Research Projects

Mathematical models serve as fundamental tools for describing physical, chemical, and biological systems in engineering and scientific research. Correct mathematical formulation related to a physical situation plays a central role in understanding, analyzing, and predicting a complex phenomenon. Moreover, there are many important physical, chemical, and biological processes that have not been adequately studied and are not understood till now. Therefore, a detailed studies need to be explored to present a more realistic mathematical model of a real situation and its accurate solution. From a mathematical point of view, most of the complex real-world problems in high dimensions are challenging because the solutions to these problems require special treatment. In the literature, the exact solutions to these problems are available for some limited cases only. Therefore, many classical numerical methods, such as the FEM, FDM, FVM, and meshless methods, are applied to solve high-dimensional nonlinear models. But these classical techniques are computationally expensive, time-consuming, and require substantial memory resources. To overcome the drawbacks of the classical numerical techniques, neural network schemes can be used to handle nonlinear models with irregular boundaries to enhance model accuracy, efficiency, and adaptability. Based on the above facts, the project aims to investigate and enhance the analysis of selected mathematical models commonly encountered in science and engineering by leveraging advanced neural network techniques. The first goal of the project focuses on the rigorous study connected to the development of mathematical models to address real-world problems across diverse domains, including heat and mass transfer, dynamical systems, fluid flow in porous media, and biological systems. By formulating governing equations based on physical laws and empirical observations, the project aims to derive/modify models that capture the essential behaviour of complex real systems. In the next step of this research project, attention will be given for establishing an efficient and accurate numerical tool for the solution of the problems. Special stress will be given to develop/apply advanced neural network schemes, like Physics-Informed Neural Networks (PINNs), Extended Physics-Informed Neural Network (XPINNs), etc., for some mathematical models, aiming to enhance analytical capabilities, improve predictive accuracy, and uncover new insights into complex scientific and engineering systems. Emphasis will also be placed on model validation and the gap between traditional modelling approaches and emerging computational frameworks, enabling the solution of nonlinear, high-dimensional, and multiscale problems that are otherwise intractable. The outcomes of this project are expected to advance both theoretical and practical understanding of selected problems in science and engineering.