📝 SummarXiv

Abstract

Neural Operators have emerged as powerful tools for learning mappings between function spaces. Among them, the kernel integral operator has been widely validated on universally approximating various operators. Although many advancements following this definition have developed effective modules to better approximate the kernel function defined on the original domain (with $d$ dimensions, $d=1, 2, 3\dots$), the unclarified evolving mechanism in the embedding spaces blocks researchers' view to design neural operators that can fully capture the target system evolution. Drawing on the Schr\"odingerisation method in quantum simulations of partial differential equations (PDEs), we elucidate the linear evolution mechanism in neural operators. Based on that, we redefine neural operators on a new $d+1$ dimensional domain. Within this framework, we implement a Schr\"odingerised Kernel Neural Operator (SKNO) aligning better with the $d+1$ dimensional evolution. In experiments, the $d+1$ dimensional evolving designs in our SKNO consistently outperform other baselines across ten benchmarks of increasing difficulty, ranging from the simple 1D heat equation to the highly nonlinear 3D Rayleigh-Taylor instability. We also validate the resolution-invariance of SKNO on mixing-resolution training and zero-shot super-resolution tasks. In addition, we show the impact of different lifting and recovering operators on the prediction within the redefined NO framework, reflecting the alignment between our model and the underlying $d+1$ dimensional evolution.