Incremental identification of hybrid dynamic models
The method generates a hybrid model by optimizing an unknown modeling part of a rigorous model using process data, enhancing model accuracy and adaptability, and facilitating PID controller tuning for improved industrial process control.
Patent Information
- Authority / Receiving Office
- WO · WO
- Patent Type
- Applications
- Current Assignee / Owner
- SIEMENS AG
- Filing Date
- 2025-10-09
- Publication Date
- 2026-05-15
AI Technical Summary
Existing data-driven modeling methods for industrial processes often lack accuracy and require excessive effort, neglecting process structure, leading to increased data demands and reduced transparency, while rigorous models are not fully utilized, and hybrid models are difficult to identify due to coupled mechanistic and data-driven components.
A method for generating a hybrid computer-implemented model by formulating an optimization problem for the unknown modeling part of a rigorous model, using process data to determine model state data, training a data-based model, and integrating it into the rigorous model, allowing for a separated identification of components.
Enables efficient and accurate hybrid model development without relying on known model components, improving model performance and adaptability, and facilitating PID controller tuning for improved control in industrial processes.
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Abstract
Description
[0001] 202418892 Foreign version 06.10.2025
[0002] 1
[0003] Description
[0004] Incremental identification of dynamic hybrid models
[0005] The invention relates to a method for generating a hybrid computer-implemented model of an industrial process. The invention also relates to a device for controlling and / or monitoring an industrial system.
[0006] Process models are required for numerous applications. These include prediction tasks, process optimization, as well as classifications and anomaly detection. In fact, many process engineering procedures cannot be rigorously modeled with sufficient accuracy, or this is only possible with excessive effort. In these cases, data-driven modeling using machine learning (ML) or deep learning (DL) methods has become increasingly common in recent years. However, even here, achieving sufficient accuracy often requires considerable effort. Furthermore, methodology experts (e.g., data scientists) sometimes lack a deeper understanding of the processes, which further complicates the application.
[0007] Often, the structure of, for example, process engineering processes is not considered in data-driven modeling. However, if the structure of the process to be modeled is known (e.g., basic process flow diagrams and process flowcharts, possibly also P&ID diagrams), this knowledge should also be taken into account in the structure of the data-driven model. The method according to the invention also significantly simplifies the often necessary online adaptation of the model (especially due to changing processes) and considerably reduces the requirements for the quantity and quality of the data needed for online adaptation.
[0008] In most cases, only data-driven modeling is used. However, "forgetting" the rigorous model and relying entirely on data-driven modeling is unsatisfactory and squanders what has already been achieved. Furthermore, the data-driven model may not achieve the desired accuracy with a reasonable amount of effort. In addition, many data-driven models are not suitable for providing transparency regarding their predictions. This undermines user confidence and, in some cases, prevents their use. Moreover, there is generally greater trust in rigorous models. 202418892 Foreign version 06.10.2025
[0009] 2
[0010] It is therefore advisable to further examine the rigorous model and combine it with machine learning (ML) and deep learning (DL) methods. ML methods are frequently used to optimize the model parameters using the data, in order to better adapt the rigorous model to actual behavior. The combination of both methods is also advantageous with regard to certification, particularly under the EU AI Act.
[0011] Alternatively, the rigorous model can be used to generate simulation data, which can then be used for pretraining the data-driven model or for parallel training. However, the model error must be appropriately accounted for. Especially when only certain parts of the rigorous model are inadequate, data-driven models are used to replace just those parts.
[0012] To create a dynamic, data-driven model, the structure of the data-driven model may need to be extensively adapted (e.g., multiple layers, etc.). This often requires considerable experience and potentially several iterations of the model structure to account for process and measurement instrument dynamics in the data-driven modeling.
[0013] Furthermore, significantly more data is required for data-driven dynamic modeling to ensure reliable dynamic modeling. Modeling becomes even more complex when the system being modeled uses closed-loop control. Additionally, changing system behavior necessitates adaptation of the data-driven model, which places considerably higher demands on the scope and quality of the process data.
[0014] Processes, for example in process engineering, can have quite complex structures. It is of course possible to neglect these structures in data-driven modeling. However, neglecting the process structure leads to significantly increased demands on the scope and quality of the process data.
