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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.

Segmentation of images from scanning electron microscope, especially multiphase, poses a drawback in their microstructure quantification process. The labeling process must be automatized due to the time consumption and irreproducibility of the manual labeling procedure. Here we show a swarm intelligence-driven filtration methodology performed on raw solid oxide fuel cell anode’s material images to improve the segmentation methods’ performance. The methodology focused on two significant parts of the segmentation process, which are filtering and labeling. During the first one, the images underwent filtering by applying a series of filters, whose operation parameters were determined using Particle Swarm Optimization upon a dedicated cost function. Next, Seeded Region Growing, k-Means Clustering, Multithresholding, and Simple Linear Iterative Clustering Superpixel algorithms were utilized to label the filtered images’ regions into consecutive phases in the microstructure. The improvement was presented for three different metrics: the Misclassification Ratio, Structural Similarity Index Measure, and Mean Squared Error. The obtained distribution of metrics’ performances was based on 200 images, with and without filtering. Results indicate an improvement up to 29%, depending on the metric and method used. The presented work contributes to the ongoing efforts to automatize segmentation processes fully for an increasing number of tomographic measurements, particularly in solid oxide fuel cell research.

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