High-fidelity simulation of complex physical systems is exorbitantly expensive and inaccessible across spatiotemporal scales. Recently, there has been an increasing interest in leveraging deep learning to augment scientific data based on the coarse-grained simulations, which is of cheap computational expense and retains satisfactory solution accuracy. However, the major existing work focuses on data-driven approaches which rely on rich training datasets and lack sufficient physical constraints. To this end, we propose a novel and efficient spatiotemporal super-resolution framework via physics-informed learning, inspired by the independence between temporal and spatial derivatives in partial differential equations (PDEs). The general principle is to leverage the temporal interpolation for flow estimation, and then introduce convolutional-recurrent neural networks for learning temporal refinement. Furthermore, we employ the stacked residual blocks with wide activation and sub-pixel layers with pixelshuffle for spatial reconstruction, where feature extraction is conducted in a low-resolution latent space. Moreover, we consider hard imposition of boundary conditions in the network to improve reconstruction accuracy. Results demonstrate the superior effectiveness and efficiency of the proposed method compared with baseline algorithms through extensive numerical experiments.

From a numerical perspective, resolving the wide range of spatiotemporal scales within such physical systems is challenging since extremely small spatial and temporal numerical We propose MeshfreeFlowNet, a novel deep learningbased stencils would be required. In order to alleviate the super-resolution framework to generate continuous computational burden of fully resolving such a wide range (grid-free) spatiotemporal solutions from the low-resolution of spatial and temporal scales, multiscale computational approaches inputs. While being computationally efficient, MeshfreeFlowNet have been developed. For instance, in the subsurface accurately recovers the fine-scale quantities flow problem, the main idea of the multiscale approach of interest. MeshfreeFlowNet allows for: (i) the output is to build a set of operators that map between the unknowns to be sampled at all spatiotemporal resolutions, (ii) a set associated with the computational cells in a fine-grid and the of Partial Differential Equation (PDE) constraints to be imposed, unknowns on a coarser grid. The operators are computed and (iii) training on fixed-size inputs on arbitrarily numerically by solving localized flow problems. The multiscale sized spatiotemporal domains owing to its fully convolutional basis functions have subgrid-scale resolutions, ensuring encoder.