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.

翻译：我们提出了一种在降维基中建模动力系统演化的计算技术，重点关注在高维非均匀网格上对部分观测偏微分方程进行建模这一具有挑战性的问题。我们针对数据驱动流形映射学习先前研究的局限性，聚焦于含噪声和有限数据，以面向实际应用中的数据采集场景。基于近期在模态空间与节点空间中对偏微分方程建模的研究，我们提出一种适用于仅能获取状态变量或计算域子集上含噪声有限数据的偏微分方程建模的神经网络结构。具体而言，空间网格点测量值通过学习的线性变换进行降维，随后在此降维基中学习动力学规律，最终再变换回节点空间。与先前节点空间学习的流形映射模型相比，该方法使神经网络的参数化规模大幅降低。这主要使得训练数据集规模得以减小，同时也能缩短训练时间。