Sumudu Neural Operator for ODEs and PDEs

Benjamin Zelenskiy, Saibilila Abudukelimu, George Flint, Kevin Zhu, Sunishchal Dev

Keywords: neural operators, Sumudu transform, polynomial regression, operator learning, ODEs, PDEs, deep learning, Fourier Neural Operator, Laplace Neural Operator

TL;DR: SNO leverages the Sumudu transform for efficient operator learning, outperforming FNO and approaching LNO accuracy on ODE and PDE benchmarks.

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.

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Submission Number: 60