Machine learning techniques are being increasingly used as flexible non-linear fitting and prediction tools in the physical sciences. Fitting functions that exhibit multiple solutions local minima can be analysed terms of corresponding machine landscape. Methods to explore visualise molecular potential energy landscapes applied these gain new insight into solution space involved training nature predictions. In particular, we define quantities analogous structure, thermodynamics, kinetics,...

The manuscript addresses the problem of finding all solutions power flow equations or other similar non-linear system algebraic equations. This arises naturally in a number systems contexts, most importantly direct methods for transient stability analysis and voltage assessment. Here, authors introduce novel form homotopy continuation method called numerical polynomial that is mathematically guaranteed to find without ever encountering bifurcation. Since real much more challenging, first...

The stationary points (SPs) of a potential-energy landscape play crucial role in understanding many the physical or chemical properties given system. However, unless they are found analytically, no efficient method is available to obtain all SPs potential. We present method, called numerical polynomial-homotopy-continuation which numerically finds SPs, and embarrassingly parallelizable. requires nonlinearity potential be polynomial-like, case for almost potentials arising systems. also...

The power flow equations are at the core of most computations for designing and operating electric grids. This system multivariate nonlinear relate injections voltages in an system. A plethora methods have been devised to solve these equations, from Newton-based homotopy continuation other optimization-based methods. Although many often efficiently find a high-voltage, stable solution, challenges remain finding low-voltage solutions, which play significant roles certain stability-related...

We argue that a generic instability afflicts vacua arise in theories whose moduli space has large dimension. Specifically, by studying with multiple scalar fields we provide numerical evidence for local minimum of the potential usual semiclassical bubble nucleation rate, Gamma = A e^{-B}, increases rapidly as function number theory. As consequence, fraction tunneling rates low enough to maintain metastability appears fall exponentially discuss possible implications landscape string Notably,...

Finding equilibria of the finite size Kuramoto model amounts to solving a nonlinear system equations, which is an important yet challenging problem. We translate this into algebraic geometry problem and use numerical methods find all for various choices coupling constants K, natural frequencies, on different graphs. note that even modest sizes (N ~ 10-20), number already more than 100,000. analyze stability each computed equilibrium as well configuration angles. Our exploration landscape...

The stationary points of the potential energy function V are studied for \phi^4 model on a two-dimensional square lattice with nearest-neighbor interactions. On basis analytical and numerical results, we explore relation to occurrence thermodynamic phase transitions. We find that transition does in general not coincide any V. This disproves earlier, allegedly rigorous, claims literature necessary conditions existence Moreover, evidence indices scale extensively system size, therefore index...

A bstract We explicitly construct all supersymmetric flux vacua of a particular Calabi-Yau compactification type IIB string theory for small number carrying cycles and given D3-brane tadpole. The analysis is performed in the large complex structure region by using polynomial homotopy continuation method, which allows to find stationary points equations that characterize vacuum solutions. as function D3 tadpole agreement with statistical studies literature. calculate available tuning...

Extraction and interpretation of intricate information from unstructured text data arising in financial applications, such as earnings call transcripts, present substantial challenges to large language models (LLMs) even using the current best practices use Retrieval Augmented Generation (RAG) (referred VectorRAG techniques which utilize vector databases for retrieval) due domain specific terminology complex formats documents. We introduce a novel approach based on combination, called...

We investigate the application of quantum cognition machine learning (QCML), a novel paradigm for both supervised and unsupervised tasks rooted in mathematical formalism theory, to distance metric corporate bond markets. Compared equities, bonds are relatively illiquid trade quote data these securities sparse. Thus, measure distance/similarity among is particularly useful variety practical applications trading bonds, including identification similar tradable alternatives, pricing with few...

The stationary points of the potential energy function ${\ensuremath{\varphi}}^{4}$ model on a two-dimensional square lattice with nearest-neighbor interactions are studied by means two numerical methods: homotopy continuation method and globally convergent Newton-Raphson method. We analyze properties points, in particular respect to number quantities that have been conjectured display signatures thermodynamic phase transition model. Although no such found for model, our study illustrates...

Finding vacua for the four dimensional effective theories supergravity which descend from flux compactifications and analyzing them according to their stability is one of central problems in string phenomenology. Except some simple toy models, it is, however, difficult find all analytically. Recently developed algorithmic methods based on symbolic computer algebra can be great help more realistic models. However, they suffer serious complexities are limited small system sizes. In this...

Active research activity in power systems areas has focused on developing computational methods to solve load flow equations where a key question is the maximum number of solutions. Although several upper bounds exist, recent studies have hinted that much sharper depend topology underlying networks may exist. This paper provides significant refinement these observations. We also develop geometric construction called adjacency polytope accurately captures network and immensely useful...

Synchronization in networks of interconnected oscillators is a fascinating phenomenon that appears naturally many independent fields science and engineering. A substantial amount work has been devoted to understanding all possible synchronization configurations on given network. In this setting, key problem determine the total number such configurations. Through an algebraic formulation for tree cycle graphs, we provide upper bounds using theory birationally invariant intersection index...

We study a model for itinerant, strongly interacting fermions where judicious tuning of the interactions leads to supersymmetric Hamiltonian. On triangular lattice this is known exhibit property called superfrustration, which characterized by an extensive ground state entropy. Using combination numerical and analytical methods we various ladder geometries obtained imposing doubly periodic boundary conditions on lattice. compare our results bounds degeneracy in literature. For all systems...

We study the three-spin spherical model with mean-field interactions and Gaussian random couplings. For moderate system sizes of up to 20 spins, we obtain all stationary points energy landscape by means numerical polynomial homotopy continuation method. On basis these points, analyze complexity other quantities related glass transition compare finite-system their exact counterparts in thermodynamic limit.