Sumudu Neural Operator for ODEs and PDEs
This work addresses solving differential equations for scientific computing, but it appears incremental as it builds on existing neural operator frameworks with a new transform-based approach.
The authors tackled the problem of solving ODEs and PDEs by introducing the Sumudu Neural Operator (SNO), which leverages the Sumudu Transform to parameterize neural operators, achieving superior performance to FNO on PDEs and competitive accuracy with LNO, including the lowest error on the Euler-Bernoulli Beam and Diffusion Equation.
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.