A detailed and comprehensive account of fractional Physics-Informed Neural Networks: From implementation to efficiency

TL;DR

This paper thoroughly examines fractional physics-informed neural networks, exploring their implementation, accuracy, and computational efficiency in solving various fractional differential equations using Caputo formalism.

Contribution

It provides a comprehensive analysis of fractional physics-informed neural networks, including strategies to enhance accuracy while reducing computational costs across different fractional differential equations.

Findings

Networks accurately solve fractional equations with minor initial discrepancies

Caputo formalism's historical data requirement increases computational burden

Strategies can improve accuracy without heavy computational costs

Abstract

Fractional differential equations are powerful mathematical descriptors for intricate physical phenomena in a compact form. However, compared to integer ordinary or partial differential equations, solving fractional differential equations can be challenging considering the intricate details involved in their numerical solutions. Robust data-driven solutions hence can be of great interest for solving fractional differential equations. In the recent years, fractional physics-informed neural network has appeared as a platform for solving fractional differential equations and till now, efforts have been made to improve its performance. In this work, we present a fully detailed interrogation of fractional physics-informed neural networks with different foundations to solve different categories of fractional differential equations: fractional ordinary differntial equation, as well as two and three dimensional fractional partial differential equations. These equations are solved employing two numerical methods based on the Caputo formalism. We show that these platforms are generally able to accurately solve the equations with minor discrepancies at initial times. Nonetheless, since in Caputo formalism, the value of a fractional derivative at each point requires the function's value in all of its previous history, it is computationally burdensome. Here, we discuss strategies to improve accuracy of fractional physics-informed neural networks solutions without imposing heavy computational costs.