Description
- Abstract:
- Fractional-order partial differential equations (FPDEs) describe macroscopic properties of systems driven by Levy processes and, more generally, Continuous-Time Random Walks (CTRWs). CTRWs form a versatile family of abstract stochastic models for anomalous diffusion/transport, in which the ``actors'' at the microscopic scale -- be they particles, organisms, or stock prices -- do not obey normal or Gaussian statistics. The fractional-order operators that are required to describe such systems are nonlocal integral operators that pose novel conceptual, theoretical, and numerical challenges for modeling and solution. These challenges are compounded when considering boundary conditions. Focusing on the most fundamental example -- the fractional Laplacian -- we consolidate several recent results for nonzero boundary conditions and organize a taxonomy in which the various fractional Laplacians on bounded domains correspond to different ways of imposing boundary conditions on isotropic alpha-stable Levy motion. We prove and implement stochastic solution (or Feynman-Kac) formulas for elliptic and parabolic problems involving the spectral fractional Laplacian with Dirichlet boundary conditions. This yields an embarrassingly parallel Monte Carlo method that we demonstrate in solving 16-dimensional benchmark problems. We then develop a Path Integral Monte Carlo method for the many-body fractional Schrodinger equation with periodic boundary conditions. Our methodology opens the door to studying fractional Hamiltonians with arbitrarily complex potentials. We find that the fractional Laplacian strongly encourages particle delocalization, suggesting it may manifest atypical forms of condensation at low temperatures. We then turn to the problem of data-driven discovery of FPDE models using machine learning. We develop fractional physics-informed Gaussian processes, compatible with Matern covariance kernels, and demonstrate how they can be used to discover and interpolate a wide variety of linear integer order and space-fractional PDEs from noisy data in multidimensions. Illustrating several themes of the thesis, we employ our methodology to describe relative stock performance in the S&P 500 by calibrating a fractional Fokker-Planck equation with empirical distribution data.
- Notes:
- Thesis (Ph. D.)--Brown University, 2019
Access Conditions
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- In Copyright
- Restrictions on Use
- Collection is open for research.
Citation
Gulian, Mamikon Armen,
"Discovering and Solving Fractional-Order Partial Differential Equations: Machine Learning and Monte Carlo Methods"
(2019).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/3fm5-wj90
