Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation

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dc.contributor.advisorde Hoop, Maarten V.en_US
dc.creatorLara Benitez, Antonioen_US
dc.date.accessioned2024-05-20T19:32:46Zen_US
dc.date.created2024-05en_US
dc.date.issued2024-01-18en_US
dc.date.submittedMay 2024en_US
dc.date.updated2024-05-20T19:32:46Zen_US
dc.descriptionEMBARGO NOTE: This item is embargoed until 2024-11-01en_US
dc.description.abstractDeep learning has emerged as an incredibly successful and versatile approach within the field of machine learning, finding applications across a diverse range of domains. Originally devised for tasks such as classification and natural language processing, deep learning has made significant inroads into scientific computing. Architectures like Deeponet and Neural Operators have showcased their potential in approximating operators defined by partial differential equations (PDEs). While these architectures have shown practical success, there remains a compelling need to delve deeper into their theoretical foundations. This thesis aims to contribute to the theoretical understanding of deep learning by applying statistical learning theory to the neural operator family. Our primary focus will be on the generalization properties of this family while addressing the challenges posed by the high-frequency Helmholtz equation. To achieve this, we propose a subfamily of neural operators, known as sequential neural operators, which not only preserves all the approximation guarantees of neural operators but also exhibits enhanced generalization properties. This design draws inspiration from the self-attention mechanism found in the ubiquitous transformer architecture. To analyze both neural operators and sequential neural operators we establish upper bounds on Rademacher complexity. These bounds are instrumental in deriving the corresponding generalization error bounds. Furthermore, we leverage Gaussian-Banach spaces to shed light on the out-of-risk bounds of traditional neural operators and sequential neural operators.en_US
dc.embargo.lift2024-11-01en_US
dc.embargo.terms2024-11-01en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.citationBenitez, Antonio Lara. Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation. (2024). Masters thesis, Rice University. https://hdl.handle.net/1911/115909en_US
dc.identifier.urihttps://hdl.handle.net/1911/115909en_US
dc.language.isoengen_US
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.en_US
dc.subjectNeural operatorsen_US
dc.subjectstatistical learning theoryen_US
dc.subjectdeep learningen_US
dc.subjectout-of-distributionen_US
dc.subjectRademacher complexityen_US
dc.subjectGaussian-Banach spacesen_US
dc.subjectrisk boundsen_US
dc.titleOut-of-distributional risk bounds for neural operators with applications to the Helmholtz equationen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentComputational and Applied Mathematicsen_US
thesis.degree.disciplineEngineeringen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelMastersen_US
thesis.degree.nameMaster of Scienceen_US
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