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.

翻译：在任意域上准确高效地学习偏微分方程解算子的任务极具挑战性，对工程与工业仿真至关重要。尽管存在多种算子学习算法来逼近此类偏微分方程，我们发现精确模型未必计算高效，反之亦然。为此，我们提出一种用于任意域上偏微分方程学习的几何感知算子Transformer（GAOT）。GAOT结合了新颖的多尺度注意力图神经算子编码器与解码器、几何嵌入以及（视觉）Transformer处理器，能够将域与输入信息准确映射为偏微分方程解的鲁棒逼近。GAOT实现中的多项创新同时确保了计算效率与可扩展性。我们在来自多种偏微分方程的大量学习任务上，通过多个基线模型验证了GAOT在精度与效率上的显著提升，包括在三个大规模三维工业计算流体力学数据集上达到最先进性能。