[0015] Furthermore, several approaches are known (e.g., https: / / agupubs.onlinelibrary.wiley.com / doi / full / 10.1029 / 2018WR023333) for learning a correction model for the rigorous model using additional data. However, this additional data is only used in training, not in subsequent application, which distinguishes this approach from the inventive method presented below. 202418892 Foreign version 06.10.2025
[0016] 3
[0017] Another well-known approach involves generating further meaningful features from the data using simple, mostly algebraic equations. This starts with applying functions to individual quantities (e.g., quadrature) and also includes combined analyses of multiple quantities, as seen, for example, at https: / / turbulentflux.com / combining-physics-and-machine-learning-to-understand-multiphase-transient-flow /
[0018] There are also known applications where the rigorous and the data-based models are run in parallel and the more reliable one is selected using weighted linear combination (https: / / towardsdatascience.com / combining-physics-and-deep-learning-54eac4afe146), which can be done using Markov models (https: / / www.sciencedirect.com / science / article / abs / pii / S1875510021005436).
[0019] There are various hybrid model structures: In serial structures, data-driven models are used to learn and compensate for deviations from the mechanistic model. In this case, the outputs of the mechanistic model (represented by a system of differential-algebraic equations) serve as inputs for the data-driven model. In parallel structures, both the mechanistic model and the data-driven model share the same inputs. Hybrid structures, on the other hand, use arbitrary model variables as inputs and predict arbitrary model variables as outputs.
[0020] The hybrid structure offers several advantages and is more versatile than parallel or serial structures. It can be applied more broadly over a larger state space because the input dimension of the black-box model is reduced, unlike the serial approach, which uses all model states as input variables. Furthermore, the hybrid structure allows the black-box component of the model to be manipulated, thus potentially leading to improvements in model quality.
[0021] Identifying a (dynamic) hybrid model with a hybrid model structure is difficult because the mechanistic model part and the data-driven model part are coupled and therefore influence each other. Consequently, identifying the hybrid model structure and the black-box model part is also coupled.
[0022] In the context of identification, the goal is not simply to replace a known part of the model with a data-driven counterpart, as is the case with model reduction techniques. Rather, the focus is on identifying unknown parts of the model. This 202418892 foreign version 06.10.2025
[0023] 4
[0024] However, parts may include elements that are known but too complex to be integrated mechanistically—either due to extensive modeling effort or mathematically complicated expressions that hinder their application in subsequent processes. In such cases, the identification approach serves as a means of model reduction.
[0025] It is known that dynamic hybrid models with a hybrid structure, in which the first-principles and data-driven model parts are coupled, can be identified using a simultaneous approach. This approach involves the simultaneous identification of both model components: the mechanistic model part and the black-box model part. In the identification process, the black-box model is embedded within the first-principles model, and the parameters of the black-box model are estimated using a dynamic parameter estimation problem. Consequently, it becomes difficult to determine whether inadequate model quality or performance is due to deficiencies in the black-box model part, an inadequate hybrid model structure (missing mechanistic model components), or both.
[0026] Alternatively, other methods use model reduction techniques, in which known model parts are replaced by black-box model parts.
[0027] From US patent 2024 / 036532 A1, a method and a system for modeling an industrial process are known.
[0028] The invention is based on the objective of providing a method for generating a model which overcomes the aforementioned disadvantages and is characterized by high efficiency and high accuracy.
[0029] The previously formulated problem is solved by a method for generating a hybrid computer-implemented model of an industrial process with the features of claim 1. Furthermore, the previously formulated problem is solved by a method for determining the output values of one or more output variables of an industrial process during operation, according to claim 10. Finally, the problem is solved by a device for controlling and / or monitoring an industrial process, according to claim 14. Advantageous embodiments are the subject of the dependent claims.
