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Sci. Comput."],"published-print":{"date-parts":[[2024,4,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Hamilton\u2013Jacobi partial differential equations (HJ PDEs) have deep connections with a wide range of fields, including optimal control, differential games, and imaging sciences. By considering the time variable to be a higher dimensional quantity, HJ PDEs can be extended to the multitime case. In this paper, we establish a novel theoretical connection between specific optimization problems arising in machine learning and the multitime Hopf formula, which corresponds to a representation of the solution to certain multitime HJ PDEs. Through this connection, we increase the interpretability of the training process of certain machine learning applications by showing that when we solve these learning problems, we also solve a multitime HJ PDE and, by extension, its corresponding optimal control problem. As a first exploration of this connection, we develop the relation between the regularized linear regression problem and the linear quadratic regulator (LQR). We then leverage our theoretical connection to adapt standard LQR solvers (namely, those based on the Riccati ordinary differential equations) to design new training approaches for machine learning. Finally, we provide some numerical examples that demonstrate the versatility and possible computational advantages of our Riccati-based approach in the context of continual learning, posttraining calibration, transfer learning, and sparse dynamics identification.<\/jats:p>","DOI":"10.1137\/23m1561397","type":"journal-article","created":{"date-parts":[[2024,3,15]],"date-time":"2024-03-15T04:02:45Z","timestamp":1710475365000},"page":"C216-C248","source":"Crossref","is-referenced-by-count":11,"title":["Leveraging Multitime Hamilton\u2013Jacobi PDEs for Certain Scientific Machine Learning Problems"],"prefix":"10.1137","volume":"46","author":[{"given":"Paula","family":"Chen","sequence":"first","affiliation":[{"name":"Division of Applied Mathematics, Brown University, Providence, RI 02912 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7467-601X","authenticated-orcid":true,"given":"Tingwei","family":"Meng","sequence":"additional","affiliation":[{"name":"Department of Mathematics, UCLA, Los Angeles, CA 90025 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zongren","family":"Zou","sequence":"additional","affiliation":[{"name":"Division of Applied Mathematics, Brown University, Providence, RI 02912 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0483-7919","authenticated-orcid":true,"given":"J\u00e9r\u00f4me","family":"Darbon","sequence":"additional","affiliation":[{"name":"Corresponding author. Division of Applied Mathematics, Brown University, Providence, RI 02912 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9713-7120","authenticated-orcid":true,"given":"George Em","family":"Karniadakis","sequence":"additional","affiliation":[{"name":"Division of Applied Mathematics, Brown University, Providence, RI 02912 USA, and Pacific Northwest National Laboratory, Richland, WA 99354 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,3,15]]},"reference":[{"key":"ref1","unstructured":"M. Abadi , \nA. Agarwal , \nP. Barham , TensorFlow: Large-scale Machine Learning on Heterogeneous Systems, https:\/\/www.tensorflow.org\/ (2015)."},{"key":"ref2","volume-title":"Optimal Control: Linear Quadratic Methods","author":"Anderson B. D.","year":"2007"},{"key":"ref3","doi-asserted-by":"crossref","unstructured":"M. Bardi  and \nI. 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