operator-learning kernel approximation

Existing operator learning methods for irregular geometric domains suffer from high memory consumption and poor geometric adaptability. Method: This paper proposes the Deep Integral Operator (DIO) framework, which employs learnable compactly supported kernel functions and a sparsity-aware parametrization scheme to jointly ensure smoothness and computational efficiency; it further incorporates adaptive numerical quadrature to achieve geometry-agnostic modeling, eliminating reliance on structured grids. Contribution/Results: On standard benchmarks, DIO achieves higher accuracy than state-of-the-art neural operators, with improved training and test accuracy. It reduces trainable parameters by over an order of magnitude, significantly enhancing memory efficiency, generalization capability, and geometric robustness.

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.

This work addresses the problem of learning operators between function spaces in two settings: (i) multi-operator learning—representing a family of operators parameterized by continuous variables using a single neural network; and (ii) multi-independent-operator learning—simultaneously modeling multiple heterogeneous operators. To this end, we propose two novel architectures: MNO and MONet. We establish the first general approximation theory for multi-operator learning and derive, for the first time, explicit scaling laws relating network size to approximation accuracy. Our method integrates deep neural networks, function-space approximation theory, and a PDE-informed multi-task learning framework, augmented with a complexity-balancing mechanism. Evaluated on parametric PDE benchmarks, our approach achieves significant improvements in both operator approximation accuracy and computational efficiency, bridging theoretical guarantees with empirical performance.

Neural operators—deep models mapping between function spaces rather than vector spaces—lack open-source, discretization-agnostic implementations with theoretical convergence guarantees. To address this gap, we introduce NeuralOperator, the first modular and extensible Python library for neural operators built on PyTorch. It systematically supports state-of-the-art architectures—including Fourier Neural Operators (FNO) and Multipole Graph Neural Operators (MGNO)—and enables training and inference with functional inputs/outputs under diverse discretizations while rigorously ensuring discretization consistency and convergence. Through a unified API, comprehensive test coverage, and an end-to-end deployment toolchain, NeuralOperator significantly lowers the barrier to adopting neural operators in scientific computing tasks such as partial differential equation solving. The library bridges cutting-edge representational capacity with production-grade engineering robustness, making it both research-ready and deployable in real-world applications.

This study addresses the limited interpretability of conventional neural operators and the difficulty of balancing accuracy with efficiency in acoustic propagation over unbounded domains. We propose an interpretable neural operator framework based on kernel functions that explicitly embeds kernels to match the structure of boundary integral expansions. Three complementary kernel types—data-driven, physics-informed, and hybrid—are designed to enable unsupervised training. By integrating techniques such as radial basis parameterization, analytical fundamental solution embedding, low-rank correction, and linear principal part decomposition, the framework effectively fuses physical priors with data fitting. Benchmark evaluations demonstrate that the proposed method surpasses DeepONet in accuracy with fewer parameters while significantly reducing inference costs, thereby providing an efficient solving paradigm for exterior acoustic propagation problems.

本文提出了一种可分离神经算子方法，用于函数对函数回归模型中一般回归算子的估计，解决了传统线性或非线性扩展模型的局限。

Existing neural operators lack reliability and theoretical guarantees when handling out-of-distribution input functions. This work proposes an extended framework grounded in reproducing kernel Hilbert spaces (RKHS), leveraging kernel approximation techniques to achieve robust approximation of both out-of-distribution functions and their derivatives. The key innovation lies in establishing a theoretical connection between kernel selection and Sobolev eigenfunction spaces, thereby providing predictable guarantees on generalization error and derivative accuracy for neural operators. When applied to solving elliptic partial differential equations—particularly on manifolds represented as point clouds—the method demonstrates significantly enhanced geometric awareness, improved extrapolation accuracy, and greater computational efficiency.

This study investigates the theoretical foundations of operator learning, with a focus on convergence rates and fundamental statistical limits. By integrating tools from statistical learning theory, approximation theory, and the framework of holomorphic operators, the work establishes a unified error analysis framework to systematically derive generalization error bounds for empirical risk minimization. Under generalized regularity conditions, it further establishes minimax-optimal statistical lower bounds, revealing an intrinsic trade-off between sample complexity and model approximation capacity. The analysis delineates current theoretical boundaries in operator learning and identifies several key open problems, offering new perspectives to guide future theoretical advances in the field.

This work investigates the generalization performance of random feature methods under operator-valued kernels, with particular emphasis on the misspecified setting where the target function lies outside the associated reproducing kernel Hilbert space (RKHS). To this end, the authors develop a unified spectral regularization framework that encompasses both neural operators and neural networks within the neural tangent kernel (NTK) perspective for theoretical analysis. They extend random feature methods to operator-valued kernels for the first time and establish minimax optimal convergence rates in both well-specified and misspecified regimes. Key contributions include deriving optimal learning rates, quantifying the number of neurons required to achieve a prescribed accuracy, and strengthening the theoretical foundations of operator-valued kernel methods.

This work addresses the lack of quantitative generalization guarantees in existing neural operator methods for multi-operator learning, particularly concerning unseen operator instances, input functions, and evaluation points. Focusing on multi-task and multi-operator settings under hierarchical sampling, the study establishes the first explicit generalization error bound—based on metric entropy—for models built upon the Multiple Neural Operator (MNO) architecture. By integrating linear combinations of products of deep ReLU subnetworks with covering number analysis and approximation theory, the authors derive an approximation–estimation trade-off expression for the expected test error. This bound precisely characterizes how sampling budgets at the three levels—operators, inputs, and evaluation points—affect generalization performance, and yields explicit sample complexity and learning rates under operator sampling complexity constraints.