Method for creating a model ensemble

Abstract

In order to create a model ensemble that accurately estimates the output (y) of a physical process over the input range (U) using a small number of available data points, and thus with a minimum of test bench trials, it is proposed that for each model (Mj) of the model ensemble (1) an empirical complexity measure (cj), which evaluates the deviation of the model output (y^j) from the output (y) of the real physical process over a given input range (U), and a model error (Ej) be determined, and from the empirical complexity measure (cj) and the model error (Ej) a surface information criterion (SICj, SIC) is formed, from which the weighting factors (wj) for the model ensemble (1) are determined.

Description

METHOD FOR CREATING A MODEL ENSEMBLE FOR CALIBRINGING A CONTROL UNIT The present invention relates to a method for creating a model ensemble that estimates at least one output variable of a physical process as a function of at least one input variable, wherein the model ensemble is formed from a sum of the model output variables of a plurality of models, each weighted with a weighting factor. When developing internal combustion engines, legal requirements regarding emissions, particularly NOx, soot, CO, CO2, etc., and fuel consumption must be considered. To this end, the engine control unit (ECU) of an internal combustion engine is calibrated during development to ensure that these requirements are met during operation. Calibration in this context means that certain control parameters of the internal combustion engine, such as the air / fuel ratio, exhaust gas recirculation into the cylinder, ignition timing, etc., are predefined depending on a specific engine condition, such as torque, speed, coolant temperature, etc. For this purpose, corresponding maps are stored in the ECU, which are read during engine operation to determine the control parameters for a given condition.Due to the many influencing factors, calibration is a very time-consuming and expensive process that has primarily been carried out on specialized test benches. For this purpose, the combustion engine was mounted on a test bench and connected to a load machine that simulates specific, predefined load conditions. Typically, predefined load cycles (driving cycles) are run on the test bench. During these load cycles, the combustion engine's emissions and / or fuel consumption are measured according to the current state. After evaluating the measured values, the control parameters in the engine control unit (ECU) are adjusted, and the process is repeated until a satisfactory calibration is achieved. However, the time spent on the test bench is very expensive and should be reduced as much as possible. Therefore, methods have already been developed to simplify calibration, particularly to save test bench time. These methods are often based on models of the combustion engine's emission or fuel consumption behavior, or more generally, on models of a physical process. The challenge here is to determine sufficiently accurate models of the physical process that can then be used to calibrate the control unit. Methods for the automated model identification of nonlinear processes (e.g., NOx emissions or fuel consumption) have already been published, as described, for example, in WO 2013 / 131836 A2. These methods are each based on a predefined model structure, such as a neural network, a kriging model, or a linear model network. The chosen model structure defines model parameters that are determined by the automated model identification methods.For this purpose, data in the form of measured values ​​are acquired on a test bench, and the model is parameterized or trained based on this data. Consequently, only a small number of test runs with a real combustion engine on a real test bench are necessary. With the trained model, the influence of specific control parameters on emissions or fuel consumption can then be investigated without the need for further test bench trials. The dissertation by Hartmann B., "Local Model Networks for the Identification and Experimental Design of Nonlinear Systems," University of Siegen, January 2014, discusses linear model networks as a model structure in detail. As is well known, valid local models are defined for sub-ranges of the input variable domain in linear model networks. The output of the linear model network over the entire input variable domain then results from the sum of the outputs of the local models, weighted by validity functions.A local model therefore only estimates a locally valid output variable, or a part of the output variable of the model network. However, selecting the best model structure for modeling a specific behavior of an internal combustion engine (e.g., NOx emissions) is already challenging and not immediately obvious. For this reason, so-called model ensembles have also been used. In this approach, various models are trained and then weighted to obtain the best possible estimate for a specific behavior of an internal combustion engine (e.g., emissions or fuel consumption). The output of the overall model (the model ensemble) is thus a weighted sum of the outputs of the individual models. Therefore, the weighting factors for a model ensemble must be determined. A frequently used method for determining the weighting factors is based on the Akaike information criterion, as described in Akaike, H., "Information theory and an extension of the maximum likelihood principle," Proceedings 2nd International Symposium on Information Theory, Budapest 1973, pp. 267-281. Hartmann's