Invariant Physics-Informed Neural Networks for Ordinary Differential Equations

Physics-informed neural networks have emerged as a prominent new method for solving differential equations. While conceptually straightforward, they often suffer training difficulties that lead to relatively large discretization errors or the failure to obtain correct solutions. In this paper we introduce invariant physics-informed neural networks for ordinary differential equations that admit a finite-dimensional group of Lie point symmetries. Using the method of equivariant moving frames, a differential equation is invariantized to obtain a, generally, simpler equation in the space of differential invariants. A solution to the invariantized equation is then mapped back to a solution of the original differential equation by solving the reconstruction equations for the left moving frame. The invariantized differential equation together with the reconstruction equations are solved using a physcis-informed neural network, and form what we call an invariant physics-informed neural network. We illustrate the method with several examples, all of which considerably outperform standard non-invariant physics-informed neural networks.Comment: 20 pages, 6 figure

Dengue shock syndrome in sickle cell disease precipitating sickle cell hepatopathy: a case report

Sickle cell disease (SCD) is a known risk factor for the development of severe dengue, however, literature documenting dengue in SCD is scarce. Dengue fever further triggers the sickling process in a patient with SCD by augmenting endothelial dysfunction, the main identifiable cause behind organ dysfunction. Hepatic involvement in SCD due to enhanced sickling can be in the form of acute viral hepatitis, cholecystitis, acute sickle hepatic crisis, and more severe sickle cell intrahepatic cholestasis (SCIC). Initially starting as an acute sickle hepatic crisis, SCIC progresses to striking jaundice, enhanced bleeding tendency coupled with mostly renal failure. We report a rare case of a female, native of Chhattisgarh with SCD and dengue shock syndrome who had fatal hepatic complications resulting from accelerated severe endothelial dysfunction due to concurrