periodic harmonic noise generated by shaft rotation etc. has great influence on the selection of the resonance frequency band, the right choice of an ap- propriate cut-off frequency for the upper- and the lower bound and the centre frequency of the bandpass filter is im- portant. In order to achieve this, the frequency band with the highest signal-to-noise ratio is determined by computing a kurtogram. The kurtogram shows a spectral kurtosis for different window widths and centre frequencies. Kurtosis is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Due to the fourth power, impulsive deviations from the mean result in large kurtosis values. Selecting the center frequency and bandwidth from the kurto- gram with the highest kurtosis value is a promising choice for the applied bandpass filter. As described above, this ap- proach requires detailed knowledge about the installed bear- ing and a high manual effort of a domain expert. An embodiment of the inventive method is described in the following with respect to Fig. 1 and explained in more detail with respect to Fig. 2 and Fig. 3A/B. 202213257 Auslandsfassung 13 A first step S1, see Fig. 1, to provide a physically explain- able fault information of a bearing built in a machine is to obtain sensor data 10, see Fig. 2, measured at the bearing as input data relating to an input data domain and a fault de- tection model 11 which is trained on sensor data related to said input data domain. Preferably, the sensor data 10 is vi- bration data or electric current data measured at or near the bearing. The sensor data 10 measured at the bearing are meas- ured in the time domain. The output of the fault detection model 11 is a predicted failure value 12 of the bearing by processing the obtained sensor data 10. The measured sensor data 10 are mapped from the input data domain to a selected data domain resulting in an augmented fault detection model which outputs augmented predicted fail- ure value related to the selected data domain, see step S2. The selected data domain has a physical meaning to the fault of the bearing and is therefore a semantic representation of the sensor data. For monitoring bearings, the mapping is per- formed preferably from time as input data domain into a fre- quency as the selected data domain. In Figure 2 the mapping and the resulting augmented fault detection model is depicted by reference sign 13. A feature attribution 14 is performed on the augmented fault detection model 13 quantifying an importance of at least one individual feature of the input data in the selected data do- main to the augmented failure value, wherein the augmented failure value is equivalent or even the same as the failure value of the obtained fault detection model, see step S3. The feature attribution is performed by any model agnostic fea- ture attribution method applicable to the fault detection model. In a last step S4, the individual features of the input data and the respective quantified importance in the selected data domain are displayed at a user interface. Fig. 2 shows an ex- ample of the displayed output as a diagram 15. A color of a 202213257 Auslandsfassung 14 vertical line 16 indicates the value of the importance of this frequency feature for the predicted failure value output by the fault detection model. The frequency feature is a ded- icated frequency or a frequency band comprising of several subsequent frequencies which contribute jointly to the indi- cated importance. Based on a predefined critical value in the selected data domain which is related to a root cause of the bearing an alarm is automatically output to the user in- terface if the quantified importance of the augmented fault detection model is detected at the predefined critical value. Preferably, the output quantified importance is displayed in a color-code with respect to quantity. Fig. 2A and Fig 2B show the domain mapping and the resulting augmented fault detection model 13 in more detail. The domain mapping is performed by applying a transformation function ϕ,631 onto the measured sensor data 10 and by applying an in- verse form of the transformation function ϕ ⁻¹ ,632 onto the fault detection model 11. Thus, obtained sensor data 10 re- lating to the input data domain t are transferred to the sen- sor data 101 in the selected data domain f. The inverse form of the transformation function ϕ ⁻¹ ,632) applied onto the fault detection model 11 results in the augmented fault detection function 13. Thus, the obtained sensor data 10 mapped into the selected data domain are input into the augmented fault detection model 13 which outputs the augmented fault value 12. The input data in the selected data domain, the augmented fault detection model 13 and the augmented fault value 12 are input to the feature attribution 14. The feature attribution 14 outputs a quantification of the importance of features in the obtained sensor data 101 in the selected data domain. Fig. 2B shows