Solving partial differential equations (PDEs) by learning the solution operators has emerged as an attractive alternative to traditional numerical methods. However, implementing such architectures presents two main challenges: flexibility in handling irregular and arbitrary input and output formats and scalability to large discretizations. Most existing architectures are limited by their desired structure or infeasible to scale large inputs and outputs. To address these issues, we introduce an attention-based model called an inducing-point operator transformer (IPOT). Inspired by inducing points methods, IPOT is designed to handle any input function and output query while capturing global interactions in a computationally efficient way. By detaching the inputs/outputs discretizations from the processor with a smaller latent bottleneck, IPOT offers flexibility in processing arbitrary discretizations and scales linearly with the size of inputs/outputs. Our experimental results demonstrate that IPOT achieves strong performances with manageable computational complexity on an extensive range of PDE benchmarks and real-world weather forecasting scenarios, compared to state-of-the-art methods.

翻译：通过学习解算子来求解偏微分方程（PDE）已成为传统数值方法之外的一种富有吸引力的替代方案。然而，实现此类架构面临两大挑战：处理不规则、任意输入输出格式的灵活性，以及适应大规模离散化的可扩展性。现有大多数架构受限于其预设结构，或难以处理大规模输入输出。为解决这些问题，我们提出一种基于注意力的模型——诱导点算子Transformer（IPOT）。受诱导点方法启发，IPOT旨在以计算高效的方式处理任意输入函数和输出查询，同时捕获全局交互。通过使用较小的隐层瓶颈将输入/输出离散化与处理器分离，IPOT具备处理任意离散化的灵活性，且计算复杂度与输入/输出规模呈线性关系。实验结果表明，与最先进方法相比，IPOT在广泛的PDE基准测试和真实天气预报场景中，以可控的计算复杂度实现了强劲性能。