Device and method for 3D tensor completion

Abstract

The invention relates to a device and a method for 3D tensor completion. The device comprises • a device (I / O) for receiving an incomplete 3-dimensional matrix (MI), • a device (CPU) for generating permutations (M1,M2,M3) of the received 3-dimensional matrix, • a device (CPU) for unfolding the permutations (M1,M2,M3) into three 2-dimensional orthogonal matrices (N1,N2,N3) • a device (CPU) for removing elements from the matrices or permutations that have a deviation greater than a predetermined value with respect to neighboring elements, • a completion facility (AI) with a variable number of columns and a variable number of rows of a variable number of values ​​based on a hierarchical adaptation of the prediction space, • a post-processing unit (CPU) for the generated predictions to create an extended 3-dimensional matrix (MO),

Description

background

[0001] With the increasing proliferation of powerful computers, predictions of all kinds are becoming possible. The increasing computing power has enabled practical access to correlations and even predictions using existing training data.

[0002] For example, drug discovery is an expensive undertaking. Therefore, predictions in this area are particularly desirable. Machine learning models have already been developed to predict drug combinations or changes in protein binding affinity.

[0003] In the area of ​​predicting the physical properties of mixtures, machine learning methods are equally promising for very complex or very hazardous mixtures for which there are few experimental data sets.

[0004] Activity coefficients (COEs) are among the data sets that play a fundamental role in the thermodynamics of mixtures and serve as an important tool for the development of reaction media or separation processes. Activity coefficients (COEs) describe, for example, the molecular interactions within a mixture. Their knowledge enables the determination of Henry coefficients, phase equilibria, and the selection of solvents for reactions and separations using physical models such as PC-SAFT.

[0005] Using the example of Fig. 1 this is further clarified.

[0006] Activity coefficients (COE) can be determined, for example, using expensive MES measurements. Measurements are generally slow and expensive, require considerable effort, and can only be performed on a specific scale. Thus, activity coefficients (COE) are subject to significant uncertainty at high to infinite dilutions.

[0007] The activity coefficients (COE) can be determined, for example, using inverse gas-liquid chromatography. These values ​​are also provided in the DECHEMA series, i.e., in a database (DB), but they are usually estimated using thermodynamic models, and experimental data are rarely provided, which makes accurate predictions even more valuable.

[0008] Activity coefficients (COE) can also be predicted using physically based thermodynamic models (ML), but the accuracy often falls outside the bounds of the experimental uncertainty. Such PM models are significantly faster and require less effort than measurements (PM), and as such, they are comparatively inexpensive. However, these models can never be considered in isolation from model parameters, which also limit their scope.

[0009] The most common method for predicting activity coefficients (COEs) is the UNIFAC method, which is a group contribution method. Despite the highly successful UNIFAC method, many (approximately <50%) group parameters are still missing.

[0010] In addition to some other group contribution models, there are also physical models (PM) such as solvation models and linear solvation models. Such models require adjustable parameters.

[0011] Both databases DB and models PM can access measurements MES, so that it is possible to extend a database DB with measurements or to give a model PM an additional support point.

[0012] As from Fig. As can be seen in Figure 1, activity coefficients (COE) can be determined in a variety of ways. In addition to the approaches already mentioned, it is also possible to determine activity coefficients (COE) using trained neural networks (ML). The advantage of trained neural networks (ML) is that they can operate relatively inexpensively and quickly, even over a wide range of sizes, with precision. For example, data from a database (DB) can be used for training or to determine models.

[0013] Machine learning models have the potential to overcome the limitations of traditional physics prediction models by using existing data as input training sets to make predictions on unavailable data sets.

[0014] There are already various machine learning methods for predicting activity coefficients (COE).

[0015] However, the effectiveness of these procedures varies and their applicability is limited to certain conditions.

[0016] For example, the method from the article “Predicting Activity Coefficients at Infinite Dilution for Varying Temperatures by Matrix Completion” by the authors Damay, J.; Jirasek, F.; Kloft, M.; Bortz, M.; Hasse, H. , published in Industrial & Engineering Chemistry Research 2021, 60 (40), 14564-14578. DOI: 10.1021 / acs.iecr.1c02039 as well as “Prediction of Activity Coefficients by Matrix Completion” by the authors Jirasek, F.; Alves, RAS; Damay, J.; Vandermeulen, RA; Bamler, R.; Bortz, M.; Mandt, S.; Kloft, M.; Hasse, H., published in The Journal of Physical Chemistry Letters 2020, 11 (3), 981-985. DOI: 10.1021 / acs.jpclett.9b03657 used a variational inference-based matrix completion method and obtained more accurate predictions for isothermal activity coefficients at 298.15 (±1) K than UNIFAC38.

