Machine Learning of Space-Fractional Differential Equations

08/02/2018
by   Mamikon Gulian, et al.
0

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 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.

READ FULL TEXT
research
08/10/2019

A comparison between Caputo and Caputo-Fabrizio fractional derivatives for modelling Lotka-Volterra differential equations

In this paper, we apply the concept of the fractional calculus to study ...
research
01/12/2021

Analysis of Anisotropic Nonlocal Diffusion Models: Well-posedness of Fractional Problems for Anomalous Transport

We analyze the well-posedness of an anisotropic, nonlocal diffusion equa...
research
10/10/2022

Multi-term fractional linear equations modeling oxygen subdiffusion through capillaries

For 0<ν_2<ν_1≤ 1, we analyze a linear integro-differential equation on t...
research
01/26/2022

Machine-learning of nonlocal kernels for anomalous subsurface transport from breakthrough curves

Anomalous behavior is ubiquitous in subsurface solute transport due to t...
research
03/27/2021

Why Do Big Data and Machine Learning Entail the Fractional Dynamics?

Fractional-order calculus is about the differentiation and integration o...
research
07/16/2020

Stability and Complexity Analyses of Finite Difference Algorithms for the Time-Fractional Diffusion Equation

Fractional differential equations (FDEs) are an extension of the theory ...
research
03/22/2018

Statistical test for fractional Brownian motion based on detrending moving average algorithm

Motivated by contemporary and rich applications of anomalous diffusion p...

Please sign up or login with your details

Forgot password? Click here to reset