Symmetry Applied in Mathematical Modeling and Computational Methods

Dear Colleagues,

Mathematical modelling has become an essential tool in practically all technical fields nowadays. Applications in all areas of research appear every day: the fields of medicine, engineering, ecology, life sciences, social sciences, economy, psychology, etc., benefit from the emergence of novel mathematical techniques and the application of old ones in creative and synergetic approaches. The relevance of this perspective is so influential at present that cannot be ignored by any researchers in cross-disciplinary fields.

The concept of Symmetry plays a key role in many mathematical and computational methods. For example, in the analytical–numerical homotopy method, the continuous transformation of a differential operator is used to find the solutions in terms of a deformation parameter. From this symmetry transformation, the solution for the null value of this parameter is finally found and this corresponds to the original problem.

On the other hand, the integration techniques known as symplectic methods are based upon the fact that the Hamiltonian equations are a symplectomorphism in the phase space. These methods have also provided a wide range of results, such as the integration of orbital equations over very large periods, that would be very difficult to obtain with other, more conventional methods.

In the last few years, we have also seen an emergence of renovated interest in artificial neural networks, spurred by the availability of faster computers with improved architectures capable of training these models efficiently. Neural networks are, after all, also mathematical models based upon the concept of artificial neurons and artificial synapses or links through which the information flows via linear combinations and activation functions. Symmetry also plays a key role in the difficult task of training the network as breaking the symmetry by choosing a random distribution of weigh values for the links is fundamental to achieving convergence.

Exploiting the symmetries in the data is also a powerful method to reduce the amount of data required for training. Moreover, symmetric neural network architectures are also amenable to special training algorithms that exploit these symmetries to reduce training time. Many interdisciplinary applications of neural networks can also benefit from these new algorithms based upon symmetry, including the successful use of machine learning and networks in medicine, biology, business, and ecology, among others.

On a very different kind of application, we must also remember that the whole concept of symmetry breaking in gauge theories for the forces of nature is at the very root of our most basic model of nature: The Standard Model of particle physics. Symmetry breaking in the very early Universe could explain also some persistent riddles in our understanding of nature such as the matter–antimatter asymmetry or the production of a stochastic background of gravitational waves.

The topics of research areas covered for this Special Issue are as follows (not being an exhaustive list):

• Numerical methods and symmetry;
• Novel techniques of neural network training;
• Mathematical models in physics and symmetry;
• Symmetric networks in epidemiology;
• Symmetric processes in ecological networks;
• Gauge theories models and spontaneous symmetry breaking in cosmology;
• Symmetries in optimal control theory and trajectory optimization for spacecraft;
• Symmetries in matrix analysis and linear algebra;
• Symmetric random walks and applications;
• Complex networks in the social sciences;
• Homotopy methods for numerical integration.

All papers submitted to the Special Issue will be thoroughly reviewed by at least two independent experts. We hope that this Special Issue will encourage change and the development of useful new tools and applications in different fields, and that it will enrich the scientific community for the researchers in the concerned fields.

Dr. Luis Acedo
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

• neural networks
• symplectic methods
• optimal control theory
• symmetries in mathematical models
• gauge theories
• symmetry breaking models