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Learn.: Sci. Technol."],"published-print":{"date-parts":[[2026,6,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    The time-dependent Schr\u00f6dinger equation (TDSE) in real space is fundamental to understanding the dynamics of many-electron quantum systems, with applications ranging from quantum chemistry to condensed matter physics and materials science. However, solving the TDSE for complex fermionic systems remains a significant challenge, particularly due to the need to capture the time-evolving many-body correlations, while the antisymmetric nature of fermionic wavefunctions complicates the function space in which these solutions must be represented. We propose a general-purpose neural network framework for solving the real-space TDSE, Fermionic Antisymmetric Spatiotemporal Network, which treats time as an explicit input alongside spatial coordinates, enabling a unified spatiotemporal representation of complex, antisymmetric wavefunctions for fermionic systems. This approach formulates the TDSE as a global optimization problem, avoiding step-by-step propagation and supporting highly parallelizable training. The method is demonstrated on five benchmark problems: a one-dimensional harmonic oscillator, interacting fermions in a time-dependent (TD) harmonic trap, three-dimensional hydrogen orbital dynamics, a laser-driven hydrogen atom, and a laser-driven H\n                    <jats:sub>2<\/jats:sub>\n                    molecule, achieving excellent agreement with reference solutions across all cases. These results demonstrate the method\u2019s accuracy and flexibility within the bound-state manifold across various dimensions and interaction regimes. While the current localized Ansatz inherently restricts the description of extensive ionization and continuum states, the method demonstrates the capability to stably simulate coherent multi-electron dynamics over extended time windows. Our framework offers a highly expressive alternative to traditional basis-dependent or mean-field methods, opening new possibilities for\n                    <jats:italic>ab initio<\/jats:italic>\n                    simulations of TD quantum systems, with applications in quantum dynamics, molecular control, and ultrafast spectroscopy.\n                  <\/jats:p>","DOI":"10.1088\/2632-2153\/ae62c8","type":"journal-article","created":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T22:54:27Z","timestamp":1776812067000},"page":"035012","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["A global spacetime optimization approach to the real-space time-dependent Schr\u00f6dinger equation"],"prefix":"10.1088","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-4185-3838","authenticated-orcid":true,"given":"Enze","family":"Hou","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0853-1933","authenticated-orcid":false,"given":"Yuzhi","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Linxuan","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6696-0707","authenticated-orcid":false,"given":"Difa","family":"Ye","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-1348-8583","authenticated-orcid":false,"given":"Lei","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5623-1148","authenticated-orcid":true,"given":"Han","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"266","published-online":{"date-parts":[[2026,5,7]]},"reference":[{"key":"mlstae62c8bib1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.95.033415","type":"journal-article","article-title":"High-order-harmonic generation in atomic and molecular systems","volume":"95","author":"Su\u00e1rez","year":"2017","journal-title":"Phys. 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