dissertation also describes model ensembles with weighting factors based on an Akaike information criterion. The Akaike information criterion is used to assess the plausibility of a model Mj. For each model Mj, the model error Ejund and the complexity of the model Mjin are determined in the form AlCj = N · log MSEj + 2 · [α] · p■ is evaluated. The model error Ej evaluates the deviation of the model output y of model Mj from the actually measured output y of the process. In the Akaike Information Criterion (AIC), the mean squared error MSEj of model Mj is used as the model error Ej, based on N different, known input variables u of model Mj. The mean squared error MSEj of the j-th model Mj is known to be calculated as MSE = - y, (u;)) <2> . In the Akaike information criterion, complexity is simply assessed as the number Pj of model parameters of the j-th model Mj. There are also known variations; for example, in the case of a neural network as the model structure, the number of effective parameters, which can be calculated in a known manner (not detailed here), is often used to assess the complexity of the j-th model Mj. It is also known to weight complexity with a factor α (represented by Cjan in the equation above for AI), the so-called risk aversion parameter. The weighting factors Wj for the individual models Mj are then determined according to the associated —AICj certain plausibility according to the Akaike information criterion AICj with w - = e <2> , calculated and normalized to 1. The problem with this Akaike Information Criterion (AIC) is that while it can be calculated quickly, it is designed for a large number N of known data points (measured output y with respect to a specific input u). These known data points are often also the data points used to train the models Mj. Furthermore, the model structures (for the number of parameters pj) of the models Mj must be known in order to calculate their plausibility and thus their weighting factors Wj. For the application in question, however, the smallest possible number of known data points is desirable, as the effort required for test bench trials and measurements on the test bench should be reduced as much as possible. For a smaller number of data points N, an adapted Akaike information crisis was used. (p, + l) - (p,. + 2) Theorium in the form AICc = AIC + 2— was proposed. But this also- N -P- - 2 The Akaike information criterion, when applied to the small number of data points present here, yields unsatisfactory results. For the purposes of this invention, a small number of data points is defined as a number N comparable to the number Pj of the model parameters, i.e., N ≤ p, where N and Pj preferably have the same order of magnitude. In particular, an objective of the invention is to minimize the number N of measured data points in order to reduce the required number of test bench trials or measurements. Furthermore, in this application, the models Mj are often already fully trained, meaning their structure is unknown. The models Mj of the model ensemble can therefore sometimes exist as an unknown black box.The well-known Akaike Information Criterion (AIC) cannot be applied to such unknown models for creating a model ensemble, as the number of model parameters is unknown. Due to these disadvantages of the Akaike Information Criterion (AIC), it cannot be used, or at least not satisfactorily, for creating a model ensemble under the given conditions (small number of available data points, possibly no knowledge of the model structures). It is therefore an object of the present invention to provide a method for creating a model ensemble that requires a small number of available data points and thus a minimum of test bench trials, and that does not require knowledge of the model structures of the models of the model ensemble. In order to determine the weighting factors for a good model ensemble for such a small number of actually measured, available data points and for partially unknown models, the invention provides an empirical complexity measure for each model, which evaluates the deviation of the model output from the output of the real physical process over a predefined input range, and a model error. A surface information criterion is then formed from the empirical complexity measure and the model error, from which the weighting factors for the model ensemble can be determined. This does not evaluate the model structure as in the Akaike information criterion, but rather uses an empirical complexity measure that assesses the complexity of a model based on the model's deviation from the underlying physical process.This approach evaluates not only the deviation at the measured data points (model error), but also the deviation between these data points, i.e., across an entire range of input variables, which is reflected in the empirical complexity measure. This eliminates the need for knowledge of the model structures of the models within the model ensemble. Using the empirical complexity measure also significantly reduces the number of required data points, thus also greatly shortening the time required on the test bench for measuring these data points. The model error of a j-th model can be easily and quickly determined as the mean squared error between the output variables (data points) of the physical process measured at input variables and the model output variables calculated at these input variables, according to the relationship MSE = -y, (u;)). <2> be-. It will be calculated. The empirical complexity measure of a j-th model becomes particularly advantageous with the formula c- = be- This