Machine Learning of Space-Fractional Differential Equations

Machine Learning of Space-Fractional Differential Equations. Gulian, M., Raissi, M., Perdikaris, P., & Karniadakis, G. arXiv:1808.00931 [cs, stat], August, 2019. 39 citations (Semantic Scholar/arXiv) [2023-02-27] arXiv: 1808.00931

Paper abstract bibtex

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 “physics informed” Gaussian Process kernel. This kernel includes the parametrized diﬀerential operators applied to a prior covariance kernel. We extend this framework to the data-driven discovery of linear space-fractional diﬀerential equations. The methodology is compatible with a wide variety of space-fractional operators in Rd and stationary covariance kernels, including the Mat´ern class, and allows for optimizing the Mat´ern parameter during training. Since fractional derivatives are typically not given by closed-form analytic expressions, the main challenges to be addressed are a user-friendly, general way to set up fractional-order derivatives of covariance kernels, together with feasible and robust numerical methods for such implementations. Making use of the simple Fourier-space representation of space-fractional derivatives in Rd, we provide a uniﬁed set of integral formulas for the resulting Gaussian Process kernels. The shift property of the Fourier transform results in formulas involving d-dimensional integrals that can be eﬃciently treated using generalized Gauss-Laguerre quadrature.

@article{gulian_machine_2019,
	title = {Machine {Learning} of {Space}-{Fractional} {Differential} {Equations}},
	url = {http://arxiv.org/abs/1808.00931},
	abstract = {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 “physics informed” Gaussian Process kernel. This kernel includes the parametrized diﬀerential operators applied to a prior covariance kernel. We extend this framework to the data-driven discovery of linear space-fractional diﬀerential equations. The methodology is compatible with a wide variety of space-fractional operators in Rd and stationary covariance kernels, including the Mat´ern class, and allows for optimizing the Mat´ern parameter during training. Since fractional derivatives are typically not given by closed-form analytic expressions, the main challenges to be addressed are a user-friendly, general way to set up fractional-order derivatives of covariance kernels, together with feasible and robust numerical methods for such implementations. Making use of the simple Fourier-space representation of space-fractional derivatives in Rd, we provide a uniﬁed set of integral formulas for the resulting Gaussian Process kernels. The shift property of the Fourier transform results in formulas involving d-dimensional integrals that can be eﬃciently treated using generalized Gauss-Laguerre quadrature.},
	language = {en},
	urldate = {2022-01-19},
	journal = {arXiv:1808.00931 [cs, stat]},
	author = {Gulian, Mamikon and Raissi, Maziar and Perdikaris, Paris and Karniadakis, George},
	month = aug,
	year = {2019},
	note = {39 citations (Semantic Scholar/arXiv) [2023-02-27]
arXiv: 1808.00931},
	keywords = {/unread, 35R11, 65N21, 62M10, 62F15, 60G15, 60G52, Computer Science - Machine Learning, Statistics - Machine Learning, ⛔ No DOI found},
}

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