the transformation function ϕ,631 in more de- tail. The transformation function ϕ,631 is an invertible, bi- jective transformation function. This means that the trans- formation function ϕ,631 applied to obtained sensor data 100 in the input data domain x outputs in sensor data 101 in the selected data domain z. On the other side, the inverse form 202213257 Auslandsfassung 15 of the transformation function ϕ ⁻¹ ,632 applied to sensor data 101 in the selected data domain z outputs in sensor data 100 in the input data domain x. The input data domain x can be any parameter, e.g., time t as shown in Fig. 2A. The selected data domain z can be any parameter, e.g., frequency as shown in Fig. 2A. In many operational cases the sensor data are measured over time. Typically, one or several sensors detect vibration of the bearing in terms of an acceleration of the whole bearing or parts of the bearing. Another parameter measured to derive defects of a bearing is an electric current data of the ma- chine. The sensors are usually located at a part of the ma- chine close to the bearing. The obtained sensor data 10 shown in Fig. 2 provide acceleration values a measured over time t. The time is the input data domain. The machine learning model for fault detection, i.e., the fault detection model 11 can be considered as a function ^^ _^^ : ^^ ^{^^} → ^{^} 0,1 ^{^} mapping input sensor data 10 to an output providing a predicted failure value, e.g., a decision whether a fault is present (value 1) or not (value 0). The fault detection model was trained by a sufficient amount of ideally labeled training data containing realistic sensor data, e.g., vibra- tion signals measured in the input data domain, e.g., over time, during actual operation of the machine of interest with healthy and defective parts. The fault detection model 11 is preferably a deep neural net- work, especially an Autoencoder, a Convolutional Neural Net- work or a Deep Belief Network which is able to learn extreme complicated patterns. Most approaches utilizing deep neural networks show good performance on test data, are flexible and able to cope with high-dimensional complex sensor data and are reliable regarding their predictions. Nevertheless, such models are inherently very complex black box models. This means that it is entirely unclear based on which logic such models form their decision. 202213257 Auslandsfassung 16 An approach to explain black box machine learning models is given by feature attribution methods. Such methods quantify to which extend individual input data features have contrib- uted to the final predicted failure value of the model. The input data feature comprises one or several adjacent data- points of the input data. A decision of a machine learning model, i.e., the predicted failure value, can be considered as a function ^^: ^^ ^ௗ → ^^ mapping d-dimensional sensor data ^^ ∈ ^^ ^ௗ to a real number ^^( ^^)expressing a predicted model decision. In case of a fault detection of a bearing, ^^( ^^)could for example indicate a logit value or a respective probability for the presence of a defect estimated by the fault detection model. The goal of feature attribution is to identify an importance vector ^^ ∈ ^^ ^ௗ such that ^^ _^^ quantifies the importance that each in- put feature ^^ _^^ had on the model prediction ^^( ^^)for fixed input ^^. Up to now, any such feature attribution method will retrieve feature attributions in units of the sensor data features ^^, i.e., in units of the input data domain of the sensor data 10. So, for example if the fault detection model 11 is trained to classify sensor data in a time domain, ^^ will specify the importance of the sensor data features in time domain, like depicted in the Figure 3A. Sensor data features which have a high importance to the predicted failure value of the bearing are marked by dashed lines 21, sensor data features which have a low importance to the predicted failure value of the bearing are marked by dotted lines 22. The im- portance can also be indicated by color, wherein the value of the importance score is encoded by a color-scale and/or the intensity of the lines 21, 22. In the case where the fault detection model cares mainly about frequency information, but feature importance is displayed in the time domain, this would cause uninterpretable results and might not be meaning- ful to domain experts at all. As one can see at Fig. 3A, it is very hard to infer any spe- cific pattern or to find concrete reasons to explain the 202213257 Auslandsfassung 17 fault detection model’s decision. On the other hand, see Fig. 3B, if the values of the feature importance are computed in the frequency domain providing a frequency representation of the measured sensor data, it is immediately clear that the presence of a single frequency peak, see line 31, has had a significant influence on the predicted failure value. Similar as in Fig. 3A the value of importance in the selected data domain, here