Fanrui Meng | Search arXiv

arXiv2026

Low Mach number limit around the planar diffusion wave for 3D Navier-Stokes-Fourier equations

We investigate the low Mach number limit of the three-dimensional full compressible Navier-Stokes-Fourier (NSF) equations on $\mathbb{R}\times\mathbb{T}^2$ for two different classes of initial data, corresponding respectively to a well-prepared regime and an ill-prepared regime. The density and temperature are allowed to approach different asymptotic states at infinity. For the well-prepared regime, the solutions of compressible NSF equations converge to a planar diffusion wave solution globally in time as the Mach number tends to zero, where the difference between the states at the far fields is independent of the Mach number. Moreover, the optimal time-decay rate can be obtained. It can be viewed as the first global-in-time result on the low Mach number limit of three-dimensional NSF equations with large temperature variations. For the ill-prepared regime, the corresponding local-in-time result is obtained by performing separate energy estimates for the zero and non-zero modes, and an auxiliary convergence lemma proposed by Métivier-Schochet in \cite{Métivier2001}. It is remarked that the difference between the states at the far fields is allowed to be arbitrarily large.

math.AP↗

arXiv2021

Unsupervised Learning of Local Discriminative Representation for Medical Images

Local discriminative representation is needed in many medical image analysis tasks such as identifying sub-types of lesion or segmenting detailed components of anatomical structures. However, the commonly applied supervised representation learning methods require a large amount of annotated data, and unsupervised discriminative representation learning distinguishes different images by learning a global feature, both of which are not suitable for localized medical image analysis tasks. In order to avoid the limitations of these two methods, we introduce local discrimination into unsupervised representation learning in this work. The model contains two branches: one is an embedding branch which learns an embedding function to disperse dissimilar pixels over a low-dimensional hypersphere; and the other is a clustering branch which learns a clustering function to classify similar pixels into the same cluster. These two branches are trained simultaneously in a mutually beneficial pattern, and the learnt local discriminative representations are able to well measure the similarity of local image regions. These representations can be transferred to enhance various downstream tasks. Meanwhile, they can also be applied to cluster anatomical structures from unlabeled medical images under the guidance of topological priors from simulation or other structures with similar topological characteristics. The effectiveness and usefulness of the proposed method are demonstrated by enhancing various downstream tasks and clustering anatomical structures in retinal images and chest X-ray images.

cs.CV↗