Neural Networks with Adaptive and Trainable Activation Functions Applied to Solve Stochastic Differential Equations

• Ingrid de Araújo Felix UFRPE
• João Henrique de Medeiros Delgado UFRPE
• Tiago A. E. Ferreira UFRPE

Resumo

This work investigates the use of artificial neural networks (ANNs) with adaptive and trainable activation functions in solving stochastic differential equations (SDEs). The study builds upon the growing application of ANNs in ordinary (ODEs) and partial differential equations (PDEs). Still, it explores the effectiveness of these networks when handling the stochastic factor present in SDEs. The activation function plays a crucial role in the performance and convergence of neural networks. In this context, we test the Universal Activation Function (UAF), comparing its results with two other adaptive and trainable functions: GLN-Mish and GNL-ReLU. Using the NeuroDiffEq library in Python, we solve equations such as the Langevin equation and the Cox-Ingersoll-Ross equation, adjusting various parameters of both the network and the equation itself. The accuracy of the solutions was evaluated using the Mean Squared Error (MSE). The Diebold-Mariano (DM) statistical test was then applied, at a confidence level 5%, confirming that differences in the predictive performance of activation functions were not statistically significant in certain cases. The results indicate that adaptive activation functions outperform in solving SDEs, offering flexibility and precision in adjusting stochastic models.