is calculated. Using these empirical complexity measures, a particularly good model ensemble can be identified, which is especially better than any single model within the model ensemble. In order to have a degree of freedom for determining the weighting factors, the empirical complexity measure is preferably weighted with a complexity aversion parameter. In a simple embodiment of the invention, the weighting factors of each Mo- --SlCj dells of the model ensemble from the formula w - = e <2> calculate. In a particularly advantageous embodiment of the invention, it is provided that for the model ensemble the surface information criterion is formed from an error matrix containing the model errors of the models and a complexity measure matrix containing the empirical complexity measures of the models Mjent, wherein the error matrix and the complexity measure matrix are calculated according to the formula SIC = |w <t>Fw + w <t>The Cw| values ​​are each weighted twice by a weighting vector containing the weighting factors of the models, and the surface information criterion of the model ensemble is minimized with respect to these weighting factors. This optimization allows the determination of weighting factors for the model ensemble that result in a particularly small error between the model's output variable and the real physical process. It is advantageous to calculate the error matrix as the product of a matrix E, where the matrix is ​​calculated using the formula E = ^(ιι;) - ^(ιι;)). It is particularly advantageous if the complexity measure matrix is ​​weighted by a complexity aversion parameter, as this provides an additional degree of freedom by which the errors between the model output of the model ensemble and the real physical process can be further reduced. In a particularly advantageous embodiment of the invention, it can be provided that the weighting vectors for different complexity aversion parameters are calculated and the weighting vector corresponding to a selected complexity aversion parameter is chosen as the optimal weighting vector for the model ensemble, or that the weighting vector for different complexity aversion parameters is calculated and the relationship J w^ Fwa+— a is used to determine the optimal weighting vector. <2> w <t>ap l regarding the various complex K <k> N <k>J The calculated weighting vectors are minimized using the xity aversion parameter. The present invention is explained in more detail below with reference to Figures 1 to 5, which show exemplary, schematic, and non-limiting advantageous embodiments of the invention. Fig. 1 shows a model ensemble with a plurality of models weighted with weighting factors, Fig. 2 shows the approximation of data points by models of varying complexity, Fig. 3 shows the training and validation error as a function of the number of model parameters, and Figs. 4 and 5 show the effect of the calculation of the weighting factors according to the invention. A model ensemble 1, as shown in Fig. 1, consists of a number j of models Mj. Each model Mj is defined by a model structure, for example, a neural network, a kriging model, a linear model network, a polynomial, etc., and by a fixed number Pj of model parameters Pj = {p-ij, ppjj}. The model parameters Pj were or are trained or determined by a suitable method and are usually known. Each model Mj maps an input vector u = {in, uk} to an estimate of the output y of the modeled physical process. The real output y of the physical process, which can be measured, for example, is approximated by the model Mj. For example, the input vector u contains the input quantities torque and speed of an internal combustion engine, as well as the coolant temperature of the internal combustion engine, and an emission or consumption quantity is estimated.The input variables u, in the input variable vector u, can vary within a given or fixed input variable range U, with ue U. Using the model ensemble 1, or the models Mj contained therein, a physical process such as an emission or consumption parameter of an internal combustion engine, like NOx emission, CO or CO2 emission, or fuel consumption, is estimated as the output parameter y of the model ensemble 1, or as the model output parameters y of the models Mj. For the sake of simplicity, the following description assumes a single output parameter y without loss of generality, although of course an output parameter vector y consisting of several output parameters y is also possible. In model ensemble 1, each model output variable y is weighted with a weighting factor Wj, and the output variable y of model ensemble 1 is the weighted sum of the model output variables y of the individual models Mjin of the form y(u) = ^w -y- i). For the sake of simplicity, y or y is used instead of j in the description. The correct notation y(u) and yj(u) is used. Regarding the weighting factors Wj, the boundary conditions w - e [0, l] and = 1 should preferably be considered. j The problem here is to determine the weighting factors Wj as accurately as possible in order to approximate the output y of the physical process as closely as possible using model ensemble 1, or rather, using its output y. The goal, of course, is for model ensemble 1 to estimate the output y of the physical process over the entire or the relevant input range U better than the best model Mj of model ensemble 1. Fig. 2 shows the model output y of a j-th model Mj as a function of a single input u (this is the simplest case without loss of generality). The points in the figure are measured data points, i.e., each a measured output y(u) of an input