Sumudu Neural Operator for ODEs and PDEs

TL;DR

The paper introduces the Sumudu Neural Operator (SNO), leveraging the Sumudu Transform to improve neural operator performance on ODEs and PDEs, demonstrating superior accuracy and zero-shot super-resolution capabilities.

Contribution

The paper presents the novel Sumudu Neural Operator, utilizing the Sumudu Transform for neural operator design, achieving improved PDE performance and zero-shot super-resolution.

Findings

SNO outperforms FNO on PDEs

SNO has competitive accuracy with LNO

SNO enables zero-shot super-resolution

Abstract

We introduce the Sumudu Neural Operator (SNO), a neural operator rooted in the properties of the Sumudu Transform. We leverage the relationship between the polynomial expansions of transform pairs to decompose the input space as coefficients, which are then transformed into the Sumudu Space, where the neural operator is parameterized. We evaluate the operator in ODEs (Duffing Oscillator, Lorenz System, and Driven Pendulum) and PDEs (Euler-Bernoulli Beam, Burger's Equation, Diffusion, Diffusion-Reaction, and Brusselator). SNO achieves superior performance to FNO on PDEs and demonstrates competitive accuracy with LNO on several PDE tasks, including the lowest error on the Euler-Bernoulli Beam and Diffusion Equation. Additionally, we apply zero-shot super-resolution to the PDE tasks to observe the model's capability of obtaining higher quality data from low-quality samples. These preliminary findings suggest promise for the Sumudu Transform as a neural operator design, particularly for certain classes of PDEs.