arXiv subjects

Albert Chern

Publications and source records attributed to Albert Chern.

At least 19 recordsLinked to original sources

arXiv2026

Elastic Curves via Geometric Mechanics

Elastic curves are the mathematical shapes of thin elastic rods in equilibrium, with deep connections to mechanics, geometry, and computer graphics. Traditionally described as stationary points of bending energy under length and torsion constraints, their rich theory admits many equivalent characterizations. We develop a new one from the viewpoint of geometric mechanics. Our main contribution relies on a lesser-known isoperimetric characterization: a curve is elastic if and only if it is a critical point of the length functional under fixed area and volume vectors. We show that these constraints transform naturally under orientation-preserving rigid body motions, identifying them as momentum variables for these symmetries. This structure suggests a new discrete theory. We show that the low-order integral quantities length, area, and volume vectors are all naturally defined for polygonal curves, leaving the same transformation laws exactly satisfied. The resulting definition of discrete elastic curves in terms of the isoperimetric characterization restricted to discrete polygonal curves is variational, structure-preserving, and requires no auxiliary discretizations of curvature or material frames. Finally, the same structure carries the Marsden--Weinstein form, a canonical (pre-)symplectic structure on the space of curves, to polygonal curves. This yields novel approaches to Hamiltonian dynamics on discrete space curves, including tangent, vortex-filament, and modified Korteweg--de Vries flows.

math.DG↗

arXiv2026

Neural Representation of Minimal Surfaces

We propose a neural representation for minimal surfaces. Unlike prior approaches based on discretization or Physics-Informed Neural Networks (PINNs), where meshes or neural fields are optimized to approximate the governing equations, our method builds on an exact representation, similar to the classical Weierstrass--Enneper parameterization, yielding minimal surfaces up to negligible quadrature error in evaluation. We formulate a training objective for the Plateau problem that optimizes over this representation.

cs.GR↗

arXiv2026

Vakonomic Fluids

We introduce a novel discretization of the incompressible Euler equations based on their interpretation as geodesic equations on the Lie group of volume-preserving diffeomorphisms. It is well known that encoding diffeomorphisms and their infinitesimal generators through a discretized Koopman representation places a nonholonomic constraint on discrete velocities, for which there is no consensus on a variational treatment. We show that taking the vakonomic perspective, as opposed to the usual perspective of Lagrange--d'Alembert, yields discrete fluid trajectories that remain geodesics on a (sub-)Riemannian manifold. In particular, the resulting vakonomic dynamics are Lie--Poisson and their solutions admit a discrete relabeling symmetry, leading to machine-precision satisfaction of Casimir invariants along with a discrete analogue of Kelvin's Circulation Theorem. Using an efficient momentum map representation based on low-rank Clebsch variables, we show that these vakonomic fluids behave stably and consistently even at low grid resolutions, leading to increased robustness and physical realism in the long term.

math-ph↗

arXiv2026

Convex Quadratic Distance Field Computation

Methods for computing distances from sources on discrete meshes commonly either compute geodesics directly on polyhedral surfaces or approximate the distance in a finite-element framework. Exact window-based polyhedral methods are highly accurate on clean manifold surfaces, but their unfolding construction does not extend to tetrahedral volumes and relies on manifold connectivity. Because their results are tied to the input polyhedron, geometric noise directly affects the computed distance field. Finite-element methods extend naturally to triangle surfaces and tetrahedral volumes, but state-of-the-art methods represent the distance field as a piecewise-linear (PL) function, limiting accuracy on coarse or poorly shaped meshes. We argue that the PL representation itself, rather than the algorithm built on top of it, limits the result. Geodesic distance exhibits cone-like behaviour at its source and is not piecewise linear even on flat domains. In contrast, the squared distance is exactly quadratic in such domains. Therefore, our Quadratic Distance Field method represents the squared distance using piecewise-quadratic (PQ) elements, reproducing flat squared distances exactly. We show that simply increasing the element order does not improve existing algorithms, and develop a convex formulation for the squared distance over PQ elements. The same formulation applies to both triangle surfaces and tetrahedral volumes, supports anisotropic metrics and nonmanifold connectivity, and remains robust under noise. Finally, we present an efficient solver based on the alternating direction method of multipliers and demonstrate its robustness and accuracy on a benchmark of thousands of real-world models.

