[Paper Review] Data-Driven Integration Kernels for Interpretable Nonlocal Operator Learning

Savannah L. Ferretti, Jerry Lin+4 more|arXiv (Cornell University)|Mar 11, 2026

Climate variability and models0 citations

TL;DR

The paper introduces data-driven integration kernels to separate nonlocal information aggregation from local prediction, yielding interpretable kernels and near-baseline predictive skill for South Asian monsoon precipitation with fewer parameters.

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 any 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-based models achieve near-baseline performance with far fewer trainable parameters, showing that much of the relevant nonlocal information can be captured through a small set of interpretable integrations when appropriate structural constraints are imposed.

Motivation & Objective

• Motivate the need for interpretable nonlocal operator learning in geophysical processes.
• Introduce integration kernel learning as a two-step framework separating nonlocal aggregation from local prediction.
• Develop a hierarchy of kernel-based models (nonparametric and parametric) to study trade-offs between skill and interpretability.
• Apply the framework to South Asian monsoon precipitation using ERA5/IMERG data to demonstrate interpretability and performance.
• Provide kernel-based summaries of nonlocal influence that can guide physically constrained parameterizations.

Proposed method

• Represent nonlocal dependencies via kernel-integrated features obtained by applying learnable kernels over horizontal space, height, and/or time.
• Approximate the nonlocal operator as a composition F ∘ 𝒦 where 𝒦 is the kernel integration step and F is a local nonlinear mapping.
• Discretize kernels on gridded predictor fields with normalization and validity masking to yield kernel weights k_i,n,m,r^(ℓ).
• Train models with a shared downstream neural network to ensure differences arise from nonlocal encoding.
• Explore a hierarchy: baseline fully flexible models, nonparametric kernels, and parametric kernels (Gaussian, mixture of Gaussians, top-hat, exponential).
• Evaluate on South Asian monsoon precipitation using ERA5 predictors (RH, θ_e, θ_e*) and local inputs (sensible/latent heat flux, land fraction).

Figure 1: Schematic of integration kernel learning. Learned kernels summarize predictor fields across horizontal space, height, and/or time into features, which are combined with local inputs and passed to a downstream nonlinear model to predict the local output

Experimental results

Research questions

• RQ1 How does architectural encoding of nonlocal information (baseline vs. kernel-based) affect predictive skill for monsoon precipitation?
• RQ2 What is the relative importance of horizontal, vertical, and temporal nonlocality for predicting precipitation in this regime?
• RQ3 Do kernel-integrated features provide interpretable, physically meaningful patterns of nonlocal influence?
• RQ4 How do nonparametric and parametric kernel models compare in accuracy and interpretability?
• RQ5 Can kernel-integrated features offer compact summaries useful for subsequent physical parameterizations?

Key findings

• Kernel models achieve near-baseline predictive performance while using far fewer trainable parameters than fully flexible baselines.
• Vertical nonlocality contributes most to predictive skill for the South Asian monsoon within the studied scales, with horizontal and temporal context being secondary.
• Nonparametric kernel models reach similar performance to full baselines at reduced input dimensionality (e.g., R^2 0.496 vs 0.528; MSE 0.582 vs 0.546).
• Parametric kernels (Gaussian, mixture of Gaussians, exponential, top-hat) provide additional regularization with only modest losses in skill (R^2 ≈ 0.481–0.488; MSE ≈ 0.592–0.599).
• Learned vertical kernels reveal predictor-specific structure across the lower troposphere, aligning with known physical controls on convection (e.g., RH weighting near 900–1000 hPa and 650–500 hPa; θ_e and θ_e* patterns).
• Kernels yield interpretable weighting patterns that can be used to constrain physically interpretable parameterizations and support symbolic regression for nonlocal mechanisms.

Figure 2: Test set R 2 (top) and MSE (bottom) for baseline (blue), nonparametric kernel (yellow), and parametric kernel (red) models, computed in standardized log1p-transformed precipitation space. Model labels indicate nonlocal dimensions, with subscript 0 denoting locality. Superscript $k$ denotes