u. A first model M-ι with the corresponding model output yj = M approximates the measured output y using a simple model. The model M2 with the corresponding model output yi = Mi represents a somewhat more complex model that approximates the measured output y better (in the sense of a smaller deviation of the model Mj from the underlying physical process). However, the better approximation by model M2 comes at the cost of greater model complexity due to more model degrees of freedom in the form of a larger number of Pjan model parameters.More complex models generally approximate the real physical process better, but require more model parameters and more data for training the model, and are also more sensitive to changes in these parameters. This fundamental relationship is illustrated in Fig. 3. This figure shows, as an example, the model error E (e.g., the MSE mentioned above) of the model Mj plotted against the number of model parameters pj. It is shown once as the model error ET when training the model Mj with the available training data (all or some of the available measured data points). It is also shown as the model error Ev, which was determined using predefined validation data (any available measured values ​​of the process output y(u) for specific input variables u). However, for the present application, there is also the problem that little or no validation data is available. To assess the complexity of the j-th model Mjzu, an empirical complexity measure c. is used according to the invention, which does not assess the model structure as is currently the case. The empirical complexity measure c is not a technical measure, but rather assesses the deviation of the model output y from the output y of the physical process over a specific input range U. In contrast to a model error E, which concerns the deviation between the model M and the physical process at specific measured data points, the empirical complexity measure c assesses the deviation over an entire input range U, and thus also specifically between the measured data points. Various approaches are available for such an assessment. In a first approach, the surface area of ​​the model output variable y over the input variable range U is used for evaluation. The underlying concept according to the invention can also be explained with reference to Fig. 2. As can be seen, (in the one-dimensional case shown here) the length of the model output variable y (which corresponds to the surface area in the generalized case) over the input variable range ue U is greater the more complex the model Mj becomes, i.e., the better the output variable y is approximated by the model Mj. This can, of course, be generalized to any dimension (number of input variables u in the input vector u). The empirical complexity measure c- for evaluating the deviation of the model Mj from the physical process over the input variable range U based on the surface area is determined according to the following relationship: Here, V is the well-known nabla operator with respect to the input variables in the input variable vector u, i.e., V. The integral is taken over a specific, preferably... The entire input range U is determined. This integral grows monotonically with the surface area of ​​the model output y. Thus, the empirical complexity measure q is the surface area of ​​the model output y over the input range U. As an alternative empirical measure of complexity q, which assesses the deviation of the model Mj or the model output y from the output y of the physical process, the variance of the model output y can be used. The variance (also known as the second moment of a random variable) is, as is well known, the expected squared deviation of a random variable from its expected value. Applied to the present invention, the variance compares the model output y at the available N data points with the model output y between these data points, which is referred to here as variability. The underlying idea is that a model Mj with increased variability generally predicts the underlying physical process less accurately over the input range U than a model Mj with lower variability.This is due to the fact that the better the model Mj approximates the measured data points, i.e., the more complex the model Mj becomes, the greater the probability of increased variability. However, if the variability becomes too large, the risk of overfitting the model Mj also increases. The typical behavior of such an overfitted or overly complex model Mj is a model output y that varies considerably over the input range U, which in turn can lead to a larger deviation between the real output y and the model output y. This variance-based variability can be represented in the empirical complexity measure q by calculating the empirical complexity measure q according to the following formula. T l <n> Cj = j yj(u) yj(u)du --Xyj(ui) <t>yj(ui) U i=l It is obvious that there are other ways to evaluate the deviation between model Mj and the physical process, or rather the output variable y of the process, and the model output variable y. The basic idea remains unchanged: the empirical complexity measure c increases with the complexity of the underlying model Mj. The empirical complexity measure q thus also assesses the complexity of the model Mj. According to the invention, a surface information criterion SICj of the j-th model Mj is derived from the empirical complexity measure q. This SICj, analogous to the prior art AIC, is again formed from the model error Ej of the model Mj and the empirical complexity measure q, i.e., SICj = (EJ + ακ - Cj). For