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This paper introduces a kernel-based encoder-decoder framework for learning operators that effectively handles multi-input and multi-output scenarios in scientific machine learning. The authors demonstrate that the convergence rate of their method is determined by the most difficult approximation problem, rather than the total number of inputs and outputs, which is a significant insight for operator learning. Their specialized approach, KernelMO, achieves competitive predictive accuracy across various parametric partial differential equations while significantly reducing training and inference costs compared to traditional neural operator architectures.
Operators can be learned more efficiently with kernel methods that outperform deep learning models while maintaining high predictive accuracy.
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.
