Estimation of the Saving Rate Function in the Spatial Solow-Swan Model by PINNs

Resumen

The Solow-Swan model, developed in 1956 [6, 7], has been a key reference for understanding economic growth, but it does not account for the spatial distribution of activities, limiting its applicability. To address this limitation, spatial dimensions have been incorporated [1, 4], allowing researchers to study the impact of location on growth. Recently, researchers have been exploring inverse problems, such as obtaining the convex-concave production function in the spatial model [2, 3], but to the best of our knowledge, no study has yet investigated the determination of the saving rate function. In this context, this work aims to determine the saving rate, which depends solely on the spatial coordinate, by solving the inverse problem in the spatial Solow-Swan model. We propose a model that describes the evolution of capital density in local economies distributed along the compact interval Ω = [0, l], with 0 < l < ∞. At each point x ∈ Ω, there is a capital density K(t, x) ≥ 0 and a labor density L(x) ≥ 0, which are used to produce an aggregate good through a Cobb-Douglas production function f(K, L) = A(x)[K^φ(t, x)L(x)^1−φ]. Here, A(x) represents the technological factor, and φ ∈ (0, 1) indicates the intensity of capital usage. The distribution of labor is initially given by the exogenous function L(x) ≥ 0. Thus, the evolution of the capital stock in the economy is governed by a reactive-diffusive partial differential equation. We propose a methodology based on Physics-Informed Neural Networks [5] with two Multi-Layer Perceptron networks. The first network solves the PDE, while the second estimates the saving rate. Both networks use the Adam optimizer with a learning rate of 10⁻³ to calibrate the parameters. Studies were conducted to determine the architecture of each network. To evaluate the methodology, we consider a case study with obtained estimations of s(x) with a mean squared error of 8 · 10⁻⁶. Further work should extend test cases and apply real data from IBGE. [...]