cs.GR↗

arXiv2026

L'Hopital rules for complex-valued functions in higher dimensions

In calculus, l'Hopital's rule provides a simple way to evaluate the limits of quotient functions when both the numerator and denominator vanish. But what happens when we move beyond real functions on a real interval? In this article, we study when the quotient of two complex-valued functions in higher dimension can be defined continuously at the points where both functions vanish. Surprisingly, the answer is far subtler than in the real-valued setting. We provide a complete characterization for the continuity of the quotient function. We also point out why extending this result to smoother quotients remains an intriguing challenge.

math.CV↗

arXiv2025

Implicit representations of codimension-2 submanifolds and their prequantum structure

This paper explores the geometry of the space of codimension-2 submanifolds. We implicitly represent these submanifolds by a class of complex-valued functions. We show that the space of all these implicit representations admits a prequantum bundle structure over the space of submanifolds, equipped with the well-known Marsden-Weinstein symplectic structure. This bundle allows a new geometric interpretation of the Marsden-Weinstein structure as the curvature of a connection form, which measures the average of volumes swept by the deformation of the S^1-family of hypersurfaces, defined as the phase level sets of the complex function implicitly representing a submanifold.

math.SG↗

arXiv2024

Penrose's eight-conic theorem

This article proves the following theorem, first enunciated by Roger Penrose about 70 years ago but never published: In $\mathbb{R}P^{2}$, if conics are assigned to seven of the vertices of a combinatorial cube such that (i) conics connected by an edge are in double contact, and (ii) the chords of contact associated to a cube face meet in a common point, then there exists an eighth conic such that the completed cube satisfies (i) and (ii). The theorem turns out to be a remarkable generalization of many well-known theorems of projective geometry -- Pappus, Desargues, Pascal, Brianchon, Monge, and Poncelet are the best-known ones. This archetypal principle provides a unifying framework in which the myriad specializations of the theorem and their interrelationships can be grasped as an organic whole, enriching the field of projective geometry and opening new vistas for research. The article begins with a series of motivational examples. It then gives a geometric proof assuming that the conics are regular, followed by an algebraic one that removes this restriction. The geometric proof is obtained as a slice of an analogous theorem for quadrics in $\mathbb{R}P^{3}$; the algebraic one is based on the determinants of a special matrix associated to the configuration of conics.

math.GM↗

arXiv2024

Stokes flow of an evolving fluid film with arbitrary shape and topology

The dynamics of evolving fluid films in the viscous Stokes limit is relevant to various applications, such as the modeling of lipid bilayers in cells. While the governing equations were formulated by Scriven in 1960, solving for the flow of a deformable viscous surface with arbitrary shape and topology has remained a challenge. In this study, we present a straightforward discrete model based on variational principles to address this long-standing problem. We replace the classical equations, which are expressed with tensor calculus in local coordinates, with a simple coordinate-free, differential-geometric formulation. The formulation provides a fundamental understanding of the underlying mechanics and directly translates to discretization. We construct a discrete analogue of the system using the Onsager variational principle, which, in a smooth context, governs the flow of a viscous medium. In the discrete setting, instead of term-wise discretizing the coordinate-based Stokes equations, we construct a discrete Rayleighian for the system and derive the discrete Stokes equations via the variational principle. This approach results in a stable, structure-preserving variational integrator that solves the system on general manifolds.

physics.flu-dyn↗

arXiv2024

Generalizable Physics-Informed Learning for Stochastic Safety-Critical Systems

Accurate estimation of long-term risk is essential for the design and analysis of stochastic dynamical systems. Existing risk quantification methods typically rely on extensive datasets involving risk events observed over extended time horizons, which can be prohibitively expensive to acquire. Motivated by this gap, we propose an efficient method for learning long-term risk probabilities using short-term samples with limited occurrence of risk events. Specifically, we establish that four distinct classes of long-term risk probabilities are characterized by specific partial differential equations (PDEs). Using this characterization, we introduce a physics-informed learning framework that combines empirical data with physics information to infer risk probabilities. We then analyze the theoretical properties of this framework in terms of generalization and convergence. Through numerical experiments, we demonstrate that our framework not only generalizes effectively beyond the sampled states and time horizons but also offers additional benefits such as improved sample efficiency, rapid online inference capabilities under changing system dynamics, and stable computation of probability gradients. These results highlight how embedding PDE constraints, which contain explicit gradient terms and inform how risk probabilities depend on state, time horizon, and system parameters, improves interpolation and generalization between/beyond the available data.

