Redefining Neural Operators in $d+1$ Dimensions
This work addresses a bottleneck in neural operator design for PDE simulations, offering a novel framework that improves performance on complex systems, though it is incremental in building on existing quantum simulation methods.
The paper tackles the challenge of designing neural operators that fully capture system evolution by redefining them on a d+1 dimensional domain, resulting in the Schrödingerised Kernel Neural Operator (SKNO) that consistently outperforms baselines across ten benchmarks, including the 3D Rayleigh-Taylor instability, and demonstrates resolution-invariance.
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 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 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.