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The materials’ consolidation, especially ceramics, is very important in advanced research development and industrial technologies. Science of sintering with all incoming novelties is the base of all these processes. A very important question in all of this is how to get the more precise structure parameters within the morphology of different ceramic materials. In that sense, the advanced procedure in collecting precise data in submicro-processes is also in direction of advanced miniaturization. Our research, based on different electrophysical parameters, like relative capacitance, breakdown voltage, and [Formula: see text], has been used in neural networks and graph theory successful applications. We extended furthermore our neural network back propagation (BP) on sintering parameters’ data. Prognosed mapping we can succeed if we use the coefficients, implemented by the training procedure. In this paper, we continue to apply the novelty from the previous research, where the error is calculated as a difference between the designed and actual network output. So, the weight coefficients contribute in error generation. We used the experimental data of sintered materials’ density, measured and calculated in the bulk, and developed possibility to calculate the materials’ density inside of consolidated structures. The BP procedure here is like a tool to come down between the layers, with much more precise materials’ density, in the points on morphology, which are interesting for different microstructure developments and applications. We practically replaced the errors’ network by density values, from ceramic consolidation. Our neural networks’ application novelty is successfully applied within the experimental ceramic material density [Formula: see text] [kg/m 3 ], confirming the direction way to implement this procedure in other density cases. There are many different mathematical tools or tools from the field of artificial intelligence that can be used in such or similar applications. We choose to use artificial neural networks because of their simplicity and their self-improvement process, through BP error control. All of this contributes to the great improvement in the whole research and science of sintering technology, which is important for collecting more efficient and faster results. 

Many recently published research papers examine the representation of nanostructures and biomimetic materials, especially using mathematical methods. For this purpose, it is important that the mathematical method is simple and powerful. Theory of fractals, artificial neural networks and graph theory are most commonly used in such papers. These methods are useful tools for applying mathematics in nanostructures, especially given the diversity of the methods, as well as their compatibility and complementarity. The purpose of this paper is to provide an overview of existing results in the field of electrochemical and magnetic nanostructures parameter modeling by applying the three methods that are “easy to use”: theory of fractals, artificial neural networks and graph theory. We also give some new conclusions about applicability, advantages and disadvantages in various different circumstances. 

The world energy crisis necessitated the cause of the research interest into new, renewable and alternative energy sources. From this point of view, there is research on phenomena and different synthetic methods and on structure and electronic property optimization expressed by important material and device advancement. Efficiency and electricity generation (batteries, fuel cells, hydrogen energy) are nowadays actual questions. Because of that research, innovations and applications require extended knowledge by fractal nature characterization. The electrochemical energy sources solutions, especially electrolytes, are in fractal nature science focus. Based on the research novelties, especially electronic materials, we presented an investigation on fractal structure influence in electrochemistry. We explore the activation energy and fundamental thermodynamic functions and values, also the electrode surface changed by complex fractal correction through fractal dimension of grains and pores, and Brownian motion of involved particles, as well. At the end, the electrochemical Arrhenius and Butler–Volmer equation fractalization is applied. All of these open new perspectives for electrochemical energy processes, within electrolyte bulk and related electrodes and more precise energy generation. This is important for semiconductor processing in solar cells and devices. So, we included the knowledge of fractal sciences advancement in this field for current–voltage equation.