{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,18]],"date-time":"2026-09-18T08:58:11Z","timestamp":1789721891453,"version":"4.0.1"},"reference-count":33,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/100006228","name":"Oak Ridge National Laboratory","doi-asserted-by":"crossref","award":["ERKJ387"],"award-info":[{"award-number":["ERKJ387"]}],"id":[{"id":"10.13039\/100006228","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100006132","name":"Office of Science","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100006132","id-type":"DOI","asserted-by":"crossref"}]},{"name":"DOE Office of Advanced Scientific Computing Research"},{"name":"DOE Office of Earth and Environmental System Modeling"},{"name":"Scientific Discovery through Advanced Computing","award":["DE-SC0022254"],"award-info":[{"award-number":["DE-SC0022254"]}]},{"DOI":"10.13039\/100006206","name":"Biological and Environmental Research","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100006206","id-type":"DOI","asserted-by":"crossref"}]},{"name":"National Science Foundation","award":["DMS-2012469"],"award-info":[{"award-number":["DMS-2012469"]}]},{"name":"National Science Foundation","award":["DMS-2038080"],"award-info":[{"award-number":["DMS-2038080"]}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2023,6,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Due to the curse of dimensionality and the limitation on training data, approximating high-dimensional functions is a very challenging task even for powerful deep neural networks. Inspired by the Nonlinear Level-set Learning (NLL) method, which uses the reversible residual network (RevNet), in this paper, we propose a new method of Dimension Reduction via Learning Level Sets (DRiLLS) for function approximation. Our method contains two major components: one is the pseudoreversible neural network (PRNN) module, which effectively transforms high-dimensional input variables to low-dimensional active variables, and the other is the synthesized regression module for approximating function values based on the transformed data in the low-dimensional space. The PRNN not only relaxes the invertibility constraint of the nonlinear transformation present in the NLL method due to the use of RevNet but also adaptively weights the influence of each sample and controls the sensitivity of the function to the learned active variables. The synthesized regression uses Euclidean distance in the input space to select neighboring samples whose projections on the space of active variables are used to perform local least-squares polynomial fitting. This helps to resolve numerical oscillation issues present in traditional local and global regressions. Extensive experimental results demonstrate that our DRiLLS method outperforms both the NLL and active subspace methods, especially when the target function possesses critical points in the interior of its input domain.<\/jats:p>","DOI":"10.1137\/21m1459198","type":"journal-article","created":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T06:14:18Z","timestamp":1685945658000},"page":"A1148-A1171","source":"Crossref","is-referenced-by-count":8,"title":["Level Set Learning with Pseudoreversible Neural Networks for Nonlinear Dimension Reduction in Function Approximation"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9195-1431","authenticated-orcid":true,"given":"Yuankai","family":"Teng","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of South Carolina, Columbia, SC 29208 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhu","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of South Carolina, Columbia, SC 29208 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6520-582X","authenticated-orcid":true,"given":"Lili","family":"Ju","sequence":"additional","affiliation":[{"name":"Corresponding author. Department of Mathematics, University of South Carolina, Columbia, SC 29208 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7107-5307","authenticated-orcid":true,"given":"Anthony","family":"Gruber","sequence":"additional","affiliation":[{"name":"Department of Scientific Computing, Florida State University, Tallahassee, FL 32306 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7256-150X","authenticated-orcid":true,"given":"Guannan","family":"Zhang","sequence":"additional","affiliation":[{"name":"Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,6,5]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1002\/wics.101"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.2009.0110"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0"},{"key":"ref4","doi-asserted-by":"crossref","unstructured":"B. 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