Refining advection–diffusion–reaction models based on data assimilation with sensitivity operators and machine learning methods

Atmospheric chemistry dynamics, described by nonlinear advection–diffusion–reaction equations, is characterized by high uncertainty and variability. A two-stage algorithm for refining an advection–diffusion–reaction model based on measurement data is investigated. Refinement is achieved by adding a specially “trained” parametric term to the model equations. 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 performed using an algorithm based on sensitivity operators and adjoint ensembles [1]. 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 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. A 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 using an operator splitting scheme, resulting in an advection–diffusion–reaction model. The developed algorithm for identifying the “true” parametric element based on measurement data from a regular monitoring network has been tested on the regional and urban inverse modeling scenarios. The resulting refined mathematical model can be used to produce forecasts and, potentially, serve as a basic model for data assimilation.

The development of a general algorithm for refining a mathematical model was carried out within the framework of the state assignment of the ICM&MG SB RAS FWNM-2025-0003; adaptation and testing for urban scenarios were carried out with the support of the Russian Science Foundation and Government of the Novosibirsk Region, grant No. 25-11-20061, https://rscf.ru/project/25-11-20061/.

References:

1. Penenko A. V., Penenko V. V., Tsvetova E. A. Data Assimilation Algorithms for Atmospheric Chemistry Models // Izvestiya, Atmospheric and Oceanic Physics. 2025. Т. 61. № 3. С. 378–390. doi: 10.1134/S0001433825700719