[0030] A method for generating a hybrid computer-implemented model of an industrial process, which is used in particular in a process plant or a manufacturing plant for 202418892 Foreign version 06.10.2025
[0031] 5
[0032] The application, in which the model comprises a rigorous model part and a data-based model part, includes the following steps: a) Using a known modeling part and an unknown modeling part for the rigorous model part, b) Formulating and solving an optimization problem for the unknown modeling part of the rigorous model part using process data derived from a run (i.e., an execution) of the industrial process, preferably also from model input data, to determine model state data and other data of the unknown modeling part of the rigorous model part, c) Subsequently training the data-based model part using the other data of the unknown modeling part of the rigorous model, and the model input data and / or the model state data of the unknown modeling part of the rigorous model part.d) Subsequent integration of the trained data-based model part into the rigorous model part to determine the unknown modeling part of the rigorous model part.
[0033] An industrial process is understood as a process in an industrial context through which energy, matter, or information are changed in their state. This change of state can, for example, occur from an initial state to a final state. The industrial process takes place primarily in an industrial plant, such as a process plant or a manufacturing plant.
[0034] A rigorous modeling (the rigorous model part) includes mathematical / physical relationships in the form of (partial) differential equations or differential-algebraic equations in order to determine one or more output variables from input variables or input parameters of the modeling.
[0035] Data-driven modeling is an approach in which mathematical models are developed to describe and predict processes. This approach is based on collected data. Data-driven modeling generally uses statistical techniques and algorithms to identify patterns and relationships within the data.
[0036] The method according to the invention assumes that a first part of the entire model to be generated is described by rigorous (physical) modeling and a second part by data-based modeling. Therefore, this model is also referred to as a hybrid model. 202418892 Foreign version 06.10.2025
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[0038] The formulation of the rigorous model component uses a known modeling component and an unknown modeling component. For the known modeling component, fundamental physical relationships such as mass conservation or energy conservation can be assumed. These can also be generated with reasonable effort within the framework of a rigorous, i.e., exact, model. The unknown modeling component is used for more complex, especially nonlinear, parts of the model.
[0039] In a subsequent step of the procedure, an optimization problem for the unknown part of the model is formulated and solved. A method for minimizing squared errors can be used for this purpose. Process data from the industrial process is used for this optimization procedure. This data can either be taken directly from the process or from a process data archive containing historical process data. During the optimization, model state data and other data from the unknown part of the rigorous model are determined. The data-based model can be implemented as a neural network, a regression model, or a self-organizing map.
[0040] This additional data, along with the model input data and / or the model state data, is then used to train the data-based part of the model.
[0041] In a final step, the trained data-driven model part is integrated into the rigorous model part to determine the initially unknown modeling part of the rigorous model part. This makes the entire model known and suitable for versatile use in industrial contexts.
[0042] Separating the identification of the rigorous model component from the identification of the black-box model component (the data-driven model component) offers several advantages. First, it enables a rapid assessment of the suitability of a hybrid model structure. Second, it decouples the process of finding a suitable machine learning model from the search for an optimal hybrid model structure. This separation simplifies and accelerates the model development process. Such a separation is not possible with a simultaneous identification approach.
[0043] The incremental approach allows for the calculation of the best possible model performance for a given model structure. In contrast, concurrent approaches cannot achieve this, as the model quality depends on the selection and training of the machine learning model. 202418892 Foreign version 06.10.2025
[0044] 7 depends on the fact that it is already embedded during identification. This advantage allows for a rapid assessment of the suitability of the hybrid model structure and contributes to efficient and accelerated model development.
[0045] The incremental approach facilitates the identification of influential factors in the state space that need to be modeled using data-driven methods. Concurrent approaches typically require either using all states as inputs for machine learning models (which can be data-intensive and lead to extrapolation risks) or reducing the input space using expert knowledge, which may be scarce. Consequently, the present method does not rely heavily on in-depth expert knowledge of the system for model building and effectively uses the available data to determine the most influential states. This advantage improves the efficiency and speed of developing hybrid dynamic models.
[0046] In comparison to model reduction approaches that use a hybrid model structure and replace known model components (e.g., thermodynamics or fluid dynamics) with data-driven models, the incremental method described here offers several key differences:
[0047] 1.) Model identification and reduction: The incremental approach goes beyond model reduction: While model reduction approaches focus on simplifying existing models, the incremental approach enables the identification of hybrid models from scratch. This means that hybrid models can be developed even without the availability of first-principles-based models. By utilizing available data, the approach enables the development of hybrid models that accurately capture system dynamics.
