Integration of Classical Mathematical Modeling with an Artificial Neural Network for the Problems with Limited Dataset

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.