Exponential-Nemytskii operators improve modeling of time-dependent fields
Exponential-Nemytskii Neural Operators: Learning Field Evolution Across Step Sizes
Machine Learning
Summary
Predicting how things change over time is hard, especially when measurements are taken at irregular time steps. This paper introduces a method that combines exact linear changes with flexible nonlinear adjustments to better model how a field evolves across different step sizes. The authors show theoretically that understanding the model’s behavior at two time scales helps predict its performance across a range of scales. They also test their approach on simulated and real data, demonstrating improved predictions compared to existing methods.
What this means in practice
- •For simulation engineers: Improve time-dependent simulations by accurately modeling dynamics across varying time step sizes.
- •For climate modelers: Enhance climate predictions by better learning how environmental fields evolve over time at different resolutions.
Authors
Jingye Li, Jinglai Li
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.