[0048] 2.) Unknown data-driven model components: In the present method, the data-driven model component to be modeled does not need to be known in advance, as is required in model reduction approaches. This is a significant advantage, as it does not rely on the availability of first-principles-based models or predetermined model components. Instead, it uses the available data to identify and train the data-driven model components, allowing for greater flexibility and adaptability in the model development process. 202418892 Foreign version 06.10.2025
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[0050] In summary, the method described here offers the possibility of developing hybrid models from scratch, without relying on known model components or the availability of first-principles-based models. By utilizing available data, it enables the identification and training of data-driven model components, thus providing flexibility and adaptability in the model development process.
[0051] In an advantageous further development of the invention, the influence of a state of the rigorous model part and / or the process data, which originate from a sequence of the industrial process, on the unknown modeling part is determined between process steps b) and c) and taken into account for the subsequent training of the data-based model part.
[0052] To determine the influence of the state of the rigorous model part and / or the process input data on the unknown modeling part, a correlation analysis and / or an influence analysis can be performed.
[0053] If the determined correlation and / or the results of the influence analysis are below a certain threshold, it can be assumed that the state of the rigorous model component and / or the process data have no influence on the unknown model component. These analyses and the assumptions made based on them can improve the robustness and extrapolation capability of the resulting model.
[0054] Preferably, regularization is applied when solving the optimization problem, in particular L1 regularization, L2 regularization, or a combination of L1 and L2 regularization. Regularization can ensure that the generated model is not too closely fitted to the training data (process data), which can improve its generalizability to new, unknown (process) data.
[0055] The rigorous model part preferably represents one or more PID controllers. PID control serves as the basic control system in various industrial processes, where it plays a crucial role in maintaining desired setpoints and regulating system variables. However, a lack of optimal PID controller settings (e.g., due to mismatched controllers or changed process operating conditions) can lead to poor control performance, which adversely affects manufacturing companies. These effects include reduced product quality, inefficient resource utilization, increased operating costs, and accelerated equipment wear. 202418892 Foreign version 06.10.2025
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[0057] A well-tuned basic control system, such as PID control, is a prerequisite for implementing advanced process control and optimization strategies. The effectiveness of higher-level control algorithms, such as model predictive control or adaptive control, depends heavily on the precise and optimal tuning of the PID controller. Therefore, improving the tuning of PID controllers is not only crucial for mitigating the aforementioned negative effects, but also a prerequisite for advanced process control and optimization techniques that can further improve process performance and optimization in industrial systems.
[0058] The hybrid model to be generated comprises, in this case, the rigorous model part, the PID controller with the current parameter values, and the data-based model part, which represents the controlled system to be identified. The hybrid model can also include multiple PID controllers or PID controller cascades. The hybrid model uses the actual PID controller structure with the current PID controller parameters. This hybrid model enables a more accurate representation of the real system dynamics. The rigorous model part of the hybrid model corresponds, as already mentioned, to the process controller (PID controller), which can be modeled precisely. The data-based model part of the hybrid model corresponds to the process itself. By using the hybrid model, this further development of the invention offers a comprehensive understanding of the system behavior.
[0059] Once the identification process is complete, the data-driven model serves as the model of the open-loop controlled system. This data-driven part of the model captures the complex and nonlinear relationships within the system and enables accurate predictions of the system's response under various operating conditions. By utilizing this data-driven part, the limitations of traditional linear models can be overcome, and a wide range of control system complexities can be addressed.
[0060] It should be noted that the resulting hybrid model can only reproduce the system dynamics as captured in the process data. That is, the process data must cover a certain range of the process dynamics to ensure that the hybrid model captures relevant process behavior for controller optimization. Techniques such as optimal experimental design can be used to determine which process data are required to capture relevant process dynamics. 202418892 Foreign version 06.10.2025
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[0062] After model identification, the open-loop control model, represented by the data-driven model component, or the hybrid model (closed-loop control model) is used for PID parameter tuning. This can be achieved by using one of the models to generate step response data or by formulating an optimization problem for PID controller tuning. The model-based approach enables systematic and precise tuning of the PID controller parameters based on the desired control performance criteria. Using the open-loop control model allows for more efficient and effective PID controller tuning.
