Kernel Methods for Learning Operators with Multiple Inputs and Outputs

Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning, particularly when dealing with parametric partial differential equations, inverse problems, and multi-fidelity simulations. This paper introduces a general kernel-based encoder-decoder framework for multi-input and multi-output operator learning. Instead of relying exclusively on deep neural networks, the approach separates observation, representation, learning, and reconstruction. Inputs and outputs from potentially distinct function spaces are first encoded into latent spaces, a surrogate is learned between these representations, and the final prediction is decoded back to the target space.

The authors establish a rigorous mathematical foundation through an encoder-decoder error decomposition. This separates the total approximation error into representation error, learning error, and data consistency error. A key theoretical insight is that for multi-input and multi-output operators, the overall convergence rate is governed by the most challenging constituent approximation problem rather than deteriorating with the total problem dimension. This ensures that the framework remains tractable even as the number of input and output function spaces increases.

To instantiate this framework for multiple operator learning, the authors introduce KernelMO, a family of kernel methods offering complementary formulations. The operator-valued formulation (KernelMO-OV) directly approximates maps from parameters to operators, while the product-space formulation (KernelMO-PS) approximates equivalent joint mappings. These methods yield closed-form solutions for training and inference via kernel ridge regression and interpolation techniques, bypassing the iterative training loops typical of neural networks.