Differential Algebraic Closure Framework for Geometric Analysis: A Unified Constructive Approach to Geometric PDEs and Exterior Differential Equations

Differential Algebraic Closure Framework for Geometric Analysis: A Unified Constructive Approach to Geometric PDEs and Exterior Differential Equations | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 28 October 2025 V1 Latest version Share on Differential Algebraic Closure Framework for Geometric Analysis: A Unified Constructive Approach to Geometric PDEs and Exterior Differential Equations Authors : Dongqi Liu 0009-0006-4018-9292 and shifa liu 0009-0003-6570-2812 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.176168567.78318265/v1 216 views 151 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract 本文建立了一个全面的微分代数框架，用于构建光滑流形上几何偏微分方程（PDE）和外微分方程（EDE）的显式解。我们定义了微分几何闭包 K DiffGeo 和线性外部微分闭包 K LEDE，这是通过包含几何对象、谐波形式、曲率张量和基本解的递归附加过程构建的微分闭场扩展。在这些闭包中，我们证明了大类几何偏微分方程的解——包括山边方程、利玛窦流、爱因斯坦场方程和线性外微分方程——允许尊重底层几何和代数结构的统一表示。该框架严格解决了非线性、几何约束和多维挑战，同时保留了分级代数结构和几何兼容性条件。我们提供详细的建设性证明，推导具有严格误差范围的显式解公式，并在适当的几何函数空间中建立收敛标准。提出了具有精确复杂性分析的综合算法，包括稳定性保证和具有认证误差边界的自适应精度控制。采用区间算术和离散外部微积分的严格验证框架证明了我们方法的实际有效性。这项工作表明，显式解析解存在于适当构造的微分代数闭包中，为几何可解性提供了新的代数视角，同时保持与经典理论的一致性。对谱理论、曲率流、拓扑不变量、几何深度学习和结构保留数值的扩展建立了跨数学学科的联系。 Supplementary Material File (differential_geometry.pdf) Download 436.37 KB Information & Authors Information Version history V1 Version 1 28 October 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords certified computation constructive mathematics curvature flows differential algebraic closure exterior differential equations geometric deep learning geometric partial differential equations harmonic forms riemannian geometry spectral theory Authors Affiliations Dongqi Liu 0009-0006-4018-9292 View all articles by this author shifa liu 0009-0003-6570-2812 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 216 views 151 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Dongqi Liu, shifa liu. Differential Algebraic Closure Framework for Geometric Analysis: A Unified Constructive Approach to Geometric PDEs and Exterior Differential Equations. Authorea . 28 October 2025. DOI: https://doi.org/10.22541/au.176168567.78318265/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . 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