Real-world scientific applications frequently encounter incomplete observational data due to sensor limitations, geographic constraints, or measurement costs. Although neural operators significantly advanced PDE solving in terms of computational efficiency and accuracy, their underlying assumption of fully-observed spatial inputs severely restricts applicability in real-world applications. We introduce the first systematic framework for learning neural operators from partial observation. We identify and formalize two fundamental obstacles: (i) the supervision gap in unobserved regions that prevents effective learning of physical correlations, and (ii) the dynamic spatial mismatch between incomplete inputs and complete solution fields. Specifically, our proposed Latent Autoregressive Neural Operator(LANO) introduces two novel components designed explicitly to address the core difficulties of partial observations: (i) a mask-to-predict training strategy that creates artificial supervision by strategically masking observed regions, and (ii) a Physics-Aware Latent Propagator that reconstructs solutions through boundary-first autoregressive generation in latent space. Additionally, we develop POBench-PDE, a dedicated and comprehensive benchmark designed specifically for evaluating neural operators under partial observation conditions across three PDE-governed tasks. LANO achieves state-of-the-art performance with 18--69$\%$ relative L2 error reduction across all benchmarks under patch-wise missingness with less than 50$\%$ missing rate, including real-world climate prediction. Our approach effectively addresses practical scenarios involving up to 75$\%$ missing rate, to some extent bridging the existing gap between idealized research settings and the complexities of real-world scientific computing.

翻译：现实世界的科学应用由于传感器限制、地理约束或测量成本，经常遇到不完整的观测数据。尽管神经算子在计算效率和精度方面显著推进了偏微分方程求解，但其完全观测空间输入的基本假设严重限制了在实际应用中的适用性。我们提出了首个从部分观测中学习神经算子的系统框架。我们识别并形式化了两个基本障碍：(i) 未观测区域的监督缺失阻碍了物理关联的有效学习，以及 (ii) 不完整输入与完整解场之间的动态空间失配。具体而言，我们提出的潜在自回归神经算子引入了两个新颖组件，专门设计用于解决部分观测的核心难点：(i) 一种掩码预测训练策略，通过策略性地掩蔽观测区域来创建人工监督，以及 (ii) 一个物理感知潜在传播器，通过在潜在空间中边界优先的自回归生成来重建解。此外，我们开发了POBench-PDE，这是一个专门且全面的基准测试集，旨在评估神经算子在三个偏微分方程控制任务下的部分观测条件性能。在缺失率低于50%的块状缺失情况下，LANO在所有基准测试中实现了最先进的性能，相对L2误差降低了18-69%，包括真实世界的气候预测。我们的方法有效解决了缺失率高达75%的实际场景，在一定程度上弥合了理想化研究设置与现实世界科学计算复杂性之间的现有差距。