[0063] In summary, this further development of the invention offers the following advantages:
[0064] - Comprehensive representation: The hybrid model captures both the regulatory actions of the
[0065] PID controllers as well as the complex dynamics of the open-loop controlled system, thus offering a more thorough understanding of the system behavior.
[0066] - Improved control performance: By integrating machine learning models, the hybrid model can handle nonlinearities, uncertainties, and time-varying behavior more effectively, resulting in improved controller performance. Compared to linear models, this approach enables higher accuracy. Compared to mechanistic models, this approach reduces modeling effort.
[0067] - Flexibility and adaptability: The hybrid model allows for the representation of a
[0068] A multitude of system complexities and the need to adapt to different operating conditions. In this case, the hybrid model can be fine-tuned using operational data from new, previously uncaptured operating regimes.
[0069] - Handling complex control structures: The approach can handle complex control structures, such as PID controller cascades, and determine the PID controller parameters.
[0070] The previously formulated task is also solved by a method for the in-process determination of output values of one or more output variables of an industrial process, comprising the following steps: i) execution of the industrial process, ii) acquisition of process data originating from the industrial process using suitable acquisition devices, iii) processing of the acquired process data and determination of output values of the output variable by a method designed as described above. 202418892 Foreign version 06.10.2025
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[0072] The input quantity to be determined can be a mass flow rate, a pH value, a pressure, a concentration of carbon monoxide, ammonium or ammonium nitrate, a fill level, a temperature, a volume flow rate, a concentration of an inorganic gaseous chlorine compound, an inorganic gaseous fluorine compound, a dust, a nitrogen oxide, a carbon monoxide, a nitrogen dioxide, a nitrogen monoxide, a dinitrogen monoxide, ammonia, a mercury, a methane, an oxygen, a humidity level, a carbon dioxide level, a formaldehyde level or a sulfur dioxide level.
[0073] The industrial process can involve the combustion of waste or fuel in a gas turbine, burner or engine, or a mixing and heating process in a chemical or biological reactor.
[0074] The initial values of the output variable, determined during operation, are particularly preferred for optimizing operations and / or controlling the industrial process and / or predicting maintenance requirements during the execution of the industrial process.
[0075] The previously formulated task is also solved by a device for controlling and / or monitoring an industrial process, which is designed to carry out a procedure as previously explained.
[0076] The properties, features, and advantages of this invention described above, as well as the manner in which they are achieved, will become clearer and more readily understandable in connection with the following description of the exemplary embodiment, which is explained in more detail in conjunction with the drawings. The drawings show:
[0077] FIG 1 shows a bioreactor in a schematic representation;
[0078] FIG 2 shows a flowchart of a method according to the invention;
[0079] FIG 3 shows a time course of a concentration within the bioreactor;
[0080] FIG 4 shows a chemical reactor in a schematic representation;
[0081] FIG 5 shows a flowchart of a method according to the invention; and 202418892 Foreign version 06.10.2025
[0082] 12
[0083] FIG 6 shows a time course of a concentration within the chemical reactor.
[0084] The process according to the invention is explained using two exemplary embodiments. FIG. 1 schematically depicts a bioreactor 1 in which a biological process takes place as an industrial process. One of the tasks during the operation of the bioreactor is to predict the time course of the concentration of a medium 2 in the bioreactor 1. For this purpose, a computer-implemented model of the biological process is required.
[0085] In the process of creating this model, an approach with a rigorous and a data-driven model component is used. The following well-known modeling component is used for the rigorous model component:
[0086] This known part of the model describes the physical relationship between a mass M of medium 2 in the biological reactor 1 and an inflow F into the reactor 1, as well as a concentration c of medium 2 in the reactor 1. k(t) denotes an unknown part of the model that describes the influence of kinetic motions within the reactor 1.