example, the mean squared error is used as the model error Ej. 1 <n> MSE =— y (y(Ui) - y, (u;)) <2> used, but of course any other N<~~>ί can also be used. Model error Ej, for example in the form of the mean absolute deviation, could be used. The preferred parameter ακe [0, °°[ in the surface information criterion SICj is used as a complexity aversion parameter. This represents the only degree of freedom with which the complexity of the models Mj of the model ensemble 1 can be further penalized. The larger the complexity aversion parameter ακ becomes, the more the complexity is reflected in the surface information criterion SICj. Small complexity aversion parameters ακbe therefore favor more complex models Mj, i.e., models Mj with more degrees of freedom (number of model parameters pj). Analogous to the well-known Akaike information criterion, the weighting factors Wj could be like- -—SIC which are determined from w - = e<2 1>, where preferably w - e [0, 1] and = 1 are considered as boundary conditions. Although this already forms a model ensemble 1 Since it can be achieved that, under the given conditions, better approximates the real process, i.e., with less error, than a model ensemble formed with the Akaike Information Criterion (AIC), the quality of the model ensemble 1 can be further improved according to the invention. This is done as explained below. It can be shown that the mean squared model error MSE and the empirical complexity measure c of the model ensemble 1 with respect to a weighting vector w containing the weighting factors Wj of the j models Mj, are each quadratic functions of the model errors Ejund of the empirical complexity measures q of the models Mjin of the form SIC = iw <t>Fw + aKw <t>Cw). The optional complexity aversion parameter ακ represents one degree of freedom in determining the weighting factors Wj of the j models Mjdar. Here, F denotes an error matrix containing the model errors Ej of the models Mj, and C a complexity measure matrix containing the empirical complexity measures q of the models Mj. In the case of the mean squared error MSEj as the model error Ejj with a matrix E = ^(ιι;) - ^(ιι;)) , for all ie N data points and j, the error matrix F is obtained as the product of matrix E with itself according to F=E <t>E. Depending on the chosen empirical complexity measure q, the complexity measure matrix C is, for example, c = i.<eweils mit einem> Model output vector ya, which contains the model output variables y of the j models, i.e. ya = {y1...yJ}■ Here, the matrices F and C can be calculated in advance and, above all, without knowledge of the models Mjo or their model structures or the number of model parameters pj. To determine the weighting factors Wj (or analogously the weighting vector w), the surface information criterion SIC of the model ensemble 1 can be optimized, in particular minimized, for a specific complexity aversion parameter αk with respect to the weighting factors Wj. From this, an optimization problem in the form wa= arg min iw <t>Fw + aKw <t>C) can be derived. As can easily be seen, this is a quadratic optimization problem that can be solved quickly and efficiently using available standard solution algorithms for a given complexity aversion parameter α. The boundary conditions for the optimization are preferably w - e [0, l] and ^w - = 1 . An arbitrary initial weighting vector w can be specified. The result of optimizing the surface information criterion SIC of model ensemble 1 to determine the weighting factors Wj is described with reference to Fig. 4. The exemplary implementation is based on models Mj trained for a small number of data points. The two diagrams on the left show the training error ETE and the validation error EVE of model ensemble 1, which was determined using the optimization of the surface information criterion SIC described above. The abscissa shows an empirical model ensemble complexity p"ff, which is derived from the complexities of the individual models Mja in the form p"ff = w <t>. Here w is the value for a specific Complexity aversion parameter αk is determined as a weighting vector, and the vector p contains the number of model parameters pj for all j models Mj. If the complexity aversion parameter αk is varied and the optimization is solved for each complexity aversion parameter αk, yielding a corresponding weighting vector wtt, the curves in the two left-hand diagrams are obtained. The points in the diagrams represent the training error Ek and the validation error Ev of the j-th model Mj of the model ensemble 1 (for this, the number pj of the model parameters of the j-th model Mj is plotted). As can easily be seen, the model ensemble 1 determined according to the invention is always better, i.e., with a smaller error, than the best individual model Mj. The diagram on the right in Fig. 4 illustrates the validation error Ev versus the training error ET. The points represent the individual models Mjdar. The diagram also compares the model ensemble 1 determined using the previously standard Akaike Information Criterion AIC with the model ensemble 1 determined according to the invention using the Surface Information Criterion SIC. As can be clearly seen, the Surface Information Criterion SIC performs not only significantly better than the Akaike Information Criterion AIC, but also better than any single model Mj. The diagram on the right in Fig. 4 also shows that there is a complexity aversion parameter aK,ot that minimizes the model error of the model ensemble 1. One can now attempt to find this optimal complexity aversion parameter ciK,ot manually, or at least