IKNO: Infinite-order Kernel Neural Operators

TL;DR

IKNO introduces an infinite-order kernel neural operator that enhances expressivity and efficiency, achieving state-of-the-art results on various scientific computing benchmarks with large-scale data.

Contribution

The paper proposes the novel IKNO framework utilizing infinite-order kernel integrals and develops two efficient variants with superior performance.

Findings

IKNO achieves state-of-the-art accuracy on multiple benchmarks.

The method scales effectively to large point clouds.

Fast computation schemes enable efficient global information aggregation.

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.