[0017] The method, known from the article "A neural network approach to predict activity coefficients" by Ramírez-Beltrán, ND; Vallés, HR; Estévez, LA; Duarte, H., published in Can. J. Chem. Eng. 2009, 87 (5), 748-760. DOI: 10.1002 / cjce.20212., demonstrates the potential of artificial neural networks for predicting isothermal activity coefficients.

[0018] Others use graph neural networks based on atoms, bonds, and connectivity to predict isothermal activity coefficients.

[0019] Such neural networks, specifically tailored to the problem, are accurate in their predictions within a limited range, but they are also expensive to develop.

[0020] Similar problems and solutions exist in other areas of technology. However, these solutions are very specific and not generally applicable, which hinders their wider dissemination and application. Task

[0021] Based on this, it would be advantageous to be able to offer solutions that allow a simple process to be used for a variety of applications and, in particular, over a wide range. Brief description of the invention

[0022] This object is achieved by a device and a method according to claim 1 and 2, respectively. Advantageous uses are also specified in the further claims. Further advantageous embodiments are the subject of the description and the figures. Brief description of the characters

[0023] The invention will be described in more detail below with reference to the figures. Fig. 1 a fundamental connection between solution approaches, Fig. 2 a schematic overview of an embodiment of a solution according to the invention, Fig. Figure 3 is a reproduction of a parity diagram of the occupancy rate prediction for car parks in Birmingham between 10 April 2016 and 19 December 2016 using the state-of-the-art 3D-DMF-H algorithm. Fig. 4a a graphical representation of the fill level of the input and output matrices according to the state of the art Fig. 4b a graphical representation of the degree of filling of the input and output matrices after using the solution according to the invention, Fig. Figure 5a shows a parity diagram of the activity coefficient predictions using the UNIFAC method; Fig. Figure 5b shows a parity plot of the activity coefficient predictions using experimental values ​​from the database DB as input to the 3D-DMF-H algorithm. Fig. 6a Vapor-liquid equilibria for the systems 1,4-dioxane / heptane, and Fig. 6b Vapor-liquid equilibria for the systems cyclohexane / ethyl acetate, Fig. 7 an exemplary method implementation according to embodiments of the invention, and Fig. 8 Details of exemplary steps according to embodiments of the invention. Detailed description

[0024] The invention will be described in more detail below with reference to the figures. It should be noted that various aspects are described, each of which can be used individually or in combination. This means that any aspect can be used with different embodiments of the invention, unless explicitly presented as a mere alternative.

[0025] Furthermore, for the sake of simplicity, reference will generally only be made to one entity in the following. Unless explicitly stated, the invention may also comprise several of the entities in question. Therefore, the use of the words "a," "an," and "another" is to be understood merely as an indication that at least one entity is used in a simple embodiment.

[0026] The following description refers to the illustrations. As a rule, identical or similar elements are designated by identical or similar reference symbols.

[0027] Although the invention is presented below in the context of activity coefficients, the invention is not limited to this. Rather, the inventors have recognized that the solution can be generalized to a variety of other problems. Therefore, reference will be made below to tensor completion, in which the determination of activity coefficients represents one application.

[0028] Tensors are an effective method for representing multidimensional real-world data, such as images, videos, numerical ratings, social networks, and knowledge bases.

[0029] However, since real-world data are often incomplete, missing values ​​pose a challenge for statistical analysis. Tensor completion is a technique for predicting and filling these missing values ​​in partially observed tensors.

[0030] Low-rank tensor factorization is a popular tensor completion technique, but it has limitations in capturing nonlinear patterns and handling sparse data.

[0031] To overcome these limitations, recent work has proposed neural network-based tensor completion methods that have high expressive power and generalization ability.

[0032] In this invention, inspired by the critical need for accurate predictions in tensor completion problems, a solution is given that is both simple and flexible, allowing for the easy inclusion of additional data points.

[0033] More generally, a neural network-based matrix completion method is provided for dealing with nonlinear data structures.

[0034] This is illustrated below using the example of a three-dimensional matrix required to predict activity coefficients at infinite dilution: temperature, solute and solvent.