[0087] In a first process step I (see FIG. 2) of a method according to the invention, an optimization problem for the unknown modeling part of the rigorous model part is formulated and solved. At least process data originating from an industrial process sequence are taken into account. The solution to the optimization problem consists of determining model state data and other data of the unknown modeling part of the rigorous model part.
[0088] In the present embodiment, the mass M, the inflow F, the concentration c, and a temperature ? of the medium 2 in the biological reactor 1 are sensor-acquired as process data and used to solve the optimization problem. The optimization problem itself is formulated here as a minimization of squared errors, with the relationships (without going into excessive detail) being as follows: 202418892 Foreign version 06.10.2025 x t+i = f(.x t ,y t , u t , k t ) o = dx t ,y t ,u t ,k t ') z t = h(x t ,y t )
[0089] This involves z t to output variables that correspond to the metrologically recorded process data, at x t to model states, at u t to model input data, at y t to algebraic model states and at k tto the unknown part of the model. f (x t , y t , u t , k t ) denote differential equations, #(x t ,y t ,u t ,k t ) algebraic equations and h(x) t ,y t Initial equations.
[0090] In a subsequent second step (II), the influence of a state of the rigorous model component and / or the process data, derived from an industrial process sequence, on the unknown model component is determined. This involves either a correlation analysis or an influence analysis. Here, a correlation analysis according to Pearson is performed, revealing that the temperature of the medium 2 flowing into reactor 1 shows no significant correlation with the unknown model component k(t). This finding is also used for the subsequent training of the data-driven model component in the third step (III). The data-driven model component is therefore trained only with k(t), which depends on the mass M, the inflow F, and the concentration c.As previously explained, it is assumed that the temperature T of the medium has no significant influence on k(t) and thus on the overall modeling of the biological (industrial) process.
[0091] In a subsequent fourth step IV, the trained, data-based model part is integrated into the rigorous model part in order to determine k(t) and thus the unknown modeling part of the rigorous model part.
[0092] To verify the quality of the generated model, the biological process was carried out in reactor 1, and the concentration c of medium 2 in reactor 1 was measured using sensors. Figure 3 shows the resulting time course of the process in a graph. The horizontal axis represents time in days, and the vertical axis represents the concentration c in mmol per liter. Curve 3, marked with dots and dashes, represents the measured course of the concentration values c, while curve 4, marked only with dashes, represents the course of the concentration values c determined using the previously generated model. 202418892 Foreign version 06.10.2025
[0093] 14
[0094] It is clearly evident that the modeled concentration values c are almost identical to the measured concentration values. The quality and accuracy of the modeling generated by the described method can therefore be considered very high.
[0095] In a second embodiment according to FIG. 4, a chemical reactor 5 is modeled using a method according to the invention. The following relationships are used for the rigorous model part:
[0096] Here, h denotes the fill level of a medium 8 in the chemical reactor 5, c a concentration of the medium 8 in the chemical reactor 5, c° an inlet concentration of the medium 8 in the chemical reactor 5, T a temperature of the medium 8 in the chemical reactor 5, T° an initial temperature, F a mass flow rate into the chemical reactor 5, F° an initial mass flow rate, and r a diameter of the chemical reactor 5. Furthermore, T ca temperature of cooling water that can be directed into chemical reactor 5.
[0097] In a first step I according to FIG 5, an optimization problem is solved by minimizing the squared errors, analogous to the first embodiment:
[0098] The correlation analysis according to Pearson in a subsequent second step II shows that k^ ) has no significant correlation with the process variables (temperature, mass flow rate, concentration), while / c2(t) and / c3(t) each show a significant correlation with the process variables. Steps III and IV are carried out analogously to the first embodiment described above. 202418892 Foreign version 06.10.2025
[0099] 15
[0100] To verify the quality of the generated model, the chemical process was carried out in chemical reactor 5, and the concentration c of medium 8 in the chemical reactor 5 was measured using sensors. Figure 3 shows the resulting time course of the process in a diagram. The horizontal axis represents time in days, and the vertical axis represents the concentration c in mol per m³. 3 . Curve 7, which is marked with dots and dashes, represents the measured course of the concentration values c, while curve 6, which is marked only with dashes, represents the course of the concentration values c determined using the previously generated model.
