{"682901":{"#nid":"682901","#data":{"type":"event","title":"PhD Defense by Harris Cobb","body":[{"value":"\u003Cp\u003EYou are cordially invited to attend my dissertation defense Monday July 7 at 11 AM in-person: Skiles 006. Virtual: https:\/\/gatech.zoom.us\/j\/99430137245\u003C\/p\u003E\u003Cp\u003E\u003Cbr\u003ETitle: Applications of Neural Networks with Locally Converging Inputs (NNLCI) for Classical and Quantum PDE Solvers\u003C\/p\u003E\u003Cp\u003EAbstract: We develop a unified framework for improving numerical solvers with Neural Networks with Locally Converging Inputs (NNLCI). First, we applied NNLCI to 2D Maxwell\u2019s equations with perfectly matched\u2010layer boundary conditions for light\u2013PEC (perfect electric conductor) interactions. A network trained on local patches around specific PEC shapes successfully predicted solutions on globally different geometries. Next, we tested NNLCI on various ODEs: it failed for chaotic systems (e.g., double pendulum) but was effective for nonchaotic dynamics, and in simple cases can be interpreted as a well\u2010defined function of its inputs. Although originally formulated for hyperbolic conservation laws, NNLCI also performed well on parabolic and elliptic problems, as demonstrated in a 1D Poisson\u2013Nernst\u2013Planck ion\u2010channel model. Building on these results, we applied NNLCI to multi\u2010asset cash\u2010or\u2010nothing options under Black\u2013Scholes. By correcting coarse\u2010 and fine\u2010mesh ADI solutions, NNLCI reduced RMSE by factors of 4\u201312 on test parameters, even when trained on a small fraction of the parameter grid. Careful treatment of far\u2010field boundary truncation was critical to maintain convergence far from the strike price. Finally, we demonstrate NNLCI\u2019s first application to quantum algorithms by improving variational quantum\u2010algorithm (VQA) outputs for the 1D Poisson equation under realistic NISQ\u2010device noise. Although noisy VQA solutions deviate from classical finite\u2010difference references and do not converge to true solutions, NNLCI effectively maps these noisy outputs toward high\u2010accuracy references. We hypothesize that NNLCI implicitly composes the map from coarse quantum outputs to a noisy convergence space, then to the true solution. We discuss conditions for NNLCI to approximate a well\u2010defined inverse of the numerical scheme and contrast this with Monte Carlo methods, which lack deterministic intermediate states. These results establish NNLCI as a versatile, data\u2010efficient tool for accelerating solvers in classical and quantum settings.\u003C\/p\u003E\u003Cp\u003ECommittee:\u003Cbr\u003E\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Dr. Yingjie Liu \u2013 School of Mathematics (advisor)\u003Cbr\u003E\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Dr. Sung Ha Kang \u2013 School of Mathematics\u003Cbr\u003E\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Dr. Wenjing Liao \u2013 School of Mathematics\u003Cbr\u003E\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Dr. Molei Tao \u2013 School of Mathematics\u003Cbr\u003E\u2022\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;Dr. Dexuan Xie \u2013 School of Mathematics, University of Wisconsin, Milwaukee\u003C\/p\u003E\u003Cp\u003EHarris Cobb\u003Cbr\u003EMathematics PhD Student\u003Cbr\u003ESchool of Mathematics\u003Cbr\u003EGeorgia Institute of Technology\u003Cbr\u003E\u0026nbsp;\u003C\/p\u003E","summary":"","format":"limited_html"}],"field_subtitle":"","field_summary":[{"value":"\u003Cp\u003EApplications of Neural Networks with Locally Converging Inputs (NNLCI) for Classical and Quantum PDE Solvers\u003C\/p\u003E","format":"limited_html"}],"field_summary_sentence":[{"value":"Applications of Neural Networks with Locally Converging Inputs (NNLCI) for Classical and Quantum PDE Solvers"}],"uid":"27707","created_gmt":"2025-06-26 15:11:09","changed_gmt":"2025-06-26 15:14:14","author":"Tatianna Richardson","boilerplate_text":"","field_publication":"","field_article_url":"","field_event_time":{"event_time_start":"2025-07-07T11:00:00-04:00","event_time_end":"2025-07-07T13:00:00-04:00","event_time_end_last":"2025-07-07T13:00:00-04:00","gmt_time_start":"2025-07-07 15:00:00","gmt_time_end":"2025-07-07 17:00:00","gmt_time_end_last":"2025-07-07 17:00:00","rrule":null,"timezone":"America\/New_York"},"location":"Skiles 006. ","extras":[],"groups":[{"id":"221981","name":"Graduate Studies"}],"categories":[],"keywords":[{"id":"100811","name":"Phd Defense"}],"core_research_areas":[],"news_room_topics":[],"event_categories":[{"id":"1788","name":"Other\/Miscellaneous"}],"invited_audience":[{"id":"78771","name":"Public"}],"affiliations":[],"classification":[],"areas_of_expertise":[],"news_and_recent_appearances":[],"phone":[],"contact":[],"email":[],"slides":[],"orientation":[],"userdata":""}}}