[0035] Accordingly, a device according to the invention for 3D tensor completion comprises a device I / O for receiving an incomplete 3-dimensional matrix MI (see Fig. 2).

[0036] The I / O device for receiving an incomplete 3-dimensional matrix MI can be a conventional interface to a wired or wireless data transmission network. In particular, the I / O device for receiving an incomplete 3-dimensional matrix MI can also be used to read an incomplete 3-dimensional matrix MI from a local or remote database DB. Likewise, the I / O device for receiving can also be used to obtain individual measured values ​​to complete an incomplete 3-dimensional matrix MI, should these become available in the meantime through measurement or from another source.

[0037] Furthermore, the device according to the invention comprises a device CPU for generating permutations M1, M2, M3 of the received 3-dimensional matrix.

[0038] In addition, the device according to the invention has a device CPU for unfolding the permutations M1,M2,M3 into three 2-dimensional orthogonal matrices N1,N2,N3.

[0039] Furthermore, the CPU device can also be designed to remove elements from the matrices or permutations which have a deviation greater than a predetermined value with respect to neighboring elements.

[0040] Furthermore, the device according to the invention comprises a device for completion A1 with a variable number of columns and a variable number of rows of a variable number of values ​​based on a hierarchical adaptation of the prediction space. This device for completion A1 is, for example, an appropriately trained neural network, which, however, is not further relevant for understanding the invention. The device for completion A1 can also be implemented using a conventional (appropriately programmed) microprocessor, but also as an ASIC or FPGA or another suitable computing unit, in particular also as part of the CPU device.

[0041] The device according to the invention further comprises a device for post-processing CPU of the generated predictions in order to create an extended 3-dimensional matrix MO.

[0042] This CPU device can be a conventional (appropriately programmed) microprocessor but also an ASIC or FPGA or another suitable computing unit.

[0043] Accordingly, a device according to the invention for 3D tensor completion has a device I / O for providing the extended 3-dimensional matrix MO (see Fig. 2).

[0044] The I / O device for providing the extended 3-dimensional matrix MO, like the I / O device for receiving an incomplete 3-dimensional matrix MI, can be a conventional interface to a wired or wireless data transmission network. In particular, an extended 3-dimensional matrix MO can also be input into a local or remote database DB via the I / O device for providing the extended 3-dimensional matrix MO. In particular, the I / O device for providing the extended 3-dimensional matrix MO can also be implemented jointly with the I / O device for receiving an incomplete 3-dimensional matrix MI.

[0045] According to one embodiment of the invention, a method for 3D tensor completion is also provided. This method can be designed in various ways and allows parallel processing, at least in part.

[0046] The method according to the invention comprises the step of receiving 100 an incomplete 3-dimensional matrix MI (a three-dimensional tensor).

[0047] In a step 200, (parallel or successive) permutations M1, M2, M3 of the received, incomplete 3-dimensional matrix MI are generated.

[0048] For example, predefined mathematical functions in common mathematical calculation programs can be used for this purpose. One example is the "permute" function in Mathlab. This function can create the following (3! = 6) permutations (e.g., [1 2 3], [1 3 2], [2 1 3], [2 3 1], [3 1 2], [3 2 1]). Note that one of the permutations can also be the incomplete 3-dimensional matrix MI.

[0049] In a further step 300, the permutations M1, M2, M3 are expanded into three 2-dimensional orthogonal matrices N1, N2, N3. It should be noted at this point that due to the expansion from three-dimensional space into two-dimensional space in the previous example of the 6 permutations corresponding to [ABC] -> [AB*C], the resulting 2D input matrices [1 2 3] and [1 3 2], [2 1 3] and [2 3 1], or [3 1 2] and [3 2 1] are identical, so that the actual number of usable permutations is reduced to 3.

[0050] In a preprocessing step 400, elements are removed from the matrices or permutations that have a deviation greater than a predetermined value with respect to neighboring elements.

[0051] In step 500, completion is performed (using a neural network A1) with a variable number of columns, a variable number of rows, and a variable number of values ​​based on a hierarchical adaptation of the prediction space.

[0052] Although the invention is described in such a way that the number of row and column entries should be identical, this is not a requirement, but it can also be provided that these values ​​are decoupled from each other so that the number of predictions can be further increased.

[0053] This hierarchical approach will be further explored in the following, focusing on the Fig. 2 as well as 7 and 8, whereby in Fig. 8 of the Fig. 7 dashed part and the elements before and after the Fig. 7 are shown.

