On Numerical Solutions of Some Nonlinear Obstacle-Type Problems Based on Modern Artificial Intelligence Tools

The project aims to study some free-boundary obstacle-type problems that arise in applied sciences. Applications of such differential equations include engineering, physics, financial mathematics, and other fields.

Currently, one of the approaches to constructing numerical solutions to these problems involves the use of AI tools to perform more accurate approximations and simulations in science and engineering.

It is planned to study free-boundary obstacle-type problems to achieve both theoretical and numerical results. In particular, for the problem of non-local financial bubbles, there is an intention to develop new numerical algorithms and study their stability and convergence. It is planned to use Physics Informed Neutral Networks (PINN), an important direction in machine learning, where neural networks learn to solve partial differential equations by incorporating physical principles into the mathematical model.