发表机构
Zhejiang University（浙江大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出PDE-JEPA框架，通过预测性表示预训练、几何投影对齐和物理结构潜预测器，在九个PDE基准上实现分布内33.4%和分布外51.4%的平均性能提升。
AI 中文摘要
物理轨迹所包含的信息远不止系统的快照：它们还揭示了系统状态在支配条件下的演化方式。然而，针对参数化偏微分方程（PDE）的表示学习在很大程度上依赖于基于重建的目标，这些目标强调恢复观测到的物理场。在本文中，我们研究将预测性表示预训练作为基于重建的学习的一种替代方案。我们发现预测性表示保留了丰富的物理信息，但仅凭这一优势并不能确保准确的场演化。基于这些观察，我们提出了PDE-JEPA用于参数化PDE动力学。具体来说，我们首先使用掩码潜变量预测训练一个编码器，以捕捉PDE动力学的潜在规律。为了将预训练表示显式地适应到更符合动力学的状态空间，我们随后引入了一个几何投影器，将潜轨迹几何与物理场的演化几何对齐。最后，在此几何对齐的潜空间基础上，我们进一步开发了一个物理结构的潜变量预测器，将动力学分解为参数无关的演化和参数相关的响应两个组成部分。在九个广泛使用的PDE基准上的大量实验表明，我们的框架在分布内平均比现有最先进方法高出33.4%，而在外推至未见过的支配参数时，平均改进达到51.4%。项目页面可在此处访问。
英文摘要
Physical trajectories contain more than snapshots of a system: they also reveal how its states evolve under governing conditions. However, representation learning for parametric partial differential equations (PDEs) has largely relied on reconstruction-based objectives that emphasize recovering observed physical fields. In this paper, we investigate predictive representation pretraining as an alternative to reconstruction-based learning. We find that predictive representations preserve rich physical information, yet this advantage alone does not ensure accurate field evolution. Based on these observations, we introduce PDE-JEPA for parametric PDE dynamics. Specifically, we first train an encoder using a masked-latent prediction to capture the underlying regularities of PDE dynamics. To explicitly adapt the pretrained representation toward a more dynamics-aligned state space, we then introduce a geometry projector that aligns latent trajectory geometry with the evolution geometry of physical fields. Finally, building on this geometry-aligned latent space, we further develop a physics-structured latent predictor that decomposes the dynamics into parameter-independent evolution and parameter-dependent response components. Extensive experiments on nine widely used PDE benchmarks demonstrate that our framework outperforms existing state-of-the-art methods by an average of 33.4\% in-distribution, while achieving an average improvement of 51.4\% when extrapolating to unseen governing parameters. The project page is available \href{https://tanpig-x.github.io/PDE-JEPA/}{here}.