[0054] Assume that the matrix after preprocessing has a maximum number of rows and columns (t num ) of matrix entries. This means that in each row and column there are at least t num Data points must be entered.

[0055] With t num entries per row and column, the resulting input matrix is ​​usually well filled, which improves the accuracy of the predictions.

[0056] We start with the maximum t num , which starts the first run. The predictions are then (temporarily) stored in a database, for example. After that, t num reduced by 1 and a new run started.

[0057] After this, all predictions that did not occur in the previous run (or in any of the previous runs) are (temporarily) stored in the database.

[0058] If this is continued, the matrices will benum larger and the ratio between existing values ​​and empty fields becomes worse.

[0059] In embodiments of the invention, when making a prediction in 2-dimensional spaces, a point can in principle be filled multiple times with created values.

[0060] In such a case, different approaches can be used. A value can be retained, overwritten, or averaged or evaluated to determine which value is (temporarily) stored in the database.

[0061] An example evaluation can be done using the formula Lj=kM DM⋅∑iH(vij−v˜j(1+tmed)v˜j)+kCVvJ¯⋅cVmax1nj⋅∑inj(vij−vj¯)2 This way, each prediction can be assigned a value, and in the case of multiple predictions, the prediction with the lowest value can be used. The calculation depends on the input matrix and is therefore performed before the predictions are calculated. This means that, in principle, one can determine where there are multiple predictions and, if there are multiple predictions, which permutation produces the best prediction without calculating the predictions. This approach also avoids outliers.

[0062] For example, the selection based on the value can also be part of the post-processing in step 600.

[0063] Subsequently, in step 600, the generated predictions are post-processed (using the CPU device) to create an extended 3-dimensional matrix MO.

[0064] The thus created extended 3-dimensional matrix MO is then provided in step 700.

[0065] The methodology of the invention will be discussed again below. This is based on the deep matrix factorization (DMF) approach proposed by Fan and Cheng (Fan, J.; Cheng, J. Matrix completion by deep matrix factorization. Neural Networks 2018, 98, 34-41. DOI: 10.1016 / j.neunet.2017.10.007), i.e., a matrix completion method (abbreviated to MCM) in which missing entries of matrices are filled.

[0066] While common MCM methods use a low rank of an incomplete matrix X ∈ ℝ m×n and assume a linear latent variable model, the DMF approach uses a nonlinear mapping f and latent variable z ∈ ℝ r , with r = rank(X). This approach proves to be more effective for creating the missing entries of the matrix X, especially when the actual data set is nonlinear.

[0067] Expressed in matrix form, with the columns of Z formed from the latent variables z, X=f(Z)

[0068] The DMF approach uses a multi-layer neural network Al, in which a high-dimensional, considered matrix X is factorized into low-dimensional latent inputs and weights. This approach uses unconventional neural networks Al, in which the input is unknown and the output is incomplete, whereas in conventional neural networks, both the input and the output are both complete and determined.

[0069] Since the nonlinear mapping f is applied individually to each column of the latent variable matrix Z, the accuracy of the prediction depends on the number of entries in the columns of X, with only one entry per row being required.

[0070] In the method according to the invention, hereinafter also referred to as 3D-DMF-H, the input matrix is ​​first transformed. In order to make predictions for a three-dimensional matrix G ∈ ℝ a×b×c To calculate the matrix, the matrix is ​​transformed into one of several different two-dimensional permutation matrices N1, N2, N3. For example, in Fig. 2 the left permutation M1 according to G in H ∈ ℝ a×bc transformed for further processing.

[0071] The input parameter—as already explained above—plays a prominent role within the inventive method. This parameter serves as a threshold for the minimum number of row and column entries in the input matrix, below which the corresponding rows or columns are deleted. By dynamically adjusting this value in the 3D-DMF-H algorithm, the calculation time is reduced compared to a static value, which is described in more detail in the post-processing section.

[0072] To improve the prediction probability, preprocessing 400 can be performed. This is due to the fact that neural networks exhibit AI properties, which lead to less accurate predictions for columns when the values ​​​​vary greatly (e.g., [0.514, 2.81, 7.36]) compared to values ​​​​that are closer together (e.g., [0.426, -0.232, 0.147]).

[0073] One way to describe deviations is the coefficient of variation c V However, this has the property that high c V -values ​​can occur if only one value deviates significantly from the others. Therefore, within the scope of the invention, a combination of c V and the deviation from the median (MD).

