The article discusses building hybrid models relating classical numerical methods for solving ordinary and partial differential equations and the universal neural network approach being developed by D Tarkhov and A Vasilyev. The different ways of constructing multilayer neural network structures based on grid methods are considered. The technique of building a continuous approximation using one simple modification of classical schemes is presented. Introduction non-linear relationships into the classic models with and without posterior learning are investigated. The numerical experiments are conducted.

The article deals with the problem of predicting of the temporal elongation law of the sample under dynamic loading. The determination of tensile behavior of samples under uniaxial loading is performed by a standard tensile method. The neural network approach is applied to construct an approximate elongation-force dependence using measurement data and posterior model of the dependence of rupture conditions on the neural network parameters. The considered approach can be used in the building industry.

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This work is devoted to the description and comparative study of some methods of mathematical modeling. We consider methods that can be applied for building cyber-physical systems and digital twins. These application areas add to the usual accuracy requirements for a model the need to be adaptable to new data and the small computational complexity allows it to be used in embedded systems. First, we regard the finite element method as one of the “pure” physics-based modeling methods and the general neural network approach as a variant of machine learning modeling with physics-based regularization (or physics-informed neural networks) and their combination. A physics-based network architecture model class has been developed by us on the basis of a modification of classical numerical methods for solving ordinary differential equations. The model problem has a parameter at some values for which the phenomenon of stiffness is observed. We consider a fixed parameter value problem statement and a case when a parameter is one of the input variables. Thus, we obtain a solution for a set of parameter values. The resulting model allows predicting the behavior of an object when its parameters change and identifying its parameters based on observational data.