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

arXiv: https://arxiv.org/abs/2412.03970 · HTML: https://mlsanity.com/paper/2412.03970/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 80 references that are in the map.
- First posted on arXiv 5 Dec 2024.

## Builds on (most cited first, 17 of 17)

- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm](https://mlsanity.com/paper/2001.07305/) (2020)
- [Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)](https://mlsanity.com/paper/2106.11927/) (2021)
- [DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data](https://mlsanity.com/paper/1908.04463/) (2019)
- [Robust discovery of partial differential equations in complex situations](https://mlsanity.com/paper/2106.00008/) (2021)
- [Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FROND](https://mlsanity.com/paper/2404.17099/) (2024)
- [Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data](https://mlsanity.com/paper/2011.11981/) (2020)
- [LLM4ED: Large Language Models for Automatic Equation Discovery](https://mlsanity.com/paper/2405.07761/) (2024)
- [Integration of knowledge and data in machine learning](https://mlsanity.com/paper/2202.10337/) (2022)
- [Machine Learning of Space-Fractional Differential Equations](https://mlsanity.com/paper/1808.00931/) (2018)
- [Physics-constrained robust learning of open-form partial differential equations from limited and noisy data](https://mlsanity.com/paper/2309.07672/) (2023)
- [Interpretable AI-Driven Discovery of Terrain-Precipitation Relationships for Enhanced Climate Insights](https://mlsanity.com/paper/2309.15400/) (2023)
- [Neural Fractional Differential Equations](https://mlsanity.com/paper/2403.02737/) (2024)

## Built on by (most cited first, 3 of 3)

- [On the post-hoc Evaluation of PDE Discovery: A Multifaceted Challenge of Scientific Advancement](https://mlsanity.com/paper/2607.23753/) (2026)
- [Robust data-driven discovery of fractional differential equations via weak formulations and Pareto-based subset selection](https://mlsanity.com/paper/2608.12879/) (2026)
- [Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fields](https://mlsanity.com/paper/2608.27475/) (2026)

## Abstract (arXiv)

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 L'evy 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.