[0074] To determine whether c V for each column of the input matrix H is above a certain threshold, a threshold t CV This value can be determined experimentally. A possible limit is t CV ≈ 1.3 If this limit is exceeded, this column is a candidate for deletion.

[0075] The median deviation MD is calculated by taking the median of a column ṽ J from an entry in column v ij subtracted and divided by the median of the column, as shown in equation (2). MDij=vij−vJ˜vJ˜ If this exceeds a threshold value t med , the Heaviside function (H) is used to calculate the median deviation indicator MDI according to equation (3) MDIij=H(vij−vJ˜⋅(1+tmed)vJ˜) which returns a 0 if the deviation of the entry is small enough, or a 1 if the deviation of the entry is too large. The threshold value t med can again be chosen appropriately. A possible threshold value is t med ≈ 0.95.

[0076] Then the results of each column are aggregated and compared with a given threshold t MD compared. If the total deviation t MD , the column is marked for possible deletion. A possible limit is t MD ≈ 0.2.

[0077] Each column that contains both t CV as well as t MD will then be deleted.

[0078] It may still be the case that certain rows contain a (larger) number of values ​​that contribute to large deviations, even though the corresponding columns do not meet the criteria for deletion according to the first method.

[0079] To avoid problems, these rows can be considered for deletion if the ratio of the sum of the median deviation indicators to the number of entries in a row exceeds a certain limit. A possible limit is approximately 0.125, i.e., for Σ i MDI ij > 0.125 the corresponding lines are deleted.

[0080] That is, if more than 12.5% ​​of the row entries have an MDI of 1, the row is eliminated. A chosen threshold of 0.125 represents a balance between the size of the resulting matrix and the prediction accuracy.

[0081] After the above deletions have been performed, it is confirmed that each row and column still contains at least t num contains entries. If this condition is not met, the preprocessing step is repeated until the threshold is reached.

[0082] Typically, this process requires no more than two iterations.

[0083] Despite its effectiveness in ensuring that the matrix t num requirement is met, this preprocessing approach may result in the loss of some predictions.

[0084] To further solve this problem, the matrix is ​​layered and resolved in two additional ways based on the Fig. approach presented.

[0085] As mentioned above, there are multiple two-dimensional permutations of the input matrix. For a three-dimensional matrix, this approach can generate three different permutations (out of a total of 3! = 6 permutations), each of which can be used as input to the algorithm. This extension allows the algorithm to predict more values ​​than would be possible with a single permutation.

[0086] After the algorithm is completed, the predictions from the three permuted input matrices are combined to create the final prediction matrix in the post-processing step.

[0087] However, since the predictive values ​​of the different permutations overlap considerably, a formula (equation (4)) was developed to determine the selection of the individual predictive values ​​from the three permutations.

[0088] The formula used to determine the appropriate selection of each predictor from the three permutations has two components. The first component is the sum of the median deviation indicators (MDI) of a column, as shown in equation (3). The second component is the coefficient of variation (c V ) of a column divided by the maximum coefficient of variation of all columns (cVmax), which represents the variance of the column. By adjusting the parameters k MDI and k cv the two components can be weighted differently.

[0089] Although it is possible e.g. k cv ≈ 0.5 as well as k MDI ≈ 0.5, within the scope of the invention, a pair of values ​​of k cv ≈ 0.25 as well as k MDI ≈ 0.75 preferred. Lj=kMDM⋅∑iH(vij−vJ˜(1+tmed)v˜J)+kCVv¯j⋅cVmax1nj⋅∑inj(vij−vJ¯)2

[0090] Using the formula, the loss function L j The prediction that has the lowest L j -value is finally selected for inclusion in the final matrix.

[0091] The 3D-DMF-H algorithm uses a dynamic t num . The maximum t num -value, which allows the creation of a matrix for each permutation, since a larger t num value generally corresponds to more accurate predictions.

[0092] The algorithm starts with tnummax for the given permutation, generates and stores predictions and then reduces t num at each subsequent iteration by 1, until t num = 2 is reached. Since at least 2 entries per row and column are required for the prediction, the process is terminated after this pass.

[0093] To ensure high accuracy, the previously predicted values ​​are omitted in each new run. In this way, each predictable matrix entry is assigned a value of - compared to a constant t num -value - best possible accuracy. The accuracy of the predictions is determined by the t num -value and decreases with decreasing t num -value.

