Redefining Neural Operators in $d+1$ Dimensions

Haoze Song, Zhihao Li, Xiaobo Zhang, Zecheng Gan, Zhilu Lai, Wei Wang

Keywords: Partial Differential Equation, Neural Operator

TL;DR: A novel neural operator framework redefining the problem domain to $d+1$ dimensions is introduced. Within that, we develop the Schrödingerised Kernel Neural Operator employing $d+1$ dimensional kernel design to capture the underlying full evolutions.

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, $d=1, 2, 3...$), 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 Schrödingerisation 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 Schrödingerised 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 more than ten increasingly challenging benchmarks, 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.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

Submission Number: 2829