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This paper introduces Bregman neural operators, a theoretical framework that reformulates operator layers as minimizers of Bregman regularized optimization problems within Banach function spaces. By interpreting activation operators as proximity operators between dual and primal spaces, the approach enables the construction of deeper, more accurate Fourier neural operator models without sacrificing expressivity. The work advances the mathematical foundations of learning infinite-dimensional mappings, offering a rigorous pathway to improve training stability and accuracy in physics-informed machine learning applications.
A Bregman Proximal Viewpoint on Neural Operators
arXiv:2610.20120v1 Announce Type: new Abstract: We present several advances on neural operators by viewing the action of operator layers as the minimizers of Bregman regularized optimization problems over Banach function spaces. The proposed framework allows interpreting the activation operators as Bregman proximity operators from dual to primal space. This novel viewpoint is general enough to recover classical neural operators as well as a new variant, coined Bregman neural operators, which includes the inverse activation operator and features the same expressivity of standard neural operators. Numerical experiments support the added benefits of the Bregman variant of Fourier neural operators for training deeper and more accurate models.
Abstract:We present several advances on neural operators by viewing the action of operator layers as the minimizers of Bregman regularized optimization problems over Banach function spaces. The proposed framework allows interpreting the activation operators as Bregman proximity operators from dual to primal space. This novel viewpoint is general enough to recover classical neural operators as well as a new variant, coined Bregman neural operators, which includes the inverse activation operator and features the same expressivity of standard neural operators. Numerical experiments support the added benefits of the Bregman variant of Fourier neural operators for training deeper and more accurate models.
Machine Learning (cs.LG)
arXiv:2610.20120 [cs.LG]
(or arXiv:2610.20120v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.20120 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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