提升神经算子对低频信息的捕捉能力，解决PDE求解中的高频误差问题。
Deep Parallel Spectral Neural Operators for Solving Partial Differential Equations with Enhanced Low-Frequency Learning Capability
- 采用并行模块增强低频信息学习能力
- 通过卷积映射平滑处理，减少高频误差
- 在多个复杂PDE数据集上表现优异，具备分辨率不变性
设计通用人工智能（AI）求解偏微分方程（PDE）是科学与工程中的开放性难题。当前，数据驱动的求解器如神经算子已取得显著进展，但各类神经算子对低频信息的学习能力仍需提升。本文提出深度并行谱神经算子（DPNO），通过并行模块增强对低频信息的建模能力。由于截断系数的存在，非线性学习过程中部分高频信息会丢失，为此我们引入卷积映射进行平滑处理，有效降低高频误差。我们在多个具有挑战性的偏微分方程数据集上进行了实验，结果显示DPNO表现卓越。作为神经算子，DPNO还具备分辨率不变性，可适应不同网格分辨率。
原文摘要 · Abstract (English)
Designing universal artificial intelligence (AI) solver for partial differential equations (PDEs) is an open-ended problem and a significant challenge in science and engineering. Currently, data-driven solvers have achieved great success, such as neural operators. However, the ability of various neural operator solvers to learn low-frequency information still needs improvement. In this study, we propose a Deep Parallel Spectral Neural Operator (DPNO) to enhance the ability to learn low-frequency information. Our method enhances the neural operator's ability to learn low-frequency information through parallel modules. In addition, due to the presence of truncation coefficients, some high-frequency information is lost during the nonlinear learning process. We smooth this information through convolutional mappings, thereby reducing high-frequency errors. We selected several challenging partial differential equation datasets for experimentation, and DPNO performed exceptionally well. As a neural operator, DPNO also possesses the capability of resolution invariance.
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