提出GAOT模型，高效高精度求解任意域上的偏微分方程。
Geometry Aware Operator Transformer as an Efficient and Accurate Neural Surrogate for PDEs on Arbitrary Domains
- 基于多尺度图注意力与几何嵌入，构建可处理任意域的神经算子。
- 在多个三维工业流体数据集上达到当前最优性能，兼顾精度与效率。
- 适合需要快速高保真模拟的工程仿真场景，如飞机设计、湍流分析。
在任意域上准确高效地学习偏微分方程（PDE）解算子是工程与工业模拟的关键挑战。尽管已有多种算子学习算法，但高精度模型往往计算效率低，反之亦然。为此，本文提出几何感知算子变换器（GAOT），结合新型多尺度注意力图神经算子编码器与解码器、几何嵌入及（视觉）变换器处理器，将域信息与输入特征映射为稳定的PDE解近似。GAOT在多个实现层面的创新保障了计算效率与可扩展性。我们在涵盖多种PDE的大规模学习任务中验证其优势，尤其在三个大规模三维工业流体动力学（CFD）数据集上达到当前最优表现。
原文摘要 · Abstract (English)
The very challenging task of learning solution operators of PDEs on arbitrary domains accurately and efficiently is of vital importance to engineering and industrial simulations. Despite the existence of many operator learning algorithms to approximate such PDEs, we find that accurate models are not necessarily computationally efficient and vice versa. We address this issue by proposing a geometry aware operator transformer (GAOT) for learning PDEs on arbitrary domains. GAOT combines novel multiscale attentional graph neural operator encoders and decoders, together with geometry embeddings and (vision) transformer processors to accurately map information about the domain and the inputs into a robust approximation of the PDE solution. Multiple innovations in the implementation of GAOT also ensure computational efficiency and scalability. We demonstrate this significant gain in both accuracy and efficiency of GAOT over several baselines on a large number of learning tasks from a diverse set of PDEs, including achieving state of the art performance on three large scale three-dimensional industrial CFD datasets.
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