Universal Integer-Order and Fractional Integro-Differential Equation Solvers

This project, titled "UniFIDES: Universal Fractional Integro-Differential Equation Solvers," is currently submitted to Nature Machine Intelligence. This README provides a guide to installing and using the TensorFlow implementation of UniFIDES, a robust, user-friendly platform to solve integer-order and fractional FIDEs in both forward and inverse directions.

What to Expect

The repository contains Python scripts that solve two examples of Fredholm and Volterra equations in the form of a forward problem.

Steps to Install

1. Clone the Repository

   git clone https://github.com/procf/UniFIDES.git
   cd UniFIDES

2. Set Up a Virtual Environment (Optional but recommended)

   python -m venv venv
   source venv/bin/activate  # On Windows use `venv\Scripts\activate`

3. Install Required Packages

      pip install -r requirements.txt

Execution

To run the solver, execute the main.py script:

python main.py

Testing

The notebook was tested on an MBP M1 Max (64 GB RAM) without GPU acceleration running macOS Sonoma v14.2.1

The UniFIDES notebook

For a more unified experience, the UniFIDES' notebook is provided in Google Colab:

Background

Solving forward and inverse problems in the form of differential equations, which govern the spatio-temporal evolution of numerous dynamical systems across scientific and engineering disciplines, has been the core of extensive research endeavors for decades. Science-aware machine learning tools, nonetheless, have proven transformative over the past few years in relaxing many traditional constraints required to solve such systems numerically, such as numerical discretization, mesh generation, stability conditions, and more.

Although machine learning platforms have achieved notable success in solving differential equations, there is still a significant gap in developing universal machine learning solutions for integro-differential equations. This gap widens further when considering FIDEs, in which the integral and derivative operators may accept any real numbers. FIDEs are particularly challenging to solve yet are fundamental in modeling critical phenomena such as the stock market, pandemic spread, and population growth of species.

Here, we introduce a physics-aware platform named UniFIDES, designed to solve FIDEs for both forward and inverse problems. Agnostic to nonlinearity and the type of problem in nature, UniFIDES is tested on a variety of Fredholm and Volterra integer-order and fractional cases across a broad range of scientific applications and remained competitive in terms of accuracy and computation cost. With its out-of-the-box functionality, UniFIDES may be readily streamlined with current computational pipelines to augment machine intelligence.

The provided notebook solves two instances of Fredholm and Volterra equations by toggling the Type switch in the code. Let's begin with the former, which is an integer-order nonlinear 1D Fredholm IDE with applications in diffusion processes and quantum mechanics:

$$\begin{equation} \left\{ \begin{aligned} &\left[ {}^{1} \mathcal{D}_x \right] u(x) = \cos{x} - x + \frac{1}{4}\left[ {}^1 \mathcal{I}_{-1/4}^{1/4} \right] xt u^2(t) \, \mathrm{d}t \\\ &x \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]\\\ &u\left( -\frac{\pi}{2} \right)=0 \end{aligned} \right. \end{equation}$$

Here, $\left[ {}^{1} \mathcal{D}_x \right]$ is the derivative operator of order $\beta=1$, and $\left[ {}^1 \mathcal{I}_{-1/4}^{1/4} \right]$ is the integral operator of order $\alpha=1$ between $-1/4$ and $1/4$. In fact, $\alpha$ and $\beta$ can be any real numbers. The exact solution reads $u(x) = 1 + \sin{x}$. This is a forward problem, and the objective is to find $u(x)$. This problem is implemented in its continuous form in the notebook, and UniFIDES' prediction is plotted below.

Case 5 in this submission is a 1D Volterra fractional integral equation (FIE):

$$\begin{equation} \left\{ \begin{aligned} &u(x) = \sqrt{\pi}\left(1+x\right)^{-1.5} - 0.02\frac{x^3}{1+x} + 0.01x^{2.5}\left[ {}^{0.5} \mathcal{I}_{0}^{x} \right] u(t) \mathrm{d} t\\\ &x \in \left[ 0, 4 \right]\\\ &u\left(0\right)=\sqrt{\pi} \end{aligned} \right. \end{equation}$$

The integral operator in this case has a fractional order ($\alpha = 0.5$), and the exact solution reads $u(x) = \sqrt{\pi}(1 + x)^{-1.5}$. Such FIEs are frequently seen in crystal growth and heat conduction. By setting Type='Volterra', this FIE is solved by UniFIDES, whose prediction is plotted below:

Here's the submission 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.

Contributors

This work was done by Milad Saadat, Deepak Mangal, and Safa Jamali. Authors acknowledge the support from the National Science Foundation’s DMREF program through Award #2118962.