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

2026-06-06 · Source: cs.AI updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Scientific Machine Learning · Depth: Expert, extended

Summary

A new d+1 dimensional framework for neural operators (NOs) is introduced, drawing inspiration from the Schrödingerisation method in quantum simulation of Partial Differential Equations (PDEs). This framework redefines NOs on an expanded d+1 dimensional domain, addressing prior limitations in capturing system evolution within embedding spaces. The proposed Schrödingerised Kernel Neural Operator (SKNO) utilizes a d+1 dimensional evolving linear block, demonstrating significantly improved performance. Experiments confirm SKNO's leading results on benchmarks including 1D Heat, Advection, Burgers, 2D Darcy Flow, and Navier-Stokes equations (low viscosity: 1e-5). It also shows strong capabilities in zero-shot super-resolution, maintaining low relative L_2 error across resolutions from 128 to 8192, and operates efficiently on a single Nvidia GeForce RTX 4090 GPU.

Key takeaway

For machine learning engineers developing PDE solvers, you should consider adopting the d+1 dimensional neural operator framework to enhance model accuracy and generalization. This approach, particularly with the Schrödingerised Kernel Neural Operator (SKNO), offers improved performance on complex fluid dynamics and diffusion problems, and superior zero-shot super-resolution capabilities. Evaluate its d+1 dimensional linear blocks for better capturing system evolution compared to traditional d-dimensional methods.

Key insights

Neural operators redefined in d+1 dimensions using Schrödingerisation better capture system evolution and achieve superior performance.

Principles

• Linear PDEs can be transformed into d+1 dimensional Hamiltonian systems.
• Operator learning in embedding spaces benefits from d+1 dimensional evolution.
• Combining global and local propagators improves signal propagation.

Method

The SKNO method involves lifting d-dimensional input to a d+1 dimensional function, evolving it via d+1 dimensional kernel integral operators with residuals, and then recovering the d-dimensional solution.

In practice

• Implement d+1 dimensional kernel integral operators for PDE solving.
• Use linear layers for lifting and MLPs for recovering operators.
• Combine spectral convolution and differential operators for signal propagation.

Topics

• Neural Operators
• Partial Differential Equations
• Schrödingerisation
• d+1 Dimensional Modeling
• Kernel Integral Operators
• Super-resolution

Best for: AI Scientist, Machine Learning Engineer, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by cs.AI updates on arXiv.org.