arXiv:2608. 11831v1 Announce Type: new Abstract: Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning.
Machine-generated by The Flow from the publisher's headline and feed description — not written or checked by a human. The full article lives at arXiv Machine Learning.
The paper introduces the Frame Kernel Method, a multiscale operator learning approach for surrogate modeling of multiscale partial differential equations. It uses a novel multiscale kernel frame function approximation to cast the learning problem as one of estimating frame coefficients, enabling automatic multiscale decomposition of outputs. The authors provide interpolation proofs, error estimates, and demonstrate that the method outperforms popular neural operators on challenging PDE problems while offering a posteriori multiscale analysis.
The paper introduces Total Sensitivity Kernels (TSKs), a weighted ANOVA kernel framework that learns the importance of individual inputs and their interactions for approximating a multivariable black-box function from limited data. By selecting an RKHS where the target function has minimum norm, the authors derive a unique solution and prove consistency for finite-data interpolation. Numerical experiments show that adapting the kernel to the learned multivariable structure can significantly improve approximation accuracy compared to a standard product kernel.
arXiv:2407. 00809v4 Announce Type: replace Abstract: This paper introduces the Kernel Neural Operator (KNO), a provably convergent operator-learning architecture that utilizes compositions of deep kernel-based integral operators for function-space approximation of operators (maps from functions to functions).
arXiv:2507.07292v2 Announce Type: replace Abstract: We develop a new and general encode-approximate-reconstruct operator learning model that leverages learned neural representations of bases for inpu...
The paper studies operator learning on function spaces using encoder–decoder architectures. It shows that as input and output resolutions grow, the induced kernels converge to a limiting kernel, enabling regularity assumptions independent of resolution. The authors derive upper and lower bounds for regularized stochastic gradient descent, extend the analysis to neural networks via the limiting neural tangent kernel, and provide error bounds and complexity guarantees for various kernel and encoding constructions.
We study kernel-based operator learning in a two-stage sampling framework, where an offline kernel regression operator learns a discretized representation of the target operator from input-output pairs and an online kernel reconstruction operator recovers the output function from predicted observations. Our main theoretical contribution is an explicit budget allocation condition relating the number $N$ of training pairs, the number $n$ of input observations, and the output resolution $m$.