About the New Paradigm of Modeling Real Objects

Abstract

The advantages and disadvantages of the classical approach to mathematical modeling of real objects and processes occurring in them are considered. Building a model in the old paradigm requires precise knowledge of the equations and related conditions that determine with a certain accuracy the solution that is taken for the model. Such models practically do not have the property of adapting to data changes. Within the framework of the new paradigm, a more promising approach is proposed: based on updated heterogeneous data – equations, laws of physics, properties of symmetry, etc. – a hierarchy of adaptive models is being built, which can be refined and rebuilt in accordance with observational data about the object. The characteristic features and stages of the new method are noted: the decomposition of the object, the characteristic of the quality of the model, the choice of basic elements, the selection of parameters and structure of the model, adaptation and the possibility of alienation and transfer of the model. The possibilities and advantages of parametric models are discussed. A new class of multilayer models based on Analytical Modification of Numerical Methods (AMNM) is proposed, which can be used to build a wide range of mathematical models without a resource-intensive complex training procedure for neural networks. In a neural network situation, this approach is closely related to deep learning (DL) issues. An adequate apparatus for constructing models in a new paradigm – physics-informed neural networks (PINN) – is discussed, and the use of accumulated machine learning (ML) results for PINN is recommended.