{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T00:16:26Z","timestamp":1778199386690,"version":"3.51.4"},"reference-count":59,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,10]],"date-time":"2025-07-10T00:00:00Z","timestamp":1752105600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002855","name":"National Key R&amp;D Program \u201cTransportation Infrastructure\u201d project","doi-asserted-by":"publisher","award":["2022YFB2603400"],"award-info":[{"award-number":["2022YFB2603400"]}],"id":[{"id":"10.13039\/501100002855","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002855","name":"National Key R&amp;D Program \u201cTransportation Infrastructure\u201d project","doi-asserted-by":"publisher","award":["L2023G008"],"award-info":[{"award-number":["L2023G008"]}],"id":[{"id":"10.13039\/501100002855","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002855","name":"National Key R&amp;D Program \u201cTransportation Infrastructure\u201d project","doi-asserted-by":"publisher","award":["2024YJ265"],"award-info":[{"award-number":["2024YJ265"]}],"id":[{"id":"10.13039\/501100002855","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100015860","name":"Technology Research and Development Plan Program of China State Railway Group Co., Ltd.","doi-asserted-by":"publisher","award":["2022YFB2603400"],"award-info":[{"award-number":["2022YFB2603400"]}],"id":[{"id":"10.13039\/100015860","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100015860","name":"Technology Research and Development Plan Program of China State Railway Group Co., Ltd.","doi-asserted-by":"publisher","award":["L2023G008"],"award-info":[{"award-number":["L2023G008"]}],"id":[{"id":"10.13039\/100015860","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100015860","name":"Technology Research and Development Plan Program of China State Railway Group Co., Ltd.","doi-asserted-by":"publisher","award":["2024YJ265"],"award-info":[{"award-number":["2024YJ265"]}],"id":[{"id":"10.13039\/100015860","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Foundation of China Academy of Railway Sciences Co., Ltd.","award":["2022YFB2603400"],"award-info":[{"award-number":["2022YFB2603400"]}]},{"name":"Foundation of China Academy of Railway Sciences Co., Ltd.","award":["L2023G008"],"award-info":[{"award-number":["L2023G008"]}]},{"name":"Foundation of China Academy of Railway Sciences Co., Ltd.","award":["2024YJ265"],"award-info":[{"award-number":["2024YJ265"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Accurate settlement prediction for high-speed railway (HSR) soft foundations remains challenging due to the irregular and dynamic nature of real-world monitoring data, often represented as non-equidistant and non-stationary time series (NENSTS). Existing empirical models lack clear applicability criteria under such conditions, resulting in subjective model selection. This study introduces a Monte Carlo-based evaluation framework that integrates data-driven simulation with geotechnical principles, embedding the concept of symmetry across both modeling and assessment stages. Equivalent permeability coefficients (EPCs) are used to normalize soil consolidation behavior, enabling the generation of a large, statistically robust dataset. Four empirical settlement prediction models\u2014Hyperbolic, Exponential, Asaoka, and Hoshino\u2014are systematically analyzed for sensitivity to temporal features and resistance to stochastic noise. A symmetry-aware comprehensive evaluation index (CEI), constructed via a robust entropy weight method (REWM), balances multiple performance metrics to ensure objective comparison. Results reveal that while settlement behavior evolves asymmetrically with respect to EPCs over time, a symmetrical structure emerges in model suitability across distinct EPC intervals: the Asaoka method performs best under low-permeability conditions (EPC \u2264 0.03 m\/d), Hoshino excels in intermediate ranges (0.03 &lt; EPC \u2264 0.7 m\/d), and the Exponential model dominates in highly permeable soils (EPC &gt; 0.7 m\/d). This framework not only quantifies model robustness under complex data conditions but also formalizes the notion of symmetrical applicability, offering a structured path toward intelligent, adaptive settlement prediction in HSR subgrade engineering.<\/jats:p>","DOI":"10.3390\/sym17071113","type":"journal-article","created":{"date-parts":[[2025,7,10]],"date-time":"2025-07-10T15:36:58Z","timestamp":1752161818000},"page":"1113","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Monte Carlo Simulation Framework for Evaluating the Robustness and Applicability of Settlement Prediction Models in High-Speed Railway Soft Foundations"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-0016-3653","authenticated-orcid":false,"given":"Zhenyu","family":"Liu","sequence":"first","affiliation":[{"name":"Railway Engineering Research Institute, China Academy of Railway Sciences Co., Ltd., Beijing 100081, China"},{"name":"State Key Laboratory of High-speed Railway Track System, China Academy of Railway Sciences Co., Ltd., Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8063-354X","authenticated-orcid":false,"given":"Liyang","family":"Wang","sequence":"additional","affiliation":[{"name":"Railway Engineering Research Institute, China Academy of Railway Sciences Co., Ltd., Beijing 100081, China"},{"name":"State Key Laboratory of High-speed Railway Track System, China Academy of Railway Sciences Co., Ltd., Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Taifeng","family":"Li","sequence":"additional","affiliation":[{"name":"Railway Engineering Research Institute, China Academy of Railway Sciences Co., Ltd., Beijing 100081, China"},{"name":"State Key Laboratory of High-speed Railway Track System, China Academy of Railway Sciences Co., Ltd., Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huiqin","family":"Guo","sequence":"additional","affiliation":[{"name":"Railway Engineering Research Institute, China Academy of Railway Sciences Co., Ltd., Beijing 100081, China"},{"name":"State Key Laboratory of High-speed Railway Track System, China Academy of Railway Sciences Co., Ltd., Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Feng","family":"Chen","sequence":"additional","affiliation":[{"name":"Railway Engineering Research Institute, China Academy of Railway Sciences Co., Ltd., Beijing 100081, China"},{"name":"State Key Laboratory of High-speed Railway Track System, China Academy of Railway Sciences Co., Ltd., Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Youming","family":"Zhao","sequence":"additional","affiliation":[{"name":"China Academy of Railway Sciences Co., Ltd., Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qianli","family":"Zhang","sequence":"additional","affiliation":[{"name":"Railway Engineering Research Institute, China Academy of Railway Sciences Co., Ltd., Beijing 100081, China"},{"name":"State Key Laboratory of High-speed Railway Track System, China Academy of Railway Sciences Co., Ltd., Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4079-0687","authenticated-orcid":false,"given":"Tengfei","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, Southwest Jiaotong University, Chengdu 610031, China"},{"name":"MOE Key Laboratory of High-speed Railway Engineering, Southwest Jiaotong University, Chengdu 610031, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1680\/geot.1957.7.4.168","article-title":"A Contribution to the settlement analysis of foundations on clay","volume":"7","author":"Skempton","year":"1957","journal-title":"G\u00e9otechnique"},{"key":"ref_2","first-page":"3463413","article-title":"Machine-learning-based road soft soil foundation treatment and settlement prediction method","volume":"2022","author":"Zhai","year":"2022","journal-title":"Sci. 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