the frequency domain, is coded by different structure or different color of a line at the respective fre- quency, see Fig. 3B. To achieve a feature attribution in the selected domain which is different from the input data domain of the measured sen- sor data 10, feature attributions is translated into the se- lected domain by applying a bijective mapping, which captures an invertible one-to-one correspondence between the input da- ta domain of the fault detection model 11 and the selected domain. For instance, the Fourier Transform can be consid- ered as such kind of a mapping translating from the time do- main into a frequency representation, which is also reverta- ble. Mathematically, a domain mapping into D is specified by a function ^^: ^^ ^ௗ → ^^ which is invertible, meaning that there ex- ists another function ^^ ^{ି^} : ^^ → ^^ ^ௗ such that ^^ ^{ି^} ൫ ^^( ^^)൯ = ^^. The do- main mapping can also consist of multiple concatenated map- pings with all of them being invertible. Since the goal of φ is to translate sensor data features in the input domain into a more meaningful selected domain, we will refer to ^^( ^^) = ^^ as the interpretable or semantic representation of x. If an appropriate domain mapping function φ is specified, it can be combined with a feature attribution method to compute feature importance values in terms of the selected domain, i.e., a semantic representation z rather than based on sensor data features x. More specifically, φ can be used to create an augmented model 202213257 Auslandsfassung 18 Any model agnostic feature attribution method can now be _{evaluated on ^} ^{^} _{^ instead of ^^ yielding values of importance in} the selected domain. This holds true because model agnostic methods are designed to work with any machine learning model. If model specific attribution methods shall be applied one _{needs to check whether ^} ^{^} _{^ still meets necessary assumption of} the method (e.g., differentiability) or whether the method needs to be further adapted to work on such models (e.g., new LRP-rule). Semantic Explanations for Bearing Fault Detection Defects of bearings induce fault signals, i.e., measured sen- sor data, having an amplitude modulating effect on a specific carrier signal. Domain experts can detect such effects by an- alysing an envelope spectrum and checking whether specific fault frequencies are present. This logic is rigorously grounded on the physical understanding of bearing fault de- fects. If a machine learning model is trained to identify bearing faults from raw or prepossessed signals, i.e., sensor data in the input data domain, the link to existing domain knowledge about the physics of the problem might be neglected or at least unknown. This is especially true for machine learning models, especially deep neural network, which have already been demonstrated to be able to succeed in bearing fault detection tasks. Such models might leverage any potential characteristic of the sensor data to base its decision on, and some of them might be spurious. This can lead to overfitting and could cause the model to perform bad in deployment. To prevent this from happening and to ensure trustworthy models with high prediction quality, it is necessary to validate to which ex- tend the model follows the physically grounded routine of do- main experts. More precisely, in the case of fault detection of bearings mounted in a machine, if one wants to check which features are important for the fault detection model 11 via 202213257 Auslandsfassung 19 feature attribution methods it would be ideal to get values of feature importance in terms of the frequency components of an envelope spectrum of sensor data measuring the vibration of the machine close to the bearing. This information would be immediately accessible to domain experts and would make it easy to check whether a fault detection model is in line with the physical understanding of bearing faults or not. In the following, domain maps are provided for three common scenarios of bearing fault detection models (FD models), de- pending on the type of measured sensor data in the input do- main used to train the obtained fault detection model 11 and which serve as input data 10 to the obtained fault detection model 11, see also Table 1 below. First, the fault detection model 11 is trained on signal en- velops in a time domain, given by the amplitude of the ana- lytic signal, e.g., the measured sensor data. In that case, the applied domain mapping φ is a Fourier Transform, so φ(x)=FT(x). Second, the fault detection model 11 is trained on sensor da- ta in the time domain. In order to derive the signal envelope of the sensor data from a time domain signal ^^ it is computed via the amplitude of its analytic signal. This means that the ^{signal envelope ^^^^௩ is mathematically given by ^^^^௩ =} _{| ^^ + ^^ ^^ ^^( ^^)|, where HT resembles the Hilbert transform. The de-} sired envelope spectrum can now be computed via ^^ ^^( ^^ _^^௩ ). The goal is to find a domain map that derives the envelope spectrum from time domain signals while being invertible. The absolute value, however, would violate the requirement. To circumvent this problem, one additionally needs to preserve the phase information of the analytic signal by computing its argument arg ( ^^ + ^^ ^^ ^^( ^^)). This yields to a domain map ^^: ^^ ^ௗ → ^^ ^ௗ × ^^ ^ௗ ^^ ^^ ^^ℎ ^^( ^^) = ( ^^ ^^(| ^^ + ^^ ^^ ^^( ^^)|), arg ( ^^ + ^^ ^^ ^^( ^^)). 