Redefining Neural Operators in $d+1$ Dimensions for Embedding Evolution

Neural Operators (NOs) have emerged as powerful tools for learning mappings between function spaces. Among them, the kernel integral operator has been widely used in universally approximating architectures. Following the original formulation, most advancements focus on designing better parameterizations for the kernel over the original physical domain (with $d$ spatial dimensions, $d\in{1,2,3,\ldots}$). In contrast, embedding evolution remains largely unexplored, which often drives models toward brute-force embedding lengthening to improve approximation, but at the cost of substantially increased computation. In this paper, we introduce an auxiliary dimension that explicitly models embedding evolution in operator form, thereby redefining the NO framework in $d+1$ dimensions (the original $d$ dimensions plus one auxiliary dimension). Under this formulation, we develop a Schr\"odingerised Kernel Neural Operator (SKNO), which leverages Fourier-based operators to model the $d+1$ dimensional evolution. Across more than ten increasingly challenging benchmarks, ranging from the 1D heat equation to the highly nonlinear 3D Rayleigh-Taylor instability, SKNO consistently outperforms other baselines. We further validate its resolution invariance under mixed-resolution training and super-resolution inference, and evaluate zero-shot generalization to unseen temporal regimes. In addition, we present a broader set of design choices for the lifting and recovery operators, demonstrating their impact on SKNO's predictive performance.