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Residual error propagation remains a fundamental problem in recurrent models, where small prediction inaccuracies compound over time and degrade long-horizon performance. Accurately modeling the correlation structure of such residuals is…

Geometrical interpretations of deep learning models offer insightful perspectives into their underlying mathematical structures. In this work, we introduce a novel approach that leverages differential geometry, particularly concepts from…

We propose a metric learning paradigm, Regression-based Elastic Metric Learning (REML), which optimizes the elastic metric for geodesic regression on the manifold of discrete curves. Geodesic regression is most accurate when the chosen…

Model merging offers a promising avenue for knowledge integration and parallel development without retraining. Yet, existing methods either ignore the geometry of the loss landscape or rely on intractable full-space Hessian approximations.…

Entropic Dynamics (ED) is a theoretical framework developed to investigate the possibility that laws of physics reflect laws of inference rather than laws of nature. In this work, a RED (Reversible Entropic Dynamics) model is considered.…

Physics-based and first-principles models pervade the engineering and physical sciences, allowing for the ability to model the dynamics of complex systems with a prescribed accuracy. The approximations used in deriving governing equations…

Latent Diffusion Models (LDMs) achieve high-fidelity synthesis but suffer from latent space brittleness, causing discontinuous semantic jumps during editing. We introduce a Riemannian framework to diagnose this instability by analyzing the…

Using nonlinear projections and preserving structure in model order reduction (MOR) are currently active research fields. In this paper, we provide a novel differential geometric framework for model reduction on smooth manifolds, which…

We present a thorough analysis on the invariance of the most widely used metrics in the Geometrothermodynamics (GTD) programme. We centre our attention in the invariance of the curvature of the space of equilibrium states under a change of…

We present an experimentally realizable, simple mechanical system with linear interactions whose geometric nature leads to nontrivial, nonlinear dynamical equations. The equations of motion are derived and their ground state structures are…

Classical elasticity is concerned with bodies that can be modeled as smooth manifolds endowed with a reference metric that represents local equilibrium distances between neighboring material elements. The elastic energy associated with a…

We develop an error-in-constitutive-relation (ECR) approach toward the full-field characterization of linear viscoelastic solids described within the framework of standard generalized materials. To this end, we formulate the viscoelastic…

Weight matrices in deep networks exhibit geometric continuity -- principal singular vectors of adjacent layers point in similar directions. While this property has been widely observed, its origin remains unexplained. Through experiments on…

This survey is written in summer, 2016. The purpose of this survey is to briefly introduce nonlinear dimensionality reduction (NLDR) in data reduction. The first two NLDR were respectively published in Science in 2000 in which they solve…

Deep neural networks have been demonstrated to achieve phenomenal success in many domains, and yet their inner mechanisms are not well understood. In this paper, we investigate the curvature of image manifolds, i.e., the manifold deviation…

Empirical modelling often aims for the simplest model consistent with the data. A new technique is presented which quantifies the consistency of the model dynamics as a function of location in state space. As is well-known, traditional…

Modified dispersion relations (MDRs) arise in many quantum-gravity approaches, often in non-polynomial or non-analytic form beyond the reach of effective field theory (EFT). Logarithmic, exponential and trigonometric MDRs appear in causal…

Understanding how latent representations evolve during generation is a central open problem in large language model interpretability. We introduce \textbf{Dynamical Manifold Evolution Theory} (DMET), a phenomenological framework that models…

A reliable language model should be able to signal, prior to generation, when a query falls outside its knowledge. We investigate whether representation geometry can provide such a pre-generation signal by measuring the deviation of hidden…

Learning dynamical models from data plays a vital role in engineering design, optimization, and predictions. Building models describing dynamics of complex processes (e.g., weather dynamics, or reactive flows) using empirical knowledge or…