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This paper presents two different approaches for parameter identification in hybrid models (HM). The hybrid model consists in the parallel or cascade connection of two blocks: an approximate first principles model (FPM) and an unknown block model. The first principles model is constructed based in the balance equations of the system that could have...
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Citations
... The fed-batch bioreactor problem was further addressed in [28][29][30]. Many more researchers attempted to solve the problem in a similar manner -identify the difficult to model parts in first principles and substitute them with machine learning techniques: Cubillos et al. [31] in solid drying process, Piron et al. in crossflow microfiltration process [32], Oliveira in fed-batch bioreactor and bakers' yeast production [30], Vieira and Mota in water gas heater system [33], Romijn et al. in energy balance for a glass melting process [34] and Chaffart and Ricardez-Sandoval in thin film growth process [35]. Cen et al. [36] presented a method for incorporating several neural networks into nonlinear dynamic systems for fault estimation problems. ...
One of the most common problems in science is to investigate a function describing a system. When the estimate is made based on a classical mathematical model (white-box), the function is obtained throughout solving a differential equation. Alternatively, the prediction can be made by an artificial neural network (black-box) based on trends found in past data. Both approaches have their advantages and disadvantages. Mathematical models were seen as more trustworthy as their prediction is based on the laws of physics expressed in the form of mathematical equations. However, the majority of existing mathematical models include different empirical parameters, and both approaches inherit inevitable experimental errors. Simultaneously, the approximation of neural networks can reproduce the solution exceptionally well if fed sufficient data. The difference is that an artificial neural network requires big data to build its accurate approximation, whereas a typical mathematical model needs several data points to estimate an empirical constant. Therefore, the common problem that developers meet is the inaccuracy of mathematical models and artificial neural networks. Another common challenge is the mathematical models’ computational complexity or lack of data for a sufficient precision of the artificial neural networks. Here we analyze a grey-box solution in which an artificial neural network predicts just a part of the mathematical model, and its weights are adjusted based on the mathematical model’s output using the evolutionary approach to avoid overfitting. The performance of the grey-box model is statistically compared to a Dense Neural Network on benchmarking functions. With the use of Shaffer procedure, it was shown that the grey-box approach performs exceptionally well when the overall complexity of a problem is properly distributed with the mathematical model and the Artificial Neural Network. The obtained calculation results indicate that such an approach could increase precision and limit the dataset required for learning. To show the applicability of the presented approach, it was employed in modeling of the electrochemical reaction in the Solid Oxide Fuel Cell’s anode. Implementation of a grey-box model improved the prediction in comparison to the typically used methodology.
... The approach termed as grey-box modeling can be found in the literature [7] [8] [9] [10]. The grey-box modeling method is a combination of white-box and black-box modeling methods. ...
... The grey-box modeling method is a combination of white-box and black-box modeling methods. Several terms are used in the literature referring to the grey-box modeling approach, e.g., semi-physical modeling [8] [11], hybrid modeling [9] [12], and semi-mechanistic modeling [13]. The definitions of these approaches differ slightly, but they all aim at bringing different advantages of white-box and blackbox modeling together into one model. ...
Operational optimization of ocean vessels, both off-line and in real-time, is becoming increasingly important due to rising fuel cost and added environmental constraints. Accurate and efficient simulation models are needed to achieve maximum energy efficiency. In this paper a grey-box modeling approach for the simulation of ocean vessels is presented. The modeling approach combines conventional analysis models based on physical principles (a white-box model) with a feed forward neural-network (a black-box model). Two different ways of combining these models are presented, in series and in parallel. The results of simulating several trips of a medium sized container vessel show that the grey-box modeling approach, both serial and parallel approaches, can improve the prediction of the vessel fuel consumption significantly compared to a white-box model. However, a prediction of the vessel speed is only improved slightly. Furthermore, the results give an indication of the potential advantages of grey-box models, which is extrapolation beyond a given training data set and the incorporation of physical phenomena which are not modeled in the white-box models. Finally, included is a discussion on how to enhance the predictability of the grey-box models as well as updating the neural-network in real-time.