[0094] Due to the different data structure of each of the three permutations, the resulting error curves are slightly different for each permutation. These individual curves can be averaged to create a comprehensive error curve for the final prediction matrix. For this purpose, the error and error prediction bounds (e.g., at the 95% confidence interval) for each permutation can be calculated using robust linear regression.

[0095] Returning to the application example, in order to make the predictions usable for PC-SAFT, for example, the data sets for the binary system under consideration can be extracted from the prediction matrix.

[0096] Depending on the availability of predictions for the binary system in question, either one or two data sets can be extracted (representing component 1 in component 2 and / or vice versa).

[0097] Since there is not a prediction for every possible temperature, a robust linear regression is used to model the temperature dependence of the activity coefficients γij∞ for the corresponding data set. The function describing the temperature dependence of the activity coefficients often has the following form, where A ij and B ij are system-specific constants for the solute i in the solvent j at temperature T in Kelvin. lnγij∞(T)=Aij+BijT

[0098] To calculate the parameter k ij a simplex algorithm can be used (for example with the Matlab function fminsearchbnd).

[0099] First, the activity coefficient for the system under consideration can be calculated using the PC-SAFT tool at a specific temperature with an initial k ij = 0. If two data sets are available, two activity coefficients can be calculated and the k ij is adjusted iteratively with the following cost function, since k ij = k ji applies: C=(1−γ1PCSAFTγ13DDMFH)2+(1−γ13DDMFHγ1PCSAFT)2+(1−γ2PCSAFTγ23DDMFH)2+(1−γ23DDMFHγ2PCSAFT)2

[0100] In case there is only one data set (i.e. only component 1 in 2 or vice versa), formula (6) can be reduced to the following formula: C=(1−γ1PCSAFTγ13DDMFH)2+(1−γ13DDMFHγ1PCSAFT)2

[0101] The optimized k ij -parameter can then be used to calculate the VLEs.

[0102] The invention has already been successfully tested.

[0103] Examples of this are the completion of activity coefficients outlined above as well as the detection of free parking spaces in congested parking areas.

[0104] In the example of congested parking areas, the following scenario can be assumed: A 3D tensor is required to represent occupancy rates in parking garages by time of day (1), different parking garages (2), and different days (3). Within this dimension, partial historical data exists, but it does not cover every point in time.

[0105] Using the invention, it is now possible to obtain predictions for unobserved data points by exploiting suitable correlations between existing data. For the prediction, for example, a publicly accessible database (see Stolfi, DH; Alba, E.; Yao, X. Predicting Car Park Occupancy Rates in Smart Cities. In Smart Cities; Alba, E., Chicano, F., Luque, G., Eds.; Lecture Notes in Computer Science; Springer International Publishing, 2017; pp. 107-117. DOI: 10.1007 / 978-3-319-59513-9_11., therein Database 1) with 104,067 data points (after data processing) can be used, from which the occupancy rate of various parking garages on different days and times between 0 and 1 can be calculated.

[0106] This dataset represents the occupancy rate of 31 different parking garages for times between 8:00 a.m. and 4:30 p.m. in the period from April 10, 2016, to December 19, 2016.

[0107] With this data, a reliable prediction model can be developed based on the invention that can accurately predict the future occupancy of these parking garages.

[0108] For this purpose, a neural network with an intermediate layer (so-called hidden layer) and a logistic function as activation function can be used.

[0109] Fig. shows the parity diagram with an R 2 of 0.975 and a mean absolute error (MAE) of 0.023, corresponding to an average error of 2.3%, demonstrating the effectiveness of the approach in predicting parking garage occupancy.

[0110] The invention was also used to predict activity coefficients. First, the complete DDB (109,435 data points – referred to as "Database 1") was used, corresponding to 1,213 different solutes, 1,498 different solvents, and 268 different temperatures in the range from 126.15 K to 1,273.15 K, to create the largest prediction tensor currently possible. Temperatures were rounded to integer values, and the activity coefficients were logarithmized, since the range of considered values ​​is 0 < γ. ∞ ij (T) < 25,000,000. The letters a, b and c were used as indices for the matrix entries for the activity coefficients γ ∞ ij (T) of the component i infinitely diluted in the solvent j at the temperature T: a stands for the solute index, b for the solvent index and c for the temperature index.

