arXiv subjects

Blaine Quackenbush

Publications and source records attributed to Blaine Quackenbush.

5 recordsLinked to original sources

arXiv2026

Extending Neural Operators: Robust Handling of Functions Beyond the Training Set

We develop a rigorous framework for extending neural operators to handle out-of-distribution input functions. We leverage kernel approximation techniques and provide theory for characterizing the input-output function spaces in terms of Reproducing Kernel Hilbert Spaces (RKHSs). We provide theorems on the requirements for reliable extensions and their predicted approximation accuracy. We also establish formal relationships between specific kernel choices and their corresponding Sobolev Native Spaces. This connection further allows the extended neural operators to reliably capture not only function values but also their derivatives. Our methods are empirically validated through the solution of elliptic partial differential equations (PDEs) involving operators on manifolds having point-cloud representations and handling geometric contributions. We report results on key factors impacting the accuracy and computational performance of the extension approaches.

cs.LG↗

arXiv2025

Transferable Foundation Models for Geometric Tasks on Point Cloud Representations: Geometric Neural Operators

We introduce methods for obtaining pretrained Geometric Neural Operators (GNPs) that can serve as basal foundation models for use in obtaining geometric features. These can be used within data processing pipelines for machine learning tasks and numerical methods. We show how our GNPs can be trained to learn robust latent representations for the differential geometry of point-clouds to provide estimates of metric, curvature, and other shape-related features. We demonstrate how our pre-trained GNPs can be used (i) to estimate the geometric properties of surfaces of arbitrary shape and topologies with robustness in the presence of noise, (ii) to approximate solutions of geometric partial differential equations (PDEs) on manifolds, and (iii) to solve equations for shape deformations such as curvature driven flows. We release codes and weights for using GNPs in the package geo_neural_op. This allows for incorporating our pre-trained GNPs as components for reuse within existing and new data processing pipelines. The GNPs also can be used as part of numerical solvers involving geometry or as part of methods for performing inference and other geometric tasks.

cs.LG↗

arXiv2024

Geometric Neural Operators (GNPs) for Data-Driven Deep Learning of Non-Euclidean Operators

We introduce Geometric Neural Operators (GNPs) for accounting for geometric contributions in data-driven deep learning of operators. We show how GNPs can be used (i) to estimate geometric properties, such as the metric and curvatures, (ii) to approximate Partial Differential Equations (PDEs) on manifolds, (iii) learn solution maps for Laplace-Beltrami (LB) operators, and (iv) to solve Bayesian inverse problems for identifying manifold shapes. The methods allow for handling geometries of general shape including point-cloud representations. The developed GNPs provide approaches for incorporating the roles of geometry in data-driven learning of operators.

cs.LG↗

arXiv2020

Periodic intermediate $β$-expansions of Pisot numbers

The subshift of finite type property (also known as the Markov property) is ubiquitous in dynamical systems and the simplest and most widely studied class of dynamical systems are $β$-shifts, namely transformations of the form $T_{β, α} \colon x \mapsto βx + α\bmod{1}$ acting on $[-α/(β- 1), (1-α)/(β- 1)]$, where $(β, α) \in Δ$ is fixed and where $Δ= \{ (β, α) \in \mathbb{R}^{2} \colon β\in (1,2) \; \text{and} \; 0 \leq α\leq 2-β\}$. Recently, it was shown, by Li et al. (Proc. Amer. Math. Soc. 147(5): 2045-2055, 2019), that the set of $(β, α)$ such that $T_{β, α}$ has the subshift of finite type property is dense in the parameter space $Δ$. Here, they proposed the following question. Given a fixed $β\in (1, 2)$ which is the $n$-th root of a Perron number, does there exists a dense set of $α$ in the fiber $\{β\} \times (0, 2- β)$, so that $T_{β, α}$ has the subshift of finite type property? We answer this question in the positive for a class of Pisot numbers. Further, we investigate if this question holds true when replacing the subshift of finite type property by the property of beginning sofic (that is a factor of a subshift of finite). In doing so we generalise, a classical result of Schmidt (Bull. London Math. Soc., 12(4): 269-278, 1980) from the case when $α= 0$ to the case when $α\in (0, 2 - β)$. That is, we examine the structure of the set of eventually periodic points of $T_{β, α}$ when $β$ is a Pisot number and when $β$ is the $n$-th root of a Pisot number.

math.DS↗

arXiv2018

On the continuity of entropy of Lorenz maps

We consider a one parameter family of Lorenz maps indexed by their point of discontinuity $p$ and constructed from a pair of bilipschitz functions. We prove that their topological entropies vary continuously as a function of $p$ and discuss Milnor's monotonicity conjecture in this setting.

math.DS↗