ICML 2025

A Bregman Proximal Viewpoint on Neural Operators
Abdel-Rahim Mezidi 1 * Jordan Patracone 1 * Saverio Salzo 2 * Amaury Habrard 1 3 Massimiliano Pontil 4 5
Remi Emonet 1 3 Marc Sebban 1
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 pro-
posed framework allows interpreting the activa-
tion operators as Bregman proximity operators
from dual to primal space. This novel viewpoint
is general enough to recover classical neural op-
erators as well as a new variant, coined Bregman
neural operators, which includes the inverse acti-
vation operator and features the same expressivity
of standard neural operators. Numerical experi-
ments support the added benefits of the Bregman
variant of Fourier neural operators for training
deeper and more accurate models.
1. Introduction
Neural operators (Kovachki et al., 2021; 2023), a recent
extension of neural networks, have emerged as a versatile
framework for learning mappings between function spaces.
These operators have shown great potential in solving par-
tial differential equations (PDEs) and simulating complex
dynamical systems. The exploration of neural architectures
for the approximation and learning of operators has led to
the development of a variety of models.
One influential contribution is the Fourier Neural Opera-
tor (FNO) (Li et al., 2021a), sketched in Figure 1, which
transforms encoded input data into frequency components
in order to learn intricate relationships in the frequency
domain. More recently, the Group-Equivariant FNO (G-
FNO) (Helwig et al., 2023) additionally leverages symme-
*Equal contribution
1Université Jean Monnet Saint-Etienne,
CNRS, Institut d’Optique Graduate School, Inria, Laboratoire
Hubert Curien UMR 5516, F-42023, SAINT-ETIENNE, France
2DIAG, Sapienza University of Rome, 00185 Rome, Italy 3Institut
Universitaire de France (IUF) 4Computational Statisti...