Solving Steady-State Elliptic Problems in Irregular Domains Using Physics-Informed Neural Networks and Fictitious Domain Methods

This paper introduces an innovative methodology for solving steady-state elliptic partial differential equations defined over irregular domains, by coupling the capabilities of Physics-Informed Neural Networks with the Fictitious Domain Method. The primary emphasis is placed on applications involving the heat equation, a fundamental model in thermal analysis where the complexity of non-standard geometries often poses significant challenges for traditional numerical methods. The proposed approach exploits the inherent strength of Physics-Informed Neural Networks in embedding the underlying physical laws directly into the learning process, enabling the model to approximate solutions without relying on mesh-based discretization. Simultaneously, the Fictitious Domain Method facilitates the treatment of irregular computational domains by embedding them within a larger, regular domain, thereby simplifying the application of boundary conditions and numerical operations. The synergy between these two techniques results in a flexible, efficient, and accurate computational framework that is well-suited for addressing heat transfer problems in complex geometrical configurations.