[0101] Here too, it is clearly evident that the modeled concentration values c are almost identical to the measured concentration values. The quality and accuracy of the modeling generated by the described method can therefore be considered very high.
[0102] Another application example, which is not explained in detail here, lies in the optimization of a model predictive control (MPC).
[0103] Although the invention has been illustrated and described in detail by the preferred embodiment and the figures, the invention is not limited by the disclosed examples and other variations can be derived from them by the person skilled in the art without leaving the scope of protection of the invention.
Claims
202418892 Foreign version 06.10.2025 16 Patent claims 1. A method for generating a hybrid computer-implemented model of an industrial process, which is used in particular in a process plant or a manufacturing plant, wherein the model has a rigorous model part and a data-based model part, comprising: a) using a known modeling part and an unknown modeling part for the rigorous model part, b) formulating and solving an optimization problem for the unknown modeling part of the rigorous model part using process data (e.g., process data). t ), preferably additionally from model input data (u t ), to determine model state data (x t) and further data of the unknown modeling part of the rigorous model part, c) Subsequent training of the data-based model part using the further data of the unknown modeling part of the rigorous model, and / or the model input data (u t ) and / or the model state data (x t ) of the unknown modeling part of the rigorous model part, d) Subsequent integration of the trained data-based model part into the rigorous model part to determine the unknown modeling part of the rigorous model part.
2. Method according to claim 1, wherein, between the method steps b) and c), an influence of a state of the rigorous model part and / or the process data, which originate from a sequence of the industrial process, on the unknown modeling part is determined and taken into account for the subsequent training of the data-based model part.
3. Method according to claim 2, wherein a correlation analysis and / or an influence analysis is performed to determine the influence of the state of the rigorous model part and / or the process input data on the unknown modeling part.
4. Method according to claim 3, wherein, if the determined correlation and / or the result values of the influence analysis are below a certain threshold, it is assumed that there is no influence of the state of the rigorous model part and / or the process data on the unknown modeling part.
5. Method according to any of the preceding claims, wherein minimum weighted squared errors are determined to solve the optimization problem. 202418892 Foreign version 06.10.2025 17 6. A method according to any of the preceding claims, wherein regularization is performed in the solution of the optimization problem, in particular L1 regularization, L2 regularization or a combination of L1 regularization and L2 regularization.
7. Method according to any of the preceding claims, wherein the data-based model part is designed as a neural network, a regression, in particular a polynomial regression, or a self-organizing map.
8. Method according to any of the preceding claims, wherein the rigorous model part represents one or more PID controllers.
9. Method according to claim 8, wherein the data-based model part simulates an influence of linear and / or non-linear components of the industrial process.
10. Method for determining the output values of one or more output variables of an industrial process during operation, comprising: i) carrying out the industrial process, ii) recording process input data originating from the industrial process (z t ) by suitable acquisition means, iii) processing of the acquired process input data and determination of output values of the output variable by a computer-implemented modeling method according to one of claims 1 to 9.
11. Method according to claim 10, wherein the input variable to be determined is a mass flow rate, a pH value, a pressure, a concentration of carbon monoxide, ammonium or ammonium nitrate, a fill level, a temperature, a volume flow rate, a concentration of an inorganic gaseous chlorine compound, an inorganic gaseous fluorine compound, a dust, a nitrogen oxide, a carbon monoxide, a nitrogen dioxide, a nitrogen monoxide, a dinitrogen monoxide, ammonia, a mercury, a methane, an oxygen, a humidity, a carbon dioxide, a formaldehyde or a sulfur dioxide.
12. The method of claim 10 or 11, wherein the industrial process involves the combustion of waste and / or fuel in a gas turbine, burner and / or 202418892 Foreign version 06.10.2025 18 represents an engine, and / or a chemical or biological reactor, a distillation column and / or a rectification column, and / or a heat exchanger.
13. Method according to one of claims 9 to 12, wherein the output values of the output variable determined during operation are used for optimizing operation and / or controlling the industrial process and / or for predicting maintenance requirements during the execution of the industrial process.
14. Device for controlling and / or monitoring an industrial process, which is configured to carry out a method according to any one of claims 1 to 13.