# Machine Learning of Space-Fractional Differential Equations

arXiv: https://arxiv.org/abs/1808.00931 · HTML: https://mlsanity.com/paper/1808.00931/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Aug 2018.

## Built on by (most cited first, 12 of 12)

- [Novel Deep neural networks for solving Bayesian statistical inverse](https://mlsanity.com/paper/2102.03974/) (2021)
- [A Data-Driven Framework for Discovering Fractional Differential Equations in Complex Systems](https://mlsanity.com/paper/2412.03970/) (2024)
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- [PDE-constrained Gaussian process surrogate modeling with uncertain data locations](https://mlsanity.com/paper/2305.11586/) (2023)
- [Learning epidemic trajectories through Kernel Operator Learning: from modelling to optimal control](https://mlsanity.com/paper/2404.11130/) (2024)
- [Physics-Informed Deep Learning for Traffic State Estimation](https://mlsanity.com/paper/2101.06580/) (2021)
- [Occupation Kernel Hilbert Spaces for Fractional Order Liouville Operators and Dynamic Mode Decomposition](https://mlsanity.com/paper/2102.13266/) (2021)
- [Learning differential equations from data](https://mlsanity.com/paper/2205.11483/) (2022)
- [A numerical approach for the fractional Laplacian via deep neural networks](https://mlsanity.com/paper/2308.16272/) (2023)
- [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)

## Abstract (arXiv)

Data-driven discovery of "hidden physics" -- i.e., machine learning of differential equation models underlying observed data -- has recently been approached by embedding the discovery problem into a Gaussian Process regression of spatial data, treating and discovering unknown equation parameters as hyperparameters of a modified "physics informed" Gaussian Process kernel. This kernel includes the parametrized differential operators applied to a prior covariance kernel. We extend this framework to linear space-fractional differential equations. The methodology is compatible with a wide variety of fractional operators in ℝd and stationary covariance kernels, including the Matern class, and can optimize the Matern parameter during training. We provide a user-friendly and feasible way to perform fractional derivatives of kernels, via a unified set of d-dimensional Fourier integral formulas amenable to generalized Gauss-Laguerre quadrature. The implementation of fractional derivatives has several benefits. First, it allows for discovering fractional-order PDEs for systems characterized by heavy tails or anomalous diffusion, bypassing the analytical difficulty of fractional calculus. Data sets exhibiting such features are of increasing prevalence in physical and financial domains. Second, a single fractional-order archetype allows for a derivative of arbitrary order to be learned, with the order itself being a parameter in the regression. This is advantageous even when used for discovering integer-order equations; the user is not required to assume a "dictionary" of derivatives of various orders, and directly controls the parsimony of the models being discovered. We illustrate on several examples, including fractional-order interpolation of advection-diffusion and modeling relative stock performance in the S&P 500 with alpha-stable motion via a fractional diffusion equation.