Solution of the Spatial Solow-Swan Model on the Brazilian Map by PINNs

This work investigates the spatial dynamics of capital using a two-dimensional extension of the Solow-Swan growth model defined on the Brazilian territory. The computational domain is obtained from an IBGE shapefile and converted to a Cartesian coordinate system using the Web Mercator projection, enabling the construction of a spatial database with sampled interior and boundary points. Capital dynamics are described by a reaction-diffusion partial differential equation derived from a Cobb-Douglas production function with spatially distributed labor and saving rate. The problem is solved using Physics-Informed Neural Networks (PINNs), where a multi-layer perceptron is trained by minimizing the residuals of the governing equation, the initial condition, and homogeneous Neumann boundary conditions. This approach aims to approximate the spatio-temporal distribution of capital over the Brazilian territory, providing a computational framework for studying spatial economic growth in irregular geographic domains.