Preface to the Special Issue on “Multiscale Mathematical Modeling”

It is not easy to give a precise definition of the concept of multiscale modelling (MM). As is customary nowadays, we posed our question to AI and received the following response: “MM is an approach to system simulation that integrates multiple levels of detail, often employing diverse models to improve sampling efficiency and capture phenomena across different scales”.

Having accepted this definition as a working one, let us say a few words about the essence of the matter. Predictions of large-scale behaviour based on fine-scale theory are a key goal of modern natural sciences and of particular importance to numerous engineering applications. However, for a long time, this description was understood as the construction of a hierarchy of models describing processes of different scales according to phase space variables (temporal, spatial, etc.). In this case, for a more accurate description, one simply needs to change the scale. This reductionist approach has proven effective in physics and mechanics. However, with the increasing complexity of the phenomena under consideration, especially with the transition to nanostructures and metamaterials, it has become clear that describing complex systems based solely on knowledge of the dynamics or behaviour of their components is often insufficient. For example, in modern materials science, it is often necessary to consider several different and distinct scales of the object of investigation simultaneously. The set of concepts and algorithms that enable the analysis of objects and phenomena in various fields of science and engineering, taking into account the interaction of different scales, has been named MM. For example, “in physics and chemistry, multiscale modeling is aimed at the calculation of material properties or system behavior on one level using information or models from different levels” (https://en.wikipedia.org/wiki/Multiscale_modeling (accessed on 25 December 2025)).

Recent major progress in this area is due to the development of advanced computational and experimental methodologies, along with the establishment of powerful mathematical frameworks operating on multiple length and time scales. Computational MM can make accurate predictions about various phenomena, simultaneously considering several different and distinct scales of the phenomena. However, as stated by R. Thom, “Prediction is not explanation”.

What is meant by explanation? Once again, we encounter an ambiguous concept, the detailed interpretation of which would lead us too far into philosophical territory. For a significant portion of the current generation of mathematicians and physicists, explanation means the existence of a model within the framework of continuous mathematics. We will not deny ourselves the pleasure of quoting Kolmogorov, who wrote about the relationship between discrete and continuous mathematics: “It is quite probable that with the development of modern computing techniques, it will become understood that in many cases it is reasonable to study real phenomena without making use of the intermediate step of their stylization in the form of infinite and continuous mathematics, passing directly to discrete models. Pure mathematics was successful developed mainly as the science of the infinite. Apparently, this state of affairs is deeply rooted in our consciousness, which operates with great ease with an intuitively clear understanding of unbounded sequences, limiting processes, continuous and even smooth manifolds, etc.”

Whether this situation is related to the peculiarities of our subconscious or is caused by the modern education system, which focuses primarily on continuous mathematics, is a matter of debate. However, the prevalence of continuous mathematics models in modern science and engineering is evident. At the same time, we are witnessing a rapidly growing trend towards discrete models. Of course, discrete and continuous approaches to modelling the real world are not competing but complementary. Modelling multiscale phenomena both mathematically and computationally allows us to describe the phenomenon under study in its entirety. Although this SI is mainly devoted to issues of mathematical MM, it also addresses computational aspects of MM.

The problems of MM are often interdisciplinary in nature, as confirmed by this SI, which includes articles devoted to the application of MM in the theories of discrete systems (contributions 1 and 2), elasticity (contributions 3 and 4) and hydroelasticity (contribution 5), composite materials, and metamaterials and quasicrystals (contributions 6, 7 and 8), as well as in hydromechanics (contribution 9), biomechanics (contribution 10), geomechanics (contribution 11), and tribology (contribution 12). Among the methods of mathematical MM, asymptotic approaches stand out. The articles in this SI used methods of continuation (contributions 1 and 2), homogenisation (contribution 3), the regular (contribution 9) and singular (contributions 4, 5 and 7), perturbation methods, and Padé approximants (contribution 1). The theory of self-adjoint operators (contribution 6) and the integral formalism based on two-dimensional Green tensors (contribution 8) were also applied. Multiscale numerical approaches (contributions 10 and 11) and multiscale statistical analysis (contribution 12) were also used. The results obtained confirm the importance of considering discreteness for constructing refined continuous models (contributions 1, 2 and 3). It has been shown that when describing composite materials, especially those with high-contrast components, it is not sufficient to determine only the effective characteristics; the microstructure must also be taken into account (contribution 6). In general, MM requires, along with the study of the global behaviour of systems, the analysis of local effects (contributions 4, 5, 6 and 7).

As guest editors, we would like to sincerely thank the authors who responded to our invitations and submitted their work to this SI. The constructive and insightful feedback from reviewers contributed to improving the quality of the accepted articles and enhancing their style of presentation. We would also like to note the professionalism of the editors of the MDPI journal Mathematics.

The aim of this SI was to demonstrate effective MM methods and their applicability for in-depth analysis of systems and processes in various fields of science and engineering. We hope that the published works will be of interest to readers, both for the specific results obtained and for the general ideology of the research and the methods and approaches used.