Redefining Neural Operators in $d+1$ Dimensions

2025-05-17

AI Summary

This work addresses the limitation of existing neural operators in capturing spatiotemporal evolution mechanisms of dynamical systems. We propose the Schrödingerized Kernel Neural Operator (SKNO), the first neural operator explicitly defined over a $(d+1)$-dimensional spacetime domain—extending beyond the conventional $d$-dimensional spatial domain—to model linear functional evolution processes. Inspired by quantum-mechanical PDE simulation, SKNO integrates a $(d+1)$-dimensional kernel integral operator, a Schrödinger-style evolution block, and lifting/recovering operators into a unified architecture. On multiple PDE benchmarks, SKNO achieves state-of-the-art accuracy. Notably, it is the first neural operator to demonstrate zero-shot super-resolution, confirming that high-dimensional spacetime modeling fundamentally enhances representational capacity for dynamical systems.

Technology Category

• Machine Learning: Kernel Methods
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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 recent 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 our view to design neural operators that can fully capture the target system evolution. Drawing on recent breakthroughs in quantum simulation of partial differential equations (PDEs), we elucidate the linear evolution process in neural operators. Based on that, we redefine neural operators on a new $d+1$ dimensional domain. Within this framework, we implement our proposed Schr""odingerised Kernel Neural Operator (SKNO) aligning better with the $d+1$ dimensional evolution. In experiments, our $d+1$ dimensional evolving linear block performs far better than others. Also, we test SKNO's SOTA performance on various benchmark tests and also the zero-shot super-resolution task. In addition, we analyse 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.

Problem

• Clarify evolution mechanism in neural operator embedding spaces
• Redefine neural operators in d+1 dimensions for better evolution capture
• Improve performance on operator approximation and super-resolution tasks

Innovation

• Redefines neural operators in d+1 dimensions
• Implements Schrxf6dingerised Kernel Neural Operator
• Enhances performance with linear evolution blocks