This work introduces a general kernel-based encoder-decoder framework for operator learning that separates observation, representation, learning, and reconstruction, and develops this framework for multi-input, multi-output operator learning, where operators map between products of potentially distinct function spaces.
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.
Convolution integrals widely exist in applications, and to enable fast and accurate computations, this paper introduces two general multi-stage neural operator learning frameworks. The first, Deep Collocation Neural Operator (DCNO), is a supervised approach that iteratively refines the operator approximation by learning residuals from input-output data pairs. The second, Deep Galerkin Neural Operator (DGNO), is an unsupervised framework applicable when the target operator can be represented by a PDE, leveraging the weak form of the PDE residual for training. Both methods progressively construct basis operators through multiple training stages to enrich the approximation space, leading to significantly improved accuracy over standard one-shot operator learning. We provide theoretical analysis for their approximation capabilities and implement them for learning convolutions. Extensive numerical experiments demonstrate that both DCNO and DGNO achieve high accuracy, approaching machine precision under single float for convolution problems, and offer substantial efficiency gains for numerous queries or parametric variations compared to traditional solvers. We also extend these frameworks to handle multi-input operator learning scenarios involving variations in both the density and kernel of a convolution.
Zhiping Mao, Zhenye Wen, Yong Zhang et al.· 0 citations
Developing nonlinear models that are both expressive and computationally efficient remains a challenge in machine learning and nonlinear system identification. Tensor network kernel machines (TNKM) address this challenge by combining nonlinear feature representations with compact low-rank tensor-network parameterizations. However, practical and extensible software frameworks for developing TNKM models remain limited. In this work, we introduce"tnkm", an open-source Python library for constructing and training TNKM models using JAX. The library provides a unified interface for combining different feature maps, tensor-network architectures, and optimization strategies, including alternating least squares and gradient-based methods. We demonstrate the capabilities of"tnkm"on nonlinear benchmark problems, showing that the implemented models achieve competitive prediction accuracy while retaining compact parameterizations and efficient training. The proposed framework facilitates reproducible development and application of tensor-network-based learning methods.
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 $m$. The condition is derived from a coupled error analysis that interprets the surrogate as a reconstruction from approximate data. This yields a decomposition of the total error into reconstruction and learning contributions that can be analyzed independently. As a consequence, we obtain quantitative scaling laws describing how $N$, $n$, and $m$ must be coupled to guarantee convergence and to balance offline learning and online reconstruction errors. The resulting estimates extend previous analyses of kernel-based operator learning. We further introduce a physics-informed extension that incorporates knowledge of the underlying PDE at evaluation time. Rather than encoding constraints directly into the kernel, we augment the online reconstruction step by penalizing PDE residuals at collocation points. The method requires no retraining for new inputs. Numerical experiments illustrate the theoretical findings and demonstrate the effectiveness of the proposed physics-informed reconstruction strategy.
Operator-theoretic generalization bounds for deep multi-output function classes are developed by representing network layers as Koopman composition operators on vector-valued reproducing kernel Hilbert spaces and derive Rademacher complexity bounds for invertible and width-expanding injective architectures.
Mahdi Mohammadigohari, Thomas Borsani, G. D. Fatta· 0 citations
The performance of Support Vector Machines (SVMs) critically depends on the kernel function choice, which enables implicit mapping of data into high-dimensional feature spaces. While classical kernels like Radial Basis Function (RBF) remain popular, orthogonal polynomial kernels offer mathematically interpretable alternatives that can incorporate structured prior knowledge. This work extends the orthogonal polynomial kernel paradigm by introducing a novel family based on discrete $q$-Hermite I polynomials, a class of $q$-orthogonal polynomials that generalize classical Hermite polynomials through a deformation parameter $q$. We formally define the q-Hermite kernel and establish its validity under Mercer's theorem. The kernel's inherent boundedness properties naturally prevent annihilation and explosion effects without requiring explicit scaling mechanisms. Extensive experiments across 20 benchmark datasets demonstrate that the proposed kernel achieves competitive performance compared to both classical kernels and other orthogonal polynomial kernels, while offering advantages in numerical stability and computational simplicity. Our results confirm that $q$-orthogonal polynomials constitute a promising direction for kernel design, bridging mathematical elegance with practical machine learning applications, that provides conceptual and algorithmic resources that may be further extended to emerging quantum computing paradigms. To facilitate full reproducibility, we provide the complete implementation and experimental pipeline in an open-access GitHub repository at https://github.com/Kokechacho/SVMs-QSVMs.
Álvaro Sánchez-Paniagua Ríos, Juan P. Llerena, Alberto Lastra et al.· 0 citations
The universal consistency of PIKS is established for linear differential constraints, proving that for universal kernels (such as Gaussian or Mat\'ern), the estimator asymptotically learns the target while satisfying physical constraints.
Joachim Bona-Pellissier, Giacomo Meanti, Matteo Santacesaria et al.· 0 citations