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The Saddle Point of Everything
arXiv:2605.30386v1 Announce Type: new Abstract: The harmonic oscillator is the universal Hamiltonian of stable equilibrium. Its counterpart, the inverted harmonic oscillator (IHO), is the Hamiltonian of unstable equilibrium: the saddle point of physical systems. It appears across disciplines, from condensed matter, quantum optics, and quantum chemistry to the Standard Model Higgs instability and quantum field theory near gravitational horizons.
Numerical analysis of the second-order time-dependent saddle point Maxwell system via a parameter-free discontinuous Galerkin method: The first optimal ${\bf L}^{2}$-norm error estimates
Announce Type: new Abstract: We present a novel parameter-free discontinuous Galerkin (dG) finite element method (FEM) for the time-dependent Maxwell system formulated as a saddle point problem. We establish the stability of the proposed semi-discrete problem and derive optimal error estimates in energy and \( {\bf L}^{2} \) norms for the electric field variable, as well as in \( L^{2} \) norm for the potential function. To the best of our knowledge, this work provides the first optimal \(...
Efficiently Escaping Saddle Points under Generalized Smoothness via Self-Bounding Regularity
Announce Type: replace-cross Abstract: We study the optimization of non-convex functions that are not necessarily smooth (gradient and/or Hessian are Lipschitz) using first order methods. Smoothness is a restrictive assumption in machine learning in both theory and practice, motivating significant recent work on finding first order stationary points of functions satisfying generalizations of smoothness with first order methods. We develop a novel framework that lets us systematically study...
Stability Analysis of Sharpness-Aware Minimization
arXiv:2301.06308v2 Announce Type: replace Abstract: Sharpness-aware minimization (SAM) is a training method that seeks to find flat minima in deep learning, resulting in state-of-the-art performance across various domains. Instead of minimizing the loss of the current weights, SAM minimizes the worst-case loss in its neighborhood in the parameter space. In this paper, we investigate the convergence instability of SAM near a saddle point.
Computer-Assisted Proofs for Geometric Optimization: From Crystallization to Carbon Nanotubes
Announce Type: replace Abstract: We present a framework based on computer-assisted proofs that turns geometry optimization simulations for atomistic structures into mathematical proofs. Starting from a numerically computed approximation of a local minimizer or saddle point, we use validated numerical computations to prove the existence of a critical point of the potential energy close to this approximation. We demonstrate this framework in two settings.
Modified augmented Lagrangian preconditioning for mixed-dimensional beam-solid coupling
Announce Type: new Abstract: This paper presents modified augmented Lagrangian block preconditioners for the mixed-dimensional coupling of three-dimensional solid bodies with embedded one-dimensional torsion-free Kirchhoff-Love beams using Lagrange multipliers for constraint enforcement. The finite element discretization of this mixed formulation leads to an indefinite saddle-point system. An augmented Lagrangian formulation is employed to regularize the linear system while maintaining exact...
A Geometric Characterization of the Stationary Plateau for Two-Layer Neural Networks
arXiv:2606.04327v1 Announce Type: new Abstract: We investigate the geometric structure of stationary plateaus that arise in the loss landscape of two-layer neural networks with smooth activation functions. We focus on the phenomenon of "neuron splitting" where duplicating a hidden neuron yields an affine set of stationary points in a wider network. We provide a comprehensive classification of all stationary points on these plateaus, determining under what conditions they constitute local...
Roman telescope's massive infrared mirror is ready to fly
Roman telescope's massive infrared mirror is ready to fly Sadie Harley Scientific Editor Andrew Zinin Lead Editor NASA has completed its final inspection of the primary mirror on the Roman Space Telescope, which measures 2.4 meters (7.9 feet) in diameter and contains a layer of silver hundreds of times thinner than a human hair, at 400 nanometers. The primary mirror will help accomplish Roman's mission objectives using near-infrared light, including studying dark matter and dark energy,...
Human-Like Neural Nets by Catapulting
Human-like Neural Nets by Catapulting Speculative proposal to create artificial neural nets with human-like performance by high-learning-rate/regularization training of overparameterized NNs to trigger catapulting/grokking. Over-parameterization as a route to true generalization would resolve many outstanding mysteries of artificial versus natural intelligence. There are many mysteries about deep learning and human intelligence, but we could describe the biggest anomaly this way: why are...
An adaptive Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) method for solving the biharmonic equation over planar multi-patch geometries
new Abstract: We present a novel adaptive isogeometric method for solving the biharmonic equation over planar multi-patch domains with possibly extraordinary vertices, parametrized by analysis-suitable G^1 multi-patch geometries. The proposed technique relies on the concept of Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP), which enforces the required C^1-smoothness of the solution across a common edge of two neighboring patches by imposing appropriate continuity conditions...