160 citations · 340 across the 17 of their papers we have counts for
19 papers
Solving High-Dimensional Inverse Problems with Auxiliary Uncertainty via Operator Learning with Limited Data
Joseph Hart, Mamikon Gulian, Indu Manickam +1
In complex large-scale systems such as climate, important effects are caused by a combination of confounding processes that are not fully observable. The identification of sources…
Error-in-variables modelling for operator learning
Ravi G. Patel, Indu Manickam, Myoungkyu Lee +1
Deep operator learning has emerged as a promising tool for reduced-order modelling and PDE model discovery. Leveraging the expressive power of deep neural networks, especially in h…
Connections between nonlocal operators: from vector calculus identities to a fractional Helmholtz decomposition
Marta D'Elia, Mamikon Gulian, Tadele Mengesha +1
Nonlocal vector calculus, which is based on the nonlocal forms of gradient, divergence, and Laplace operators in multiple dimensions, has shown promising applications in fields suc…
Fractional Modeling in Action: A Survey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials
Jorge Suzuki, Mamikon Gulian, Mohsen Zayernouri +1
Modeling of phenomena such as anomalous transport via fractional-order differential equations has been established as an effective alternative to partial differential equations, du…
Probabilistic partition of unity networks: clustering based deep approximation
Nat Trask, Mamikon Gulian, Andy Huang +1
Partition of unity networks (POU-Nets) have been shown capable of realizing algebraic convergence rates for regression and solution of PDEs, but require empirical tuning of trainin…
Distribution and pressure of active Lévy swimmers under confinement
Tingtao Zhou, Zhiwei Peng, Mamikon Gulian +1
Many active matter systems are known to perform Lévy walks during migration or foraging. Such superdiffusive transport indicates long-range correlated dynamics. These behavior patt…
Partition of unity networks: deep hp-approximation
Kookjin Lee, Nathaniel A. Trask, Ravi G. Patel +2
Approximation theorists have established best-in-class optimal approximation rates of deep neural networks by utilizing their ability to simultaneously emulate partitions of unity…
Analysis of Anisotropic Nonlocal Diffusion Models: Well-posedness of Fractional Problems for Anomalous Transport
Marta D'Elia, Mamikon Gulian
We analyze the well-posedness of an anisotropic, nonlocal diffusion equation. Establishing an equivalence between weighted and unweighted anisotropic nonlocal diffusion operators i…
Gaussian Process Regression constrained by Boundary Value Problems
Mamikon Gulian, Ari Frankel, Laura Swiler
We develop a framework for Gaussian processes regression constrained by boundary value problems. The framework may be applied to infer the solution of a well-posed boundary value p…
A block coordinate descent optimizer for classification problems exploiting convexity
Ravi G. Patel, Nathaniel A. Trask, Mamikon A. Gulian +1
Second-order optimizers hold intriguing potential for deep learning, but suffer from increased cost and sensitivity to the non-convexity of the loss surface as compared to gradient…
A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challenges
Laura Swiler, Mamikon Gulian, Ari Frankel +2
Gaussian process regression is a popular Bayesian framework for surrogate modeling of expensive data sources. As part of a broader effort in scientific machine learning, many recen…
Data-driven learning of robust nonlocal physics from high-fidelity synthetic data
Huaiqian You, Yue Yu, Nathaniel Trask +2
A key challenge to nonlocal models is the analytical complexity of deriving them from first principles, and frequently their use is justified a posteriori. In this work we extract…
Towards a Unified Theory of Fractional and Nonlocal Vector Calculus
Marta D'Elia, Mamikon Gulian, Hayley Olson +1
Nonlocal and fractional-order models capture effects that classical partial differential equations cannot describe; for this reason, they are suitable for a broad class of engineer…
Robust Training and Initialization of Deep Neural Networks: An Adaptive Basis Viewpoint
Eric C. Cyr, Mamikon A. Gulian, Ravi G. Patel +2
Motivated by the gap between theoretical optimal approximation rates of deep neural networks (DNNs) and the accuracy realized in practice, we seek to improve the training of DNNs.…
Stochastic Solution of Elliptic and Parabolic Boundary Value Problems for the Spectral Fractional Laplacian
Mamikon Gulian, Guofei Pang
We prove and implement stochastic solution (or Feynman-Kac) formulas for boundary value problems involving the spectral fractional Laplacian with nonzero Dirichlet boundary conditi…
Machine Learning of Space-Fractional Differential Equations
Mamikon Gulian, Maziar Raissi, Paris Perdikaris +1
Data-driven discovery of "hidden physics" -- i.e., machine learning of differential equation models underlying observed data -- has recently been approached by embedding the discov…
What Is the Fractional Laplacian?
Anna Lischke, Guofei Pang, Mamikon Gulian +8
The fractional Laplacian in R^d has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in math…
Fractional Path Integral Monte Carlo
Mamikon Gulian, Haobo Yang, Brenda M. Rubenstein
Fractional derivatives are nonlocal differential operators of real order that often appear in models of anomalous diffusion and a variety of nonlocal phenomena. Recently, a version…
Free Boundary Minimal Surfaces in the Unit Ball With Low Cohomogeneity
Brian Freidin, Mamikon Gulian, Peter McGrath
We study free boundary minimal surfaces in the unit ball of low cohomogeneity. For each pair of positive integers such that and , we construct a free b…