Joseph Klobusicky

Benjamin David Shaffer, Brooks Kinch, Joseph Klobusicky et al.

We present a machine learning framework for adaptive source localization in which agents use a structure-preserving digital twin of a coupled hydrodynamic-transport system for real-time trajectory planning and data assimilation. The twin is constructed with conditional neural Whitney forms (CNWF), coupling the numerical guarantees of finite element exterior calculus (FEEC) with transformer-based operator learning. The resulting model preserves discrete conservation, and adapts in real time to streaming sensor data. It employs a conditional attention mechanism to identify: a reduced Whitney-form basis; reduced integral balance equations; and a source field, each compatible with given sensor measurements. The induced reduced-order environmental model retains the stability and consistency of standard finite-element simulation, yielding a physically realizable, regular mapping from sensor data to the source field. We propose a staggered scheme that alternates between evaluating the digital twin and applying Lloyd's algorithm to guide sensor placement, with analysis providing conditions for monotone improvement of a coverage functional. Using the predicted source field as an importance function within an optimal-recovery scheme, we demonstrate recovery of point sources under continuity assumptions, highlighting the role of regularity as a sufficient condition for localization. Experimental comparisons with physics-agnostic transformer architectures show improved accuracy in complex geometries when physical constraints are enforced, indicating that structure preservation provides an effective inductive bias for source identification.

Alex Dolce, Ryan Lavelle, Bernard Scott et al.

The turning distance is a well-studied metric for measuring the similarity between two polygons. This metric is constructed by taking an $L^p$ distance between step functions which track each shape's tangent angle of a path tracing its boundary. In this study, we introduce \textit{turning disorders} for polygonal planar networks, defined by averaging turning distances between network faces and "ordered" shapes (regular polygons or circles). We derive closed-form expressions of turning distances for special classes of regular polygons, related to the divisibility of $m$ and $n$, and also between regular polygons and circles. These formulas are used to show that the time for computing the 2-turning distances reduces to $O((m+n) \log(m+n))$ when both shapes are regular polygons, an improvement from $O(mn\log(mn))$ operations needed to compute distances between general polygons of $n$ and $m$ sides. We also apply these formulas to several examples of network microstructure with varying disorder. For Archimedean lattices, a class of regular tilings, we can express turning disorders with exact expressions. We also consider turning disorders applied to two examples of stochastic processes on networks: spring networks evolving under T1 moves and polygonal rupture processes. We find that the two aspects of defining different turning disorders, the choice of ordered shape and whether to apply area-weighting, can capture different notions of network disorder.