[0111] A total of 7,587,105 data points were predicted using the invention, resulting in a database with 7,696,540 data points. Fig. 4a, the filling level of the input matrix is ​​shown only with experimental values ​​(109,435 entries), while in Fig. 4b shows the filling level of the final prediction matrix with 7,696,540 entries, including experimental values.

[0112] The input "Database 2" represents only 1.42% of the prediction database. This figure is sufficient, as a similar model based on deep learning and collaborative filtering was validated with only 0.07% of experimental values.

[0113] To validate the activity coefficients predicted by the invention, a subset of the Dortmund Database (DDB) containing 42,851 data points was selected (= Database 3). The activity coefficients were predicted using both the invention and the thermodynamic benchmark model UNIFAC for Database 3, which contained 832 different solvents, 980 solutes, and 239 temperatures between 213 K and 576 K. Database 3 was determined as a subset of Database 2 based on the available UNIFAC group contribution values, i.e., 42,851 of the original 109,435 data points can be predicted by UNIFAC, which corresponds to approximately 40%. After completing the data preprocessing by the invention, the number of usable data points was reduced by 14% from 42,851 to 37,111, mainly to achieve higher accuracy (see previous discussion). These data points were used as input for the invention, resulting in 3,036.461 predicted values ​​for γ. ∞ ij can be provided.

[0114] Thus, the data points from DDB database 3 only account for 1.22% of the complete prediction matrix. Fig. shows the parity plot for this ~ 3 million predicted data points with the invention compared to the corresponding predictions with UNIFAC.

[0115] The Fig. 5a the parity diagram of the activity coefficient predictions using the UNIFAC method during Fig. Figure 5b shows the parity diagram of the activity coefficient predictions using experimental values ​​from database 3 as input to the invention.

[0116] The parity diagram for the invention has a coefficient of determination of 0.99 and an MAE of 0.109, while the calculations using the UNIFAC method have a significantly higher MAE of 0.647 and a significantly lower coefficient of determination of 0.836.

[0117] For about 73% of the 37,111 data points, the logarithmic activity coefficients In γ ∞ ij predicted with an absolute error of less than 0.1 (using k-fold cross-validation as described previously); about 92% of the 37,111 data points were predicted with an absolute error of less than 0.3. These results are promising, as the experimental uncertainty for In γ ∞ ij is usually 0.1 to 0.2 and predictions of the invention therefore provide accuracies within the experimental uncertainty.

[0118] The Fig. 4a, Fig. 4b and Fig. 5a, Fig. 5b show that a variety of In γ ∞ ij (T) values ​​can be determined with the invention and that the results are of satisfactory accuracy.

[0119] Furthermore, a hybrid modeling approach can be applied in the invention, in which the In γ predicted by the invention ∞ ij (T) values ​​are used as input data to adjust the model parameters of a physically based thermodynamic model to ultimately model phase equilibria.

[0120] The results of the VLE modeling could be investigated using the PC-SAFT equation of state, and the binary interaction parameter (k ij ) was calculated using the In γ predicted by 3D-DMF-H ∞ ij (T) values ​​were estimated. The results are shown in Fig. 6a and Fig. 6b for two selected mixtures. Fig. 6a the vapor-liquid equilibria for the system 1,4-dioxane / heptane and Fig. 6b shows the vapor-liquid equilibria for the cyclohexane / ethyl acetate system. The VLEs were measured with PC-SAFT either with the adjusted kij (solid line) from the 3D-DMF-H predicted activity coefficients or with k ij = 0 (dotted line). For comparison, the corresponding literature data (circles) are provided for each system.

[0121] The selected mixtures show azeotropic phase behavior, and the occurrence and position of azeotropes in mixtures ( Fig. 6a and Fig. 6b) can be accurately predicted using PC-SAFT, where k ij to the In γ predicted by the invention ∞ ij (T) values ​​are adjusted.

[0122] Both the cyclohexane / ethyl acetate system in Fig. 6a as well as the system 1,4-dioxane / heptane in Fig. 6b show a significant improvement in the prediction accuracy, with the occurrence and position of the azeotrope as well as the vapor-liquid region being adjusted with k ij could be accurately predicted, while PC-SAFT predictions with k ij= 0, neither the azeotrope nor the vapor-liquid region can be correctly predicted. Comparison with literature data supports the value of machine learning for the chemical industry, as VLE predictions can be made for various mixtures without experimental effort. However, future work should conduct a systematic analysis of the algorithm's structure and parameters to identify and improve current limitations.