cs.LG↗

arXiv2024

Fluid Implicit Particles on Coadjoint Orbits

We propose Coadjoint Orbit FLIP (CO-FLIP), a high order accurate, structure preserving fluid simulation method in the hybrid Eulerian-Lagrangian framework. We start with a Hamiltonian formulation of the incompressible Euler Equations, and then, using a local, explicit, and high order divergence free interpolation, construct a modified Hamiltonian system that governs our discrete Euler flow. The resulting discretization, when paired with a geometric time integration scheme, is energy and circulation preserving (formally the flow evolves on a coadjoint orbit) and is similar to the Fluid Implicit Particle (FLIP) method. CO-FLIP enjoys multiple additional properties including that the pressure projection is exact in the weak sense, and the particle-to-grid transfer is an exact inverse of the grid-to-particle interpolation. The method is demonstrated numerically with outstanding stability, energy, and Casimir preservation. We show that the method produces benchmarks and turbulent visual effects even at low grid resolutions.

cs.GR↗

arXiv2024

Active nematic fluids on Riemannian 2-manifolds

Recent advances in cell biology and experimental techniques using reconstituted cell extracts have generated significant interest in understanding how geometry and topology influence active fluid dynamics. In this work, we present a comprehensive continuous theory and computational method to explore the dynamics of active nematic fluids on arbitrary surfaces without topological constraints. The fluid velocity and nematic order parameter are represented as the sections of the complex line bundle of a 2-manifold. We introduce the Levi-Civita connection and surface curvature form within the framework of complex line bundles. By adopting this geometric approach, we introduce a gauge-invariant discretization method that preserves the continuous local-to-global theorems in differential geometry. We establish a nematic Laplacian on complex functions that can accommodate fractional topological charges through the covariant derivative on the complex nematic representation. We formulate advection of the nematic field based on a unifying definition of the Lie derivative, resulting in a stable geometric semi-Lagrangian discretization scheme for transport by the flow. In general, the proposed surface-based method offers an efficient and stable means to investigate the influence of local curvature and global topology on the 2D hydrodynamics of active nematic systems. Moreover, the complex line representation of the nematic field and the unifying Lie advection present a systematic approach for generalizing our method to active $k$-atic systems.

cond-mat.soft↗

arXiv2024

Lifting Directional Fields to Minimal Sections

Directional fields, including unit vector, line, and cross fields, are essential tools in the geometry processing toolkit. The topology of directional fields is characterized by their singularities. While singularities play an important role in downstream applications such as meshing, existing methods for computing directional fields either require them to be specified in advance, ignore them altogether, or treat them as zeros of a relaxed field. While fields are ill-defined at their singularities, the graphs of directional fields with singularities are well-defined surfaces in a circle bundle. By lifting optimization of fields to optimization over their graphs, we can exploit a natural convex relaxation to a minimal section problem over the space of currents in the bundle. This relaxation treats singularities as first-class citizens, expressing the relationship between fields and singularities as an explicit boundary condition. As curvature frustrates finite element discretization of the bundle, we devise a hybrid spectral method for representing and optimizing minimal sections. Our method supports field optimization on both flat and curved domains and enables more precise control over singularity placement.

cs.GR↗

arXiv2024

Myopically Verifiable Probabilistic Certificates for Safe Control and Learning

This paper addresses the design of safety certificates for stochastic systems, with a focus on ensuring long-term safety through fast real-time control. In stochastic environments, set invariance-based methods that restrict the probability of risk events in infinitesimal time intervals may exhibit significant long-term risks due to cumulative uncertainties/risks. On the other hand, reachability-based approaches that account for the long-term future may require prohibitive computation in real-time decision making. To overcome this challenge involving stringent long-term safety vs. computation tradeoffs, we first introduce a novel technique termed 'probabilistic invariance'. This technique characterizes the invariance conditions of the probability of interest. When the target probability is defined using long-term trajectories, this technique can be used to design myopic conditions/controllers with assured long-term safe probability. Then, we integrate this technique into safe control and learning. The proposed control methods efficiently assure long-term safety using neural networks or model predictive controllers with short outlook horizons. The proposed learning methods can be used to guarantee long-term safety during and after training. Finally, we demonstrate the performance of the proposed techniques in numerical simulations.

