2026 , Volume 31, № 4, p.31-44

A two-stage algorithm for refining an advection – diffusion – reaction mathematical model based on measurement data is numerically investigated. Refinement is achieved by adding a specially “trained” parametric element to the model equation. The first stage involves identifying a general, non-stationary uncertainty function based on measurement data, corresponding to the source function of the basic advection – diffusion mathematical model. Identification is processed using an algorithm based on sensitivity operators and adjoint ensembles. In the second stage, the identification results, which include the state function and the uncertainty function values, are considered as a training sample to determine the functional relationship between them, which is implemented by a parametric element. This raises the problem of choosing the type (architecture) of this parametric element and subsequent selecting its parameters (“training”). The parametric element is defined as a production – destruction element with non-negative piecewise-linear production – destruction operators that include constant and neural network elements. The PyTorch machine learning framework is used to fit the parameters of the parametric element on a training dataset. The trained element is then incorporated into a basic advection – diffusion model implemented in C++ using an operator splitting scheme, resulting in an advection – diffusion – reaction model. The splitting scheme step for a parametric element is implemented using an explicit scheme. The developed algorithm for identifying the “true” parametric element based on measurement data from a regular monitoring network has been tested on the Baikal region scenarios. The resulting refined mathematical model can be used in other time intervals and, potentially serve as a basic model for data assimilation.