IKNO: Infinite-order Kernel Neural Operators

2026-05-21

AI Summary

This work addresses the limited expressivity of existing neural operators, which typically rely on first-order kernel integral approximations. To overcome this limitation, the authors introduce, for the first time, an infinite-order kernel integral formulation for constructing neural operators, proposing the Infinite-order Kernel Neural Operator (IKNO) with two variants: IKNO-Vanilla and IKNO-TP. The framework leverages Kronecker feature decomposition, tensor product operators, axis-wise resolvent representations, and an efficient computational scheme to achieve a closed-form finite approximation. This approach substantially enhances global information aggregation capabilities, achieving state-of-the-art accuracy across diverse time-varying and time-invariant benchmarks as well as large-scale industrial datasets. The method significantly outperforms current approaches and scales efficiently to extremely large point cloud scenarios.

Abstract

Neural operators have achieved significant success in modern scientific computing due to their flexibility and strong generalization capabilities. Existing models, however, primarily rely on first-order kernel integral approximations, which severely limit their expressivity. To address this, we propose the Infinite-order Kernel Neural Operator (IKNO), which constructs neural operators via infinite-order kernel integrals and admits an elegant closed-form finite approximation. We develop two complementary infinite-order neural operator constructions: IKNO-Vanilla, which applies the full-kernel resolvent on the product grid via Kronecker eigendecomposition, and IKNO-TP, an alternative tensor-product operator that composes per-axis resolvents. Furthermore, we develop fast computation schemes for both variants of IKNO, which achieve outstanding global information aggregation while maintaining high computational efficiency. Empirically, we evaluate our IKNO on both time-dependent and time-independent benchmarks with arbitrary input shapes, including large-scale industrial datasets. Extensive experiments demonstrate that the IKNO method consistently achieves the SOTA accuracy with significant improvements on nearly all benchmark datasets while maintaining scalability to very large point clouds.

Problem

• neural operators
• kernel integral
• expressivity
• infinite-order approximation
• scientific computing

Innovation

• Infinite-order kernel
• Neural operators
• Resolvent
• Tensor-product
• Global information aggregation