A physics-guided deep neural network approach for simulating variable-order fractional chaotic financial models

Significance

Constant fractional-order mathematical models have highlighted key aspects of real-life systems, but a major advancement occurred when scientists began exploring variable-order models. This study holds two main considerable aspects, firstly, it addresses a variable-order fractional financial mathematical system, and secondly, it solves this model using the PINNs methodology.

Purpose

This study explores the numerical solution of variable order fractional financial mathematical model through Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) optimizer-based Physics-Informed Neural Networks (PINNs). This framework captures the nonlinear, complex and memory-dependent features inherent in fractional financial model with Caputo variable-order fractional differential operator. This scheme ensures fast convergence and provides robust approximations of chaotic trajectories under varying fractional orders. This study shows better representation of the intricacies of financial systems. There are discussed two main cases of integer and variable order fractional system. Furthermore, there are three subcases of variable case with different functions. For each case different plots are displayed, which show chaotic behavior.

Findings

In this study, it is focused on a generalized variable-order fractional financial system that incorporates key economic components: interest rate, investment demand, and price index. Results confirm that the method achieves a convergence order consistent with theoretical expectations. Our findings suggest that the function employed in Case 2c, $\left(O_3\left(t\right)=\left(\frac{1}{15}\right)e^{\sin \left(\frac{1}{25}\right)t}+0.76\right)$ is the most effective in capturing the complex and non-stationary nature of real financial dynamics.