Exponential-Nemytskii Neural Operators: Learning Field Evolution Across Step Sizes
Abstract
In variable-step operator learning, agreement at observed step sizes does not by itself determine predictions at other temporal resolutions. We propose an operator whose step-size dependence is prescribed by a first-order Duhamel approximation of semilinear dynamics. It combines exact linear propagation with a local nonlinear function of the field and its derivatives, allowing evaluation below the observation interval without supervision densely sampled across step sizes. Under suitable conditions and at fixed spatial resolution, we prove that agreement between models at two step sizes controls their prediction differences throughout a prescribed range. We also give conditions for local identifiability of both components from two step sizes, supporting interpretation of the learned structure. A latent extension applies the same construction to learned fields. We evaluate the base model on two synthetic problems and the extension on two real datasets against several baselines.
Related
- Numerical Neural Operator: Connection Between Numerical Analysis and the Operator Learning and Building Discretized Neural Operators from Finite Element Methods
- Analysis of split exponential integrators for semilinear problems with noncommuting operators
- Order Reduction of Exponential Runge--Kutta Methods: Fourth-Order Schemes for Non-Commuting Operators
- Korovkin type theorems for operators acting on functions of polynomial and exponential growth on [0,∞)
- Non-centered maximal operators over ℓq balls for exponential and Laguerre-type measures
BibTeX
@misc{li2026exponentialnemytskii,
title = {{Exponential-Nemytskii Neural Operators: Learning Field Evolution Across Step Sizes}},
author = {Jingye Li and Jinglai Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39041},
eprint = {2610.39041},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39041},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exponential-Nemytskii Neural Operators: Learning Field Evolution Across Step Sizes
AU - Jingye Li
AU - Jinglai Li
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.39041
UR - https://arxiv.org/abs/2610.39041
AB - In variable-step operator learning, agreement at observed step sizes does not by itself determine predictions at other temporal resolutions. We propose an operator whose step-size dependence is prescribed by a first-order Duhamel approximation of semilinear dynamics. It combines exact linear propagation with a local nonlinear function of the field and its derivatives, allowing evaluation below the observation interval without supervision densely sampled across step sizes. Under suitable conditions and at fixed spatial resolution, we prove that agreement between models at two step sizes controls their prediction differences throughout a prescribed range. We also give conditions for local identifiability of both components from two step sizes, supporting interpretation of the learned structure. A latent extension applies the same construction to learned fields. We evaluate the base model on two synthetic problems and the extension on two real datasets against several baselines.
ER -