Background
In variable-step operator learning, agreement at observed step sizes does not by itself determine predictions at other temporal resolutions.
Proposed Operator
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.
Theoretical Guarantee
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.
Identifiability
We also give conditions for local identifiability of both components from two step sizes, supporting interpretation of the learned structure.
Latent Extension
A latent extension applies the same construction to learned fields.
Experiments
We evaluate the base model on two synthetic problems and the extension on two real datasets against several baselines.