Sumudu Neural Operator for ODEs and PDEs

TL;DR

The paper introduces the Sumudu Neural Operator (SNO), a transform-based neural operator leveraging the Sumudu transform to learn mappings between function spaces for ODEs and PDEs. SNO integrates a lifting step, Sumudu-transform-based kernel parameterization, and a projection step, with polynomial regression enabling efficient forward transforms via a Vandermonde-based approach. Empirical results show SNO achieving state-of-the-art or competitive accuracy on several PDE tasks (notably Euler-Bernoulli Beam and Diffusion) and offering substantial runtime advantages over Laplace Neural Operator on diffusion-type problems, while lagging on several ODE tasks. The work suggests the Sumudu transform as a promising design choice for neural operators in certain problem classes, particularly those with transient and damped dynamics, and outlines future directions for improving accuracy and time-efficiency analyses.

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.

Figures (24)

• Figure 1: Diagram of the Sumudu Neural Operator (SNO) architecture. Given an input function $f(t)$, we1 lift the input function to a higher dimension using a shallow, linear neural network, $L$, (2) we apply a Sumudu transform, and parametrize weights in the Sumudu space, while applying a local linear transform $W$ as a bias, (3) we normalize results, add our activation function, and project back into the original dimension through a second neural network, $P$, to get our output.
• Figure 2:
• Figure 3: PDE ground truths and pointwise prediction errors between FNO, LNO, and SNO.
• Figure 4: Heatmap comparison of baseline and super-resolved predictions on the Duffing system with zero forcing.
• Figure 5: Heatmap comparison of baseline and super-resolved predictions on the Duffing system with moderate forcing.