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.

A novel type of neural network with an architecture based on physics is proposed. The network structure builds on a body of analytical modifications of classical numerical methods. A feature of the constructed neural networks is defining parameters of the governing equations as trainable parameters. Constructing the network is carried out in three stages. In the first step, a neural network solution to an equation corresponding to a numerical scheme is constructed. It allows for forming an initial low-fidelity neural network solution to the original problem. At the second stage, the network with physics-based architecture (PBA) is further trained to solve the differential equation by minimising the loss function, as is typical in works devoted to physics-informed neural networks (PINNs). In the third stage, the physics-informed neural network with architecture based on physics (PBA-PINN) is trained on high-fidelity sensor data, parameters are identified, or another task of interest is solved. This approach makes it possible to solve insufficiently studied PINN problems: selecting neural network architecture and successfully initialising network weights corresponding to the problem being solved that ensure rapid convergence to the loss function minimum. It is advisable to use the devised PBA-PINNs in the problems of surrogate modelling and modelling real objects with multi-fidelity data. The effectiveness of the approach proposed is demonstrated using the problem of modelling processes in a chemical reactor. Experiments show that subsequent retraining of the initial low-fidelity PBA model based on a few high-accuracy data leads to the achievement of relatively high accuracy.

An algorithm based on continuous measurement of multiphase flows of oil well production has been designed to improve the efficiency of the technical control of oil production processes in the field. Separation-free, non-contact measurement of multiphase flows of oil well products allows increasing the efficiency of managing oil production processes in the field. Monitoring the current density using radioisotope measuring transducers (RMTs) allows obtaining information about the structure of the flow in the form of the distribution of gas inclusions and the speed of movement of liquid and gas in a two-phase flow. Fluid velocity measurement is based on digital processing of RMT signals, applying a continuous or discrete undecimated wavelet transform to them, and assessing the cross-correlation of wavelet coefficients in individual subspaces of the wavelet decomposition. The cross-correlation coefficients of two RMT signals located at a base distance, calculated in the subspaces of the wavelet decomposition, characterize the speed of movement of gas bubbles of different sizes in a vertical pipe. The measurement assumes that the velocity of the liquid phase of the oil flow in a vertical pipe mainly corresponds to the velocity of small bubbles. This speed should be determined by the maximum cross-correlation of wavelet coefficients in the corresponding decomposition subspace. Computer modeling made it possible to evaluate the characteristics of the algorithm for controlling the speed of liquid movement in the gas–liquid flow of oil well products and determine the mass flow rate of the liquid and the relative value of the gas content. The implementation of the algorithm in a multi-channel version of the device allows monitoring an entire cluster of wells in the field.