Partial differential equations (PDEs) are fundamental for modeling complex natural and physical phenomena. In many real-world applications, however, observational data are extremely sparse, which severely limits the applicability of both classical numerical solvers and existing neural approaches. While neural methods have shown promising results under moderately sparse observations, their inference efficiency at high resolutions is limited, and their accuracy degrades substantially in the extremely sparse regime. In this work, we propose the Di-BiLPS, a unified neural framework that effectively handle both forward and inverse PDE problems under extremely sparse observations. Di-BiLPS combines a variational autoencoder to compress high-dimensional inputs into a compact latent space, a latent diffusion module to model uncertainty, and contrastive learning to align representations. Operating entirely in this latent space, the framework achieves efficient inference while retaining flexible input-output mapping. In addition, we introduce a PDE-informed denoising algorithm based on a variance-preserving diffusion process, which further improves inference efficiency. Extensive experiments on multiple PDE benchmarks demonstrate that Di-BiLPS consistently achieves SOTA performance under extremely sparse inputs (as low as 3%), while substantially reducing computational cost. Moreover, Di-BiLPS enables zero-shot super-resolution, as it allows predictions over continuous spatial-temporal domains.

翻译：偏微分方程是建模复杂自然与物理现象的基础。然而在实际应用中，观测数据往往极度稀疏，严重限制了传统数值求解器及现有神经方法的适用性。尽管神经方法在适度稀疏观测条件下已展现出良好性能，但其在高分辨率下的推理效率受限，且在极端稀疏场景中精度显著下降。本文提出Di-BiLPS——一个统一的神经框架，能够在极度稀疏观测条件下有效处理正问题和反问题偏微分方程。该框架融合变分自编码器将高维输入压缩至紧凑潜在空间、潜在扩散模块建模不确定性，以及对比学习对齐表征。通过完全在潜在空间中运行，该框架实现了高效推理并保持灵活的输入-输出映射。此外，我们提出基于方差保持扩散过程的PDE-信息去噪算法，进一步提升了推理效率。在多个偏微分方程基准上的大量实验表明，Di-BiLPS在极度稀疏输入（低至3%）条件下持续取得最先进性能，同时大幅降低计算成本。更重要的是，Di-BiLPS支持零样本超分辨率，可在连续时空域进行预测。