[0123] Without limiting its generality, the invention in its embodiment of a device or in its embodiments of the method can be used for a variety of purposes, but in particular for the completion of tensors in the chemical field, in particular in the pharmacological field.

[0124] Other purposes are not explicitly excluded by this, but are included in the invention.

[0125] The inventive solution shows a high accuracy in predicting a large number of activity coefficients by hierarchical (-H) adaptation of the prediction space.

[0126] The invention enables accurate predictions using a simple method. The inventive solution is not limited to a specific application, such as activity coefficients, but rather is suitable for all tensor completion problems.

[0127] The inventive solution is easily extendable to include more data points, making predictions more accurate and, for example, generating new mixture data.

[0128] The inventive solution shows a high accuracy in the prediction through a hierarchical adaptation of the prediction space.

[0129] Without loss of generality, it should be noted that processing in two-dimensional spaces, ie preprocessing, completion and postprocessing, can in principle also be carried out in three-dimensional space, but places considerably higher demands on performance. QUOTES CONTAINED IN THE DESCRIPTION

[0000] This list of documents submitted by the applicant was generated automatically and is included solely for the convenience of the reader. This list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions. Cited non-patent literature

[0000] Predicting Activity Coefficients at Infinite Dilution for Varying Temperatures by Matrix Completion" by Damay, J.; Jirasek, F.; Kloft, M.; Bortz, M.; Hasse, H., published in Industrial & Engineering Chemistry Research 2021, 60 (40), 14564-14578. DOI: 10.1021 / acs.iecr.1c02039

[0016] Prediction of Activity Coefficients by Matrix Completion" by the authors Jirasek, F.; Alves, RAS; Damay, J.; Vandermeulen, RA; Bamler, R.; Bortz, M.; Mandt, S.; Kloft, M.; Hasse, H., published in The Journal of Physical Chemistry Letters 2020, 11 (3), 981-985. DOI: 10.1021 / acs.jpclett.9b03657

[0016] A neural network approach to predict activity coefficients" by the authors Ramírez-Beltrán, ND; Vallés, HR; Estévez, LA; Duarte, H. , published in Can. J. Chem. Eng. 2009, 87 (5), 748-760

[0017] Fan, J.; Cheng, J. Matrix completion by deep matrix factorization. Neural Networks 2018, 98, 34-41. DOI: 10.1016 / j.neunet.2017.10.007

[0065] Stolfi, D. H.; Alba, E.; Yao, X. Predicting Car Park Occupancy Rates in Smart Cities. In Smart Cities; Alba, E., Chicano, F., Luque, G., Eds.; Lecture Notes in Computer Science; Springer International Publishing, 2017; pp 107-117. DOI: 10.1007 / 978-3-319-59513-9_11

[0105]

Claims

[1] Apparatus for 3D tensor completion, comprising • A device (I / O) for receiving an incomplete 3-dimensional matrix (MI), • A device (CPU) for generating permutations (M1,M2,M3) of the received 3-dimensional matrix, • A device (CPU) for unfolding the permutations (M1,M2,M3) into three 2-dimensional orthogonal matrices (N1,N2,N3) • A device (CPU) for removing elements from the matrices or permutations that have a deviation greater than a predetermined value with respect to neighboring elements, • A completion facility (AI) with a variable number of columns and a variable number of rows of a variable number of values ​​based on a hierarchical adaptation of the prediction space, • A post-processing unit (CPU) for the generated predictions to create an extended 3-dimensional matrix (MO), • A device (I / O) for providing the extended 3-dimensional matrix (MO). [2] A method for 3D tensor completion, comprising the steps • Receiving an incomplete 3-dimensional matrix (MI), • Generating permutations (M1,M2,M3) of the received 3-dimensional matrix, • Unfolding the permutations (M1,M2,M3) into three 2-dimensional orthogonal matrices (N1,N2N3) • Removing elements from the matrices or permutations that have a deviation greater than a predetermined value with respect to neighboring elements, • Completion (Al) with a variable number of columns and a variable number of rows of a variable number of values ​​based on a hierarchical adaptation of the prediction space, • Post-processing (CPU) of the generated predictions to create an extended 3-dimensional matrix (MO), • Providing the extended 3-dimensional matrix (MO). [3] Use of a device for 3D tensor completion according to claim 1 for the completion of tensors in the chemical field, in particular in the pharmacological field. [4] Use of a method for 3D tensor completion according to claim 2 for the completion of tensors in the chemical field, in particular in the pharmacological field.

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