A new method to solve the forward and inverse problems for the spatial Solow model by using Physics Informed Neural Networks (PINNs)

The spatial Solow model can take into account the geographical interdependence and the spatial organization of economic activities, and offers a better understanding of economic growth. In this work, governing equations of the spatial Solow model were solved by using the Physics Informed Neural Networks (PINNs) method, and both the forward and inverse problems were considered. For the forward problems, the conditions with and without considering the technology progress were solved, and the results were validated against the existing ones and good agreement can be found. For the inverse problems, the parameter identification of the production function was conducted by using very sparse data points. For the data without noise, two parameters of the production function can be estimated by using only 2 data points, where the errors can be below 3 %. For the low level noisy data, the parameters can also be inversed with 30 data points, and the errors for the two parameters were both less than 1 %.