This work presents a physics-informed deep learning-based super-resolution framework to enhance the spatio-temporal resolution of the solution of time-dependent partial differential equations (PDE). Prior works on deep learning-based super-resolution models have shown promise in accelerating engineering design by reducing the computational expense of traditional numerical schemes. However, these models heavily rely on the availability of high-resolution (HR) labeled data needed during training. In this work, we propose a physics-informed deep learning-based framework to enhance the spatial and temporal resolution of coarse-scale (both in space and time) PDE solutions without requiring any HR data. The framework consists of two trainable modules independently super-resolving the PDE solution, first in spatial and then in temporal direction. The physics based losses are implemented in a novel way to ensure tight coupling between the spatio-temporally refined outputs at different times and improve framework accuracy. We analyze the capability of the developed framework by investigating its performance on an elastodynamics problem. It is observed that the proposed framework can successfully super-resolve (both in space and time) the low-resolution PDE solutions while satisfying physics-based constraints and yielding high accuracy. Furthermore, the analysis and obtained speed-up show that the proposed framework is well-suited for integration with traditional numerical methods to reduce computational complexity during engineering design.

翻译：这项工作提出了一个基于物理学的深层次学习超分辨率框架,以加强时间和空间上对基于时间的局部偏差方程解决方案的时空解决方案。关于深层次基于学习的超分辨率模型的先前工作显示,通过降低传统数字方法的计算成本,有望加快工程设计;然而,这些模型在很大程度上依赖培训期间所需的高分辨率(HR)标签数据。在这项工作中,我们提议了一个基于物理的深层次学习框架,以加强(在空间和时间上)粗尺度(空间和时间上)PDE解决方案的空间和时间解决方案。这个框架包括两个可独立独立地超解PDE解决方案的模块,首先在空间上,然后在时间方向上。基于物理的损失以一种新颖的方式实施,以确保在不同时间将高分辨率(HR)改良产出紧密结合,并提高框架的准确性。我们通过对先进框架的能力进行分析,通过调查其在弹性动力学问题上的绩效。我们发现,拟议的框架可以成功地(在空间和时间上)超解解的超分辨率模块,首先在空间和时间上独立地超分辨率解决PDE解决方案。基于低分辨率和低分辨率的计算方法在计算方法上展示了低分辨率分析。