approximate it manually. In a second step, following the optimization of the surface information criterion SIC, one can also attempt to determine the optimal complexity aversion parameter aK,0pt, and thus also the associated optimal weighting vector wopt, using the procedure described below. For this purpose, the corresponding weighting vectors wttKer are first determined for a plurality of complexity aversion parameters ακ. This yields a set of weighting vectors |wa}. Using the well-known Mallow equation, the complexity The aversion parameter ακ is chosen as the optimal complexity aversion parameter aK,ot, which solves the following optimization problem. In this, F is again the error matrix (F=E). <t>E) and σ is the standard deviation of the available data points, which is usually unknown. However, there are known methods (e.g., as described in Hansen, BE, “Least Squares model averaging”, Econometrica, 75(4), 2007, pp. 1175–1189) to estimate the standard deviation σ from the available data points. The vector p again contains the number of model parameters Pj for all j models Mj. For this second step, therefore, knowledge of the models Mj or their model structures is required. This optimization is not solved directly, but rather with respect to the first determined set of weighting vectors |wa}. That is, the weighting vector w associated with a specific complexity-aversion parameter ακ is selected as the optimal weighting vector wopt, which yields the minimal expression J w^ Fwa+— a <2> w <t>ap l results. K <k> N <k>J Figure 5 further demonstrates the effect of the inventive method for determining the weighting factors Wj for a small number N of available data points using an example. For this purpose, 5,000 data points each were measured for NOx and soot emissions on a specific internal combustion engine, i.e., one measurement each for NOx and soot at 5,000 input vectors u, in order to have sufficient data for the demonstration example. The input vector u comprised, for example, five input variables u, namely torque, speed, coolant temperature, position of a variable-geometry turbocharger, and position of an exhaust gas recirculation system. Fifteen different models Mj (different model structures and / or different numbers of model parameters and / or different model parameters) were trained with a random selection of data points from the 5,000 available data points.The random selection was used as the available small number N of data points. The number N of available data points was then increased from ten to 150, i.e., 10 < N < 150. The remaining data points (5,000 - N) were used as validation data for the example to verify the effectiveness of the invention. Figure 5 shows the validation error for NOx emission Ev,NOx and soot emission Ev,Soot. The validation error Ev is the mean square error between the model output y, or the output y of the model ensemble 1, and the validation data. For each number N of data points, a best and a worst model Mj (dashed lines in Figure 5) are obtained. This range of model Mj is shown in Figure 5.In addition, for each number N of data points, a model ensemble was determined according to the known Akaike Information Criterion AIC and a model ensemble 1 with the surface information criterion SIC according to the invention. The validation errors are also shown in the diagrams of Fig. 5. It is immediately apparent from this that the model ensemble 1 determined according to the invention is not only generally better than the best model Mjist, but also better than the model ensemble determined using the Akaike Information Criterion AIC. A model ensemble determined according to the invention is used, for example, in the calibration of a technical system, such as an internal combustion engine. During calibration, in a specific operating state of the technical system, defined by state variables or...A state variable vector, control variables of the technical system used to control the system, is varied to optimize at least one output variable of the technical system. Optimizing the output variables by varying the control variables is generally formulated and solved as an optimization problem. Well-established methods exist for this. The control variables determined in this way are stored as a function of the respective operating state, e.g., in the form of characteristic curves or tables. This relationship can then be used to control the technical system depending on the current operating state (which is measured or otherwise determined, e.g., estimated). That is, the control variables stored for the respective operating state are read from the stored relationship and used to control the technical process.In the case of an internal combustion engine as a technical system, the operating state is often described by measurable quantities such as rotational speed and torque, although other quantities such as coolant temperature, ambient temperature, etc., can also be used. Control variables in an internal combustion engine often include the position of a variable-geometry turbocharger, the position of an exhaust gas recirculation system, or the injection timing. The output variable to be optimized in an internal combustion engine is typically fuel consumption and / or an emission variable (e.g., NOx, CO, CO2, etc.). The calibration of the internal combustion engine, therefore, aims to ensure minimal fuel consumption and / or emissions during operation by specifying the correct control variables.< / k> < / k> < / t> < / t> < / t> < / t> < / t> < / t> < / t> < / t> < / n> < / t> < / n> < / k> < / k> < / t> < / t> < / t>