202213257 Auslandsfassung 20 Its inverse is given by ^^−1( ^^1, ^^2) = ^^ ^^ ^^( ^^1) cos ( ^^2). This finally enables us to attain a valid semantic representation of time domain signals in terms of their envelope spectrum. Lastly, the fault detection model 11 is trained on sensor da- ta in the frequency domain. In this case the sensor data are first transformed using the inverse Fourier Transform into the time domain and then apply φ specified above for the map- ping from the time domain into frequency domain. The different domain mappings and their inverse transfor- mation for different input data domains to the frequency do- main as selected data domain are summarized in the table be- low. Table 1 Such, the proposed method provides tools to evaluate how well machine learning model trained to detect bearing fault are aligned with existing prior knowledge about the underlying physics. The domain mappings φ can be combined with existing feature attribution methods to estimate to which extend the model has utilized the presence of characteristic fault fre- quencies. Such information is immediately accessible to do- main expert in contrast to uninterpretable importance values on the input domain, i.e., also called raw data, produced by feature attribution methods alone. This is visualized in Fig. 4. 202213257 Auslandsfassung 21 This is visualized in Fig. 4. A fault detection model was trained on measured sensor data signal in the time domain and has detected a bearing fault in the presented signal. On the left, importance values 41, 42 of feature attributions evalu- ated in the time domain are depicted. Again, it is hard to infer any useful information regarding the potential reason of the predicted bearing fault. It is especially not clear whether the fault detection model is aligned with existing domain knowledge, that in case of a fault, a particular peak in the envelop spectrum should be present. However, if the feature attributions are computed based on a semantic repre- sentation, i.e., a selected data domain related to a physical explanation, see right hand side of Fig. 4, it can be veri- fied that the fault detection model put strong emphasis on the relevant fault frequency indicated with a vertical line 46. At least one importance values indicating high importance 45 coincides with the relevant fault frequency 46, wherein the importance values indicating low importance 44 are clear- ly separated. This information is extreme useful to domain experts and can be used to validate or improve the fault de- tection model accordingly. An embodiment of the fault detection apparatus 50 is shown in Fig. 5. The fault detection apparatus 50 comprises a data in- terface 51, configured to obtain sensor data measured at the bearing mounted at a machine 40 as input data. The input data relate to an input data domain. Further, a fault detection model is obtained via the data interface 51. Said fault de- tection model is trained on sensor data related to said input data domain to output a predicted failure value of the bear- ing by processing the obtained sensor data. The machine 10 is a rotating machine, especially a motor, turbine, pump and press. The fault detection apparatus 50 comprises a data mapping unit 52, configured to map the measured sensor data from the input data domain to a selected data domain resulting in an augmented fault detection model which outputs augmented pre- 202213257 Auslandsfassung 22 dicted failure value related to the selected data domain. The selected data domain is configured such that it has a physi- cal meaning to the fault of the bearing. The fault detection apparatus 50 comprises a feature attribu- tion unit 53, configured to perform a feature attribution on the augmented fault detection model quantifying an importance of at least one individual feature of the input data to the augmented failure value related to the selected data domain. The fault detection apparatus 50 comprises a user interface 54, configured to displaying the individual feature of the input data and the respective quantified importance in the selected data domain. It is to be understood that the above description of examples is intended to be illustrative and that the illustrated com- ponents are susceptible to various modifications. For exam- ple, the illustrated concepts could be applied for different technical systems and especially for different sub-types of ^{the respective technical system with only minor adaptions.}