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

Neural Operators (NOs) are powerful architectures for learning mappings between function spaces. While most advances focus on refining kernel parameterizations over the $d$-dimensional physical domain, the evolution of lifted embeddings remains underexplored, which often drives models toward computationally expensive embedding-scaling designs to improve approximation. In this paper, we introduce an auxiliary function dimension that models embedding evolution in operator form, thereby reformulating the NO pipeline in $d+1$ dimensions. We instantiate this framework via Fourier-based operators acting jointly on the physical and auxiliary domains, yielding a basis-diversified auxiliary evolution module as an alternative to brute-force embedding scaling. Across more than ten increasingly challenging benchmarks, ranging from the 1D heat equation to the highly nonlinear 3D Rayleigh-Taylor instability, our model consistently achieves the lowest relative $L_2$ error among the evaluated baselines. Crucially, this advantage is empirically supported by (1) controlled budget-aware comparisons against scaled and ablated baselines; (2) robustness under mixed-resolution training and super-resolution inference; and (3) zero-shot generalization to unseen temporal regimes. In addition, we present a broader set of design choices for lifting and recovery operators, demonstrating their impact on our model's predictive performance.