eess.SY↗

arXiv2023

Force-Free Fields are Conformally Geodesic

In this paper, we establish an equivalence between force-free fields and conformally geodesic fields, and between harmonic fields and conformally eikonal fields in the context of conformal geometry. In contrast to previous work, our approach and equivalence results generalize to arbitrary dimensions. In accordance with three-dimensional theory, our defining equations emerge as the Euler-Lagrange equations of hierarchies of variational principles - distinguished by the topological constraints they impose - and retain the known inclusions of the special cases from each other. Specifically, we relate stationary points of hierarchies of $L^2$ resp. $L^1$-optimization problems by a conformal change of metric, provide an explicit construction of the conformal factors relating the relevant metrics and identify the field lines of physical vector fields fields as conformal geodesics. Despite the allowed topological complexity of the fields under consideration, these observations reveal geometric order which is obtained by merely pointwise rescaling of the metric.

math.DG↗

arXiv2023

Adaptive Surface Meshes from Harmonic Maps

We present a novel shape-approximating anisotropic re-meshing algorithm as a geometric generalization of the adaptive moving mesh method. Conventional moving mesh methods reduce the interpolation error of a mesh that discretizes a given function over a planar domain. Our algorithm, in contrast, optimizes the mesh's approximation of a curved surface; surfaces can be represented in various formats, such as a signed distance field. The optimization is achieved by continuously flowing the mesh without altering its topology, making the implementation simpler compared to other adaptive surface meshing techniques. The resulting optimal mesh can be interpreted as a harmonic map with respect to a metric using the shape operator. Furthermore, our approach can be tailored to target height fields by utilizing isotropic geometry.

cs.CG↗

arXiv2023

Wave Simulations in Infinite Spacetime

Solving the wave equation on an infinite domain has been an ongoing challenge in scientific computing. Conventional approaches to this problem only generate numerical solutions on a small subset of the infinite domain. In this paper, we present a method for solving the wave equation on the entire infinite domain using only finite computation time and memory. Our method is based on the conformal invariance of the scalar wave equation under the Kelvin transformation in Minkowski spacetime. As a result of the conformal invariance, any wave problem with compact initial data contained in a causality cone is equivalent to a wave problem on a bounded set in Minkowski spacetime. We use this fact to perform wave simulations in infinite spacetime using a finite discretization of the bounded spacetime with no additional loss of accuracy introduced by the Kelvin transformation.

math.NA↗

arXiv2023

Area formula for spherical polygons via prequantization

We present a formula for the signed area of a spherical polygon via prequantization. In contrast to the traditional formula based on the Gauss-Bonnet theorem that requires measuring angles, the new formula mimics Green's theorem and is applicable to a wider range of degenerate spherical curves and polygons.

math.DG↗

arXiv2022

Hidden Degrees of Freedom in Implicit Vortex Filaments

This paper presents a new representation of curve dynamics, with applications to vortex filaments in fluid dynamics. Instead of representing these filaments with explicit curve geometry and Lagrangian equations of motion, we represent curves implicitly with a new co-dimensional 2 level set description. Our implicit representation admits several redundant mathematical degrees of freedom in both the configuration and the dynamics of the curves, which can be tailored specifically to improve numerical robustness, in contrast to naive approaches for implicit curve dynamics that suffer from overwhelming numerical stability problems. Furthermore, we note how these hidden degrees of freedom perfectly map to a Clebsch representation in fluid dynamics. Motivated by these observations, we introduce untwisted level set functions and non-swirling dynamics which successfully regularize sources of numerical instability, particularly in the twisting modes around curve filaments. A consequence is a novel simulation method which produces stable dynamics for large numbers of interacting vortex filaments and effortlessly handles topological changes and re-connection events.

cs.GR↗