UniFIDES: Universal Fractional Integro-Differential Equation Solvers

Abstract

The development of data-driven approaches for solving differential equations has been followed by a plethora of applications in science and engineering across a multitude of disciplines and remains a central focus of active scientific inquiry. However, a large body of natural phenomena incorporates memory effects that are best described via fractional integro-differential equations (FIDEs), in which the integral or differential operators accept non-integer orders. Addressing the challenges posed by nonlinear FIDEs is a recognized difficulty, necessitating the application of generic methods with immediate practical relevance. This work introduces the Universal Fractional Integro-Differential Equation Solvers (UniFIDES), a comprehensive machine learning platform designed to expeditiously solve a variety of FIDEs in both forward and inverse directions, without the need for ad hoc manipulation of the equations. The effectiveness of UniFIDES is demonstrated through a collection of integer-order and fractional problems in science and engineering. Our results highlight UniFIDES' ability to accurately solve a wide spectrum of integro-differential equations and offer the prospect of using machine learning platforms universally for discovering and describing dynamical and complex systems.

Figures (9)

• Figure 1: The general architecture of UniFIDES for solving the forward problem of partial fractional integro-differential equations (PFIDEs). The same architecture is slightly modified to solve inverse problems. The integer-order derivatives are handled using AD (the top narrow box), while integrals and fractional derivatives are calculated through the numerical scheme introduced in \ref{['sec:Methods']}. The training is halted once the relative error plateaus or if the maximum number of iterations ($N_\mathrm{it}$) is reached. The training process and hyperparameters are described in detail in \ref{['sec:SI-train']}.
• Figure 2: The solution by UniFIDES and the exact solution for (a) 1D Fredholm IDE (Case 1) and (b) 3D Fredholm IE (Case 2). The absolute squared error ($\epsilon^2$) for line graphs is shown with red shades and is multiplied by a constant in this work (1.0e5 in this case). In panel (b), the prediction MSE is 1.07e-6.
• Figure 3: The solution by UniFIDES and the exact solution for (a) 1D Volterra IDE (Case 3) and (b) 2D Volterra IE (Case 4).
• Figure 4: The solution by UniFIDES and the exact solution for (a) 1D Volterra FIDE (Case 5) and (b) 2D Volterra partial FIDE (Case 6) for $\beta=0.7$.
• Figure 5: The solution by UniFIDES and the exact solution for the system of nonlinear Volterra FIDEs defined in Case 7 and for $\beta=0.5$.