We present a computational technique for modeling the evolution of dynamical systems in a reduced basis, with a focus on the challenging problem of modeling partially-observed partial differential equations (PDEs) on high-dimensional non-uniform grids. We address limitations of previous work on data-driven flow map learning in the sense that we focus on noisy and limited data to move toward data collection scenarios in real-world applications. Leveraging recent work on modeling PDEs in modal and nodal spaces, we present a neural network structure that is suitable for PDE modeling with noisy and limited data available only on a subset of the state variables or computational domain. In particular, spatial grid-point measurements are reduced using a learned linear transformation, after which the dynamics are learned in this reduced basis before being transformed back out to the nodal space. This approach yields a drastically reduced parameterization of the neural network compared with previous flow map models for nodal space learning. This primarily allows for smaller training data sets, but also enables reduced training times.

我们提出了一种用于在简化基础上模拟动态系统演化的计算技术，重点研究了在高维非均匀网格上对部分观测到的偏微分方程（PDEs）进行建模的难题。

不完整、受限和嘈杂数据中的偏微分方程主成分流图学习