{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T13:17:39Z","timestamp":1771939059348,"version":"3.50.1"},"reference-count":89,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,9,25]],"date-time":"2024-09-25T00:00:00Z","timestamp":1727222400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["MAKE"],"abstract":"<jats:p>This paper investigates the feasibility of downscaling within high-dimensional Lorenz models through the use of machine learning (ML) techniques. This study integrates atmospheric sciences, nonlinear dynamics, and machine learning, focusing on using large-scale atmospheric data to predict small-scale phenomena through ML-based empirical models. The high-dimensional generalized Lorenz model (GLM) was utilized to generate chaotic data across multiple scales, which was subsequently used to train three types of machine learning models: a linear regression model, a feedforward neural network (FFNN)-based model, and a transformer-based model. The linear regression model uses large-scale variables to predict small-scale variables, serving as a foundational approach. The FFNN and transformer-based models add complexity, incorporating multiple hidden layers and self-attention mechanisms, respectively, to enhance prediction accuracy. All three models demonstrated robust performance, with correlation coefficients between the predicted and actual small-scale variables exceeding 0.9. Notably, the transformer-based model, which yielded better results than the others, exhibited strong performance in both control and parallel runs, where sensitive dependence on initial conditions (SDIC) occurs during the validation period. This study highlights several key findings and areas for future research: (1) a set of large-scale variables, analogous to multivariate analysis, which retain memory of their connections to smaller scales, can be effectively leveraged by trained empirical models to estimate irregular, chaotic small-scale variables; (2) modern machine learning techniques, such as FFNN and transformer models, are effective in capturing these downscaling processes; and (3) future research could explore both downscaling and upscaling processes within a triple-scale system (e.g., large-scale tropical waves, medium-scale hurricanes, and small-scale convection processes) to enhance the prediction of multiscale weather and climate systems.<\/jats:p>","DOI":"10.3390\/make6040107","type":"journal-article","created":{"date-parts":[[2024,9,27]],"date-time":"2024-09-27T09:16:10Z","timestamp":1727428570000},"page":"2161-2182","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Exploring Downscaling in High-Dimensional Lorenz Models Using the Transformer Decoder"],"prefix":"10.3390","volume":"6","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0750-0625","authenticated-orcid":false,"given":"Bo-Wen","family":"Shen","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182, USA"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"530","DOI":"10.1177\/030913339702100403","article-title":"Downscaling general circulation model output: A review of methods and limitations","volume":"21","author":"Wilby","year":"1997","journal-title":"Prog. 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