MODIFICATION OF THE SEPARABLE PHYSICS-INFORMED NEURAL NETWORK METHOD FOR THE APPROXIMATE SOLUTION OF A MIXED PROBLEM FOR THE HEAT CONDUCTION EQUATION IN AN ELLIPTICAL DOMAIN

Abstract

Background. Parabolic differential equations are mathematical models in engineering designed to study wide range of applied models. This necessitates the development of approximate methods for solving them. Classical numerical methods based on finite difference and finite elements are difficult to apply to non-rectangular geometries of the domain and to high-dimensional domains. This becomes a problem even in the two-dimensional case. Neural networks methods, such as the Physics-informed neural network (PINN), can be applied to problems with complex domain geometry, but they still suffer from the problem of increasing domain dimension. Objective. Develop a modified approach based on Separable PINN (SPINN) for the approximate solution of a mixed problem for the heat conduction equation with Dirichlet boundary conditions in an elliptical domain. Methods. Application of the SPINN method after first converting the boundary value problem to a cylindrical coordinate system and applying modifications to the loss function and the point generation method to avoid problems associated with the singularity at zero in the Laplace operator in polar coordinates. Results. A modification of the SPINN method has been developed for the approximate solution of a mixed problem for the heat conduction equation with Dirichlet boundary conditions in an elliptical domain. The proposed approach is demonstrated on examples alongside the finite difference method and the PINN method to compare the results obtained. In each case, SPINN demonstrated lower relative errors and shorter computation times compared to both the finite difference method and the PINN method. Conclusions. The described application of the SPINN architecture to solving a mixed problem for the heat conduction equation allows for the optimization of engineering modeling of thermophysical processes in elliptical-shaped plates. The developed modification of the SPINN method demonstrated, through examples, better relative error values (1.25 times smaller in the first example and 1.15 times smaller in the second) with significantly shorter computation times (5 times shorter in the first example and 2 times shorter in the second) compared to PINN. Compared to the finite difference method, the difference is even more noticeable. This result confirms the effectiveness of the method and opens prospects for its further application to boundary value problems in curvilinear domains with arbitrary smooth boundaries.