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It utilizes a coupled system derived from the first-order optimality system of the optimal control problem and employs deep neural networks to represent the solutions to the reduced system. We present an error analysis of the scheme and provide\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$L^2(\\Omega )$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>\u03a9<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    error bounds on the state, control, and adjoint in terms of neural network parameters (e.g., depth, width, and parameter bounds) and the numbers of sampling points. The main tools in the analysis include offset Rademacher complexity and boundedness and Lipschitz continuity of neural network functions. We present several numerical examples to illustrate the method and compare it with two existing ones.\n                  <\/jats:p>","DOI":"10.1007\/s10444-025-10241-z","type":"journal-article","created":{"date-parts":[[2025,6,26]],"date-time":"2025-06-26T02:56:15Z","timestamp":1750906575000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Solving elliptic optimal control problems via neural networks and optimality system"],"prefix":"10.1007","volume":"51","author":[{"given":"Yongcheng","family":"Dai","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3775-9155","authenticated-orcid":false,"given":"Bangti","family":"Jin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ramesh Chandra","family":"Sau","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhi","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,6,26]]},"reference":[{"key":"10241_CR1","volume-title":"Sobolev Spaces","author":"RA Adams","year":"2003","unstructured":"Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. 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