Machine Learning of Space-Fractional Differential Equations
This work addresses the challenge of modeling systems with heavy tails or anomalous diffusion in physics and finance, offering a practical tool for fractional-order PDE discovery, though it is incremental as it builds on existing Gaussian Process methods.
The paper tackles the problem of discovering space-fractional differential equations from data by extending a Gaussian Process framework to handle fractional operators, enabling the learning of fractional-order models for systems like anomalous diffusion and financial data. It demonstrates applications in advection-diffusion interpolation and modeling S&P 500 stock performance with fractional diffusion equations.
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 $\mathbb{R}^d$ 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.