Data-Driven Integration Kernels for Interpretable Nonlocal Operator Learning

TL;DR

This paper introduces data-driven integration kernels that enhance interpretability in nonlocal operator learning models for climate processes by explicitly separating nonlocal information aggregation from local nonlinear prediction.

Contribution

It proposes a novel framework that uses learnable kernels to make nonlocal information aggregation interpretable and structurally constrained, improving model efficiency and understanding.

Findings

Kernel models achieve near-baseline performance with fewer parameters.

The framework reveals which locations, levels, and times contribute most to predictions.

Structural constraints enable capturing relevant nonlocal information efficiently.

Abstract

Machine learning models can represent climate processes that are nonlocal in horizontal space, height, and time, often by combining information across these dimensions in highly nonlinear ways. While this can improve predictive skill, it makes learned relationships difficult to interpret and prone to overfitting as the extent of nonlocal information grows. We address this challenge by introducing data-driven integration kernels, a framework that adds structure to nonlocal operator learning by explicitly separating nonlocal information aggregation from local nonlinear prediction. Each spatiotemporal predictor field is first integrated using learnable kernels (defined as continuous weighting functions over horizontal space, height, and/or time), after which a local nonlinear mapping is applied only to the resulting kernel-integrated features and optional local inputs. This design confines nonlinear interactions to a small set of integrated features and makes each kernel directly interpretable as a weighting pattern that reveals which horizontal locations, vertical levels, and past timesteps contribute most to the prediction. We demonstrate the framework for South Asian monsoon precipitation using a hierarchy of neural network models with increasing structure, including baseline, nonparametric kernel, and parametric kernel models. Across this hierarchy, kernel models achieve near-baseline performance with far fewer trainable parameters, indicating that much of the relevant nonlocal information can be captured through a small set of interpretable integrations when appropriate structural constraints are imposed.