用积分变换设计新神经算子，解微分方程更准更快
Sumudu Neural Operator for ODEs and PDEs
- 基于Sumudu变换分解输入，将问题映射到变换空间中学习
- 在多个偏微分方程上优于FNO，部分任务误差最低
- 支持零样本超分辨率，适合需要高精度解的科研场景
我们提出基于Sumudu变换的神经算子（SNO）。利用变换对的多项式展开关系，将输入空间分解为系数，并映射至Sumudu空间，在其中参数化神经算子。在常微分方程（杜芬振子、洛伦兹系统、受驱摆）和偏微分方程（欧拉-伯努利梁、伯格方程、扩散方程、扩散-反应方程、布鲁塞尔振子）上进行评估。SNO在偏微分方程任务上表现优于FNO，与LNO相比具有竞争力，尤其在欧拉-伯努利梁和扩散方程上取得最低误差。此外，通过零样本超分辨率实验，验证了模型从低质量输入生成高质量解的能力。初步结果表明Sumudu变换在神经算子设计中具有潜力，尤其适用于特定类别的偏微分方程。
原文摘要 · Abstract (English)
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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