A Data-Driven Framework for Discovering Fractional Differential Equations in Complex Systems

TL;DR

This work addresses the challenge of modeling complex systems with memory effects by introducing a data-driven framework to discover fractional differential equations (FDEs) directly from data. It fuses deep neural networks for denoising and super-resolution with Gauss-Jacobi quadrature to robustly compute fractional derivatives, and employs an alternating optimization strategy to identify both the equation structure and fractional orders. The method is demonstrated on three fronts: experimental frozen-soil creep revealing a fractional Kelvin model, synthetic fractional advection-diffusion equations, and Lévy-motion–based stable laws, showing successful recovery of fractional orders and coefficients under noise. This white-box approach advances interpretable modeling of nonlocal dynamics and memory effects, with potential impact across geophysics, transport in heterogeneous media, and complex systems exhibiting anomalous diffusion.

Abstract

In complex physical systems, conventional differential equations often fall short in capturing non-local and memory effects, as they are limited to local dynamics and integer-order interactions. This study introduces a stepwise data-driven framework for discovering fractional differential equations (FDEs) directly from data. FDEs, known for their capacity to model non-local dynamics with fewer parameters than integer-order derivatives, can represent complex systems with long-range interactions. Our framework applies deep neural networks as surrogate models for denoising and reconstructing sparse and noisy observations while using Gaussian-Jacobi quadrature to handle the challenges posed by singularities in fractional derivatives. To optimize both the sparse coefficients and fractional order, we employ an alternating optimization approach that combines sparse regression with global optimization techniques. We validate the framework across various datasets, including synthetic anomalous diffusion data, experimental data on the creep behavior of frozen soils, and single-particle trajectories modeled by Lévy motion. Results demonstrate the framework's robustness in identifying the structure of FDEs across diverse noise levels and its capacity to capture integer-order dynamics, offering a flexible approach for modeling memory effects in complex systems.

Figures (7)

• Figure 1: Workflow for data processing and generating the library of derivative terms. MaxIt in this figure represents the predefined maximum iteration number for optimization.
• Figure 2: (a) Evolution of sparse coefficients estimated via our alternating optimization approach. The y-coordinate represents the best sparse coefficients $\xi$ estimated by STRidge under fixed fractional orders, the x-coordinate represents the iterations of the global optimization algorithm. (b) Evolution of the loss function \ref{['eq:reg']} with fractional orders. The specific example presented in this figure involves a fractional advection-diffusion equation that incorporates 5% uniform noise, as discussed in Section \ref{['sec:fade_discovery']}.
• Figure 3: Reconstruction of experimental data for clay and silt.
• Figure 4: The upper panel illustrates the spatiotemporal distribution of synthetic data, governed by a fractional advection-diffusion equation, under varying noise levels. The lower panel shows the reconstructed data for the upper panel, generated using a DNN.
• Figure 5: Single particle trajectory of Lévy motions.