Projection Methods for Operator Learning and Universal Approximation
<|reference_start|>Projection Methods for Operator Learning and Universal Approximation: We obtain a new universal approximation theorem for continuous operators on arbitrary Banach spaces using the Leray-Schauder mapping. Moreover, we introduce and study a method for operator learning in Banach spaces $L^p$ of functions with multiple variables, based on orthogonal projections on polynomial bases. We derive a universal approximation result for operators where we learn a linear projection and a finite dimensional mapping under some additional assumptions. For the case of $p=2$, we give some sufficient conditions for the approximation results to hold. This article serves as the theoretical framework for a deep learning methodology whose implementation will be provided in subsequent work.<|reference_end|>
@article{zappala2024projection,
    title={Projection Methods for Operator Learning and Universal Approximation},
    author={Emanuele Zappala},
    journal={arXiv preprint arXiv:2406.12264},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.12264},
    primaryClass={math.NA cs.AI cs.LG cs.NA}
}