Claims

Patent claims 1. Method for creating a model ensemble (1) that estimates at least one output variable (y) of a physical process as a function of at least one input variable (u), wherein the model ensemble (1) is formed from a sum of the model output variables (y.) of a plurality (j) of models (Mj), each weighted with a weighting factor (Wj), characterized in that for each model (Mj) an empirical complexity measure (q), which evaluates the deviation of the model output variable (y.) from the output variable (y) of the real physical process over a given input variable range (U), and a model error (Ej) are determined, and a surface information criterion (SICj, SIC) is formed from the empirical complexity measure (q) and the model error (Ej), from which the weighting factors (Wj) for the model ensemble (1) are determined.

2. Method according to claim 1, characterized in that the model error (Ej) of a model (Mj) is the mean squared error (MSEj) between the output quantities (y) of the physical process measured at N input quantities (u) and the model output quantities calculated at these N input quantities according to the relationship 1 <n> MSE - =— y(y(Ui) - y, (u;)) <2> is used. N<~~>ί 3. Method according to claim 1, characterized in that the empirical complement- r <t> π measure (q) of a model (Mj) with the formula Cj = Vy^u) Vy^i^du or the formula u T 1 <n> Cj = j yj(u) yj (u)du -— ^ yj- Oii^yj-Cu;) is calculated.

4. Method according to claim 1, characterized in that the empirical complexity measure (q) is weighted with a complexity aversion parameter (ακ).

5. Method according to one of claims 1 to 4, characterized in that the weighting factors (Wj) of each model (Mj) of the model ensemble (1 ) are combined with the surface --SICj Information criterion (SICj) of the model (Mj) from the formula w - = e <2> will be calculated.

6. Method according to one of claims 1 to 3, characterized in that for the model ensemble (1) the surface information criterion (SIC) is formed from an error matrix (F) containing the model errors (Ej) of the models (Mj) and a complexity measure matrix (C) containing the empirical complexity measures (q) of the models (Mj), wherein the error matrix (F) and the complexity measure matrix (C) are calculated according to the formula SIC = |w <t>Fw + w <t>Cw is weighted twice by a weighting vector (w) containing the weighting factors (Wj) of the models (Mj) and the surface information criterion (SIC) of the model ensemble (1) is minimized with respect to the weighting factors (Wj).

7. Method according to claim 6, characterized in that the error matrix (F) is calculated as a matrix product of a matrix (E), wherein the matrix (E) is calculated using the formula E = (y(ui) - yj(ui)).

8. Method according to claim 6 or 7, characterized in that the complexity measure matrix (C) is weighted by a complexity aversion parameter (ακ).

9. Method according to claim 8, characterized in that the weighting factors (Wj) for different complexity aversion parameters (ακ) are calculated and the weighting vector (wttK) associated with a selected complexity aversion parameter (ακ) is selected as the optimal weighting vector (wopt) for the model ensemble (1 ).

10. Method according to claim 8, characterized in that weighting vectors (wttK) are calculated for different complexity aversion parameters (ακ) and the relationship is derived from them to determine the optimal weighting vector (wopt). J w^ Fwa+— a <2> w <t>ap l regarding the various complexity-aversion- K <k> N <k>J The calculated weighting vectors (wa) are minimized by parameter (ακ).< / k> < / k> < / t> < / t> < / t> < / n> < / t> < / n>