The modeling of complicated time-evolving physical dynamics from partial observations is a long-standing challenge. Particularly, observations can be sparsely distributed in a seemingly random or unstructured manner, making it difficult to capture highly nonlinear features in a variety of scientific and engineering problems. However, existing data-driven approaches are often constrained by fixed spatial and temporal discretization. While some researchers attempt to achieve spatio-temporal continuity by designing novel strategies, they either overly rely on traditional numerical methods or fail to truly overcome the limitations imposed by discretization. To address these, we propose CoPS, a purely data-driven methods, to effectively model continuous physics simulation from partial observations. Specifically, we employ multiplicative filter network to fuse and encode spatial information with the corresponding observations. Then we customize geometric grids and use message-passing mechanism to map features from original spatial domain to the customized grids. Subsequently, CoPS models continuous-time dynamics by designing multi-scale graph ODEs, while introducing a Markov-based neural auto-correction module to assist and constrain the continuous extrapolations. Comprehensive experiments demonstrate that CoPS advances the state-of-the-art methods in space-time continuous modeling across various scenarios.

翻译：从部分观测数据中建模复杂的时间演化物理动力学是一个长期存在的挑战。特别地，观测数据可能以看似随机或无结构的方式稀疏分布，这使得在各种科学与工程问题中捕捉高度非线性特征变得困难。然而，现有的数据驱动方法通常受限于固定的空间和时间离散化。尽管一些研究者试图通过设计新颖策略来实现时空连续性，但他们要么过度依赖传统的数值方法，要么未能真正克服离散化带来的限制。为解决这些问题，我们提出了CoPS，一种纯粹的数据驱动方法，用于从部分观测数据中有效建模连续物理仿真。具体而言，我们采用乘法滤波器网络来融合和编码空间信息及其对应的观测数据。接着，我们定制几何网格，并利用消息传递机制将特征从原始空间域映射到定制的网格上。随后，CoPS通过设计多尺度图常微分方程来建模连续时间动力学，同时引入一个基于马尔可夫过程的神经自校正模块来辅助和约束连续外推。全面的实验表明，CoPS在各种场景下的时空连续建模方面均优于当前最先进的方法。