Inpractice, the manifold isoften represented by triangle mesh: acollection ofvertices, edges and2 faces. B.1 ProofofTheorem110 Theorem1 (i) IfN is even, there is no such real representation ρN of SO(2) that satisfies Eqn.11 (9). Partition the matrix A>1A1 into a block matrix by exactly the same way that56 the block diagonal matrix is partitioned, then Eqn. Express the manifold equation asw = w g, then we have94 ψ(fw) = ψ(fw), as is shown by (i). Here, we provide the detailed process of computing solution basis of Eqn.(15) for allΘ CN.101 Firstly,weshowthatEqn.

In Computational Fluid Dynamics (CFD), coarse mesh simulations offer computational efficiency but often lack precision. Applying conventional super-resolution to these simulations poses a significant challenge due to the fundamental contrast between downsampling high-resolution images and authentically emulating low-resolution physics. The former method conserves more of the underlying physics, surpassing the usual constraints of real-world scenarios. We propose a novel definition of super-resolution tailored for PDE-based problems. Instead of simply downsampling from a high-resolution dataset, we use coarse-grid simulated data as our input and predict fine-grid simulated outcomes. Employing a physics-infused UNet upscaling method, we demonstrate its efficacy across various 2D-CFD problems such as discontinuity detection in Burger's equation, Methane combustion, and fouling in Industrial heat exchangers. Our method enables the generation of fine-mesh solutions bypassing traditional simulation, ensuring considerable computational saving and fidelity to the original ground truth outcomes. Through diverse boundary conditions during training, we further establish the robustness of our method, paving the way for its broad applications in engineering and scientific CFD solvers.

Abstract: Masked image modeling (MIM) has achieved promising results on various vision tasks. However, the limited discriminability of learned representation manifests there is still plenty to go for making a stronger vision learner. Towards this goal, we propose Contrastive Masked Autoencoders (CMAE), a new self-supervised pre-training method for learning more comprehensive and capable vision representations. By elaboratively unifying contrastive learning (CL) and masked image model (MIM) through novel designs, CMAE leverages their respective advantages and learns representations with both strong instance discriminability and local perceptibility. Specifically, CMAE consists of two branches where the online branch is an asymmetric encoder-decoder and the target branch is a momentum updated encoder. During training, the online encoder reconstructs original images from latent representations of masked images to learn holistic features.