Title: Geometric Phase Field Analysis for Non-Linear Tensor Networks

Jincheng Zhang

Published Sep 4, 2026• 0 citations

Abstract

This paper explores a novel framework for analyzing and manipulating tensor networks, centered around the geometric properties of phase fields. Traditional tensor network analysis often overlooks the inherent geometric structure of these networks, hindering effective manipulation and understanding. We propose a "phase-field gradient" – a dynamically evolving field representing the phase relationships within a tensor network – that explicitly incorporates differential geometry to provide a more precise and insightful method for analyzing and manipulating network topology. This approach allows for the direct manipulation of network structure, opening avenues for new research in areas such as graph theory, optimization, and machine learning. The paper details the mathematical foundations of the phase-field gradient, provides a concrete implementation demonstrating its utility, and discusses potential applications and future directions.

DOI: 10.5281/zenodo.22295137

Related papers

Title: Dynamical Topology in Tensor Networks

Jincheng Zhang

This paper explores the potential of extending tensor network methods to incorporate continuous topology, a crucial element in representing and analyzing complex, non-linear dynamical systems. Traditional tensor network representations are limited in their ability to capture the intricate interplay of topological defects and their dynamics. We propose a novel tensor network structure that explicitly encodes topological features, enabling a more comprehensive understanding of evolving systems. This work will focus on developing a framework for representing and analyzing topological defects within the context of tensor network analysis. The core mechanism involves introducing a dynamic 'twist' parameter that governs the formation and evolution of topological features, offering a significant advancement over existing approaches. This research will investigate the practical implications of this new approach, providing a foundation for studying diverse dynamical systems and highlighting the potential for enhanced modeling capabilities.

Sep 6, 20260 citations

Title: Non-Local Geometric Phase Transitions

Jincheng Zhang

This paper investigates the complexities of non-local geometric phase transitions, a rapidly growing field of research with significant implications for various scientific disciplines. The core claim centers on the development of a novel framework for analyzing these transitions, leveraging a dynamically evolving 'geometry' to predict and control transition rates. Traditional approaches often rely on static models, limiting our ability to effectively understand and manipulate these phenomena. This work proposes a fundamentally new approach that actively shapes the system's geometry, offering a significant advancement in predictive capabilities. We introduce a 'geometric coherence field' that adjusts the geometry based on the phase transition, providing a more precise and adaptable modeling strategy. The research explores the practical application of this framework through simulations and analysis, demonstrating the potential for improved control over transition rates. This paper offers a foundational contribution to the understanding and management of non-local geometric phase transitions.

Sep 6, 20260 citations

Title: Neuromorphic Tensor Network Analysis (NTTA)

Jincheng Zhang

This paper introduces Neuromorphic Tensor Network Analysis (NTTA), a novel method for analyzing tensor networks leveraging neural networks. Traditional approaches to tensor network analysis are computationally demanding, limiting their applicability to complex network structures. NTTA employs a trainable neural network to directly predict network connectivity and properties, offering a significant acceleration and efficiency improvement. The core mechanism centers on learning a representation of the tensor network structure, enabling rapid assessment of network properties without the need for exhaustive manual analysis. We present a framework for training and evaluating this network, demonstrating its effectiveness in analyzing diverse tensor network configurations. The resulting system facilitates the computationally efficient exploration of tensor network properties, potentially accelerating research and applications across various fields, including machine learning, materials science, and neuroscience.

Sep 3, 20260 citations

Title: Phase-Field Quantum Geometry (PFQ)

Jincheng Zhang

The field of quantum geometry is experiencing a resurgence, driven by the desire to understand and manipulate the fundamental structure of quantum systems. This paper introduces the Phase-Field Quantum Geometry (PFQ) framework, a novel approach to representing and manipulating geometric configurations within quantum systems. PFQ leverages the concept of phase fields to dynamically construct complex geometric structures, offering a powerful tool for studying quantum phenomena and potentially enabling new quantum technologies. The core mechanism focuses on an automated "Phase Field Mapping" algorithm, which constructs intricate geometric representations from quantum data. This paper presents a detailed explanation of the PFQ framework, including its mathematical foundations, algorithm implementation, and potential applications across diverse quantum systems. The research explores the advantages of PFQ compared to existing geometric techniques and highlights its potential to unlock deeper insights into quantum mechanics.

Sep 4, 20260 citations

Title: Quantum Geometry – Formalizing the Geometry of Quantum Systems Using Topological Concepts

Jincheng Zhang

Quantum geometry, a nascent field, seeks to formalize the geometric properties of quantum systems through the lens of topology. This paper explores the development of a novel mathematical framework for defining and analyzing these geometric properties, explicitly incorporating topological concepts to enhance understanding and provide a more robust and insightful approach compared to traditional geometric methods. We propose a new methodology predicated on the observation that certain geometric features exhibit topological behavior, offering a fundamentally different perspective on the structure of quantum systems. The paper outlines the core mechanism, detailing how topological aspects will be integrated into the definition of geometric properties and how this will lead to a deeper comprehension of quantum phenomena. The investigation will focus on establishing a rigorous mathematical foundation for this approach, demonstrating its potential for improving the analysis of complex quantum systems.

Sep 4, 20260 citations

Title: Universal Computational Geometry via Tensor Networks

Jincheng Zhang

Universal Computational Geometry (UCG) represents the task of finding geometric primitives that satisfy a set of constraints. This paper proposes a novel approach utilizing tensor networks to represent and manipulate geometric primitives, offering a unified framework for geometric computation across diverse domains. We develop a method for encoding geometric primitives as tensors, leveraging tensor network techniques to implement fundamental operations like concatenation, dilation, and inversion. The core mechanism centers around constructing a tensor network representing the geometric primitives, allowing for efficient and scalable computation. The proposed method provides a significant step towards a generalized UCG solver, addressing a long-standing challenge in the field. We provide a detailed implementation demonstrating the effectiveness of our approach, showcasing its ability to handle complex geometric transformations and provide a foundation for future advancements.

Sep 6, 20260 citations