Numerical Simulation of Direct and Inverse Problems of Pollutant Diffusion Using Advanced Numerical Methods

This paper investigates numerical modeling of transport and diffusion processes of harmful impurities governed by the advection–diffusion–reaction equation, along with the solution of corresponding direct and inverse problems. Particular emphasis is placed on identifying pollution source parameters and reconstructing spatiotemporal concentration distributions from limited and noisy observational data. Classical numerical methods, including stable finite-difference schemes, are employed for solving direct problems. Inverse problems are tackled using modern approaches such as regularization techniques, global optimization, and machine learning methods. In particular, evolutionary optimization algorithms and physics-informed neural networks (PINNs) are considered, enabling the integration of physical laws, observational data, and prior information within a unified computational framework. Computational experiments demonstrate that hybrid approaches combining classical numerical methods with machine learning significantly enhance the accuracy and stability of inverse problem solutions, especially under incomplete or noisy data conditions. Neural network-based methods exhibit strong approximation capabilities and effectively recover unknown model parameters. The results highlight the potential of integrating numerical and intelligent methods for environmental monitoring and pollutant dispersion forecasting, and can be applied in the development of operational analysis and environmental management systems.