Kernel Methods for Learning Operators with Multiple Inputs and Outputs

2026-08-12

AI Summary

This work addresses the challenge of learning infinite-dimensional, multi-input multi-output operator mappings in scientific machine learning by proposing a kernel-based encoder-decoder framework—the first systematic extension of kernel methods to this setting. By leveraging operator-valued kernels and product-space kernels, the approach enables closed-form training and inference while decoupling observation, representation, learning, and reconstruction stages, thereby ensuring both mathematical interpretability and computational efficiency. Theoretical analysis reveals that the overall convergence rate is governed by the most difficult subproblem rather than the total dimensionality. The resulting KernelMO family of methods achieves state-of-the-art or competitive accuracy across five benchmark parametric partial differential equation tasks while significantly reducing both training and inference costs compared to existing neural operator models.

Abstract

Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning. We introduce a general kernel-based encoder-decoder framework for operator learning that separates observation, representation, learning, and reconstruction. We develop this framework for multi-input, multi-output operator learning, where operators map between products of potentially distinct function spaces. Our approximation theory shows that, although the number of inputs and outputs can increase, the convergence rate is governed by the most challenging constituent approximation problem rather than the overall problem dimension. The framework leads to practical kernel methods with closed-form training and inference, combining mathematical tractability with computational efficiency. We further specialize the approach to multiple operator learning by introducing KernelMO, a family of kernel methods with complementary operator-valued and product-space formulations. Across five families of parametric partial differential equations, the proposed methods achieve competitive or state-of-the-art predictive accuracy while reducing training and inference costs relative to neural operator architectures and deep learning based models, offering an efficient and lightweight alternative.

Problem

• operator learning
• multi-input multi-output
• kernel methods
• infinite-dimensional mappings
• function spaces

Innovation

• kernel methods
• operator learning
• multi-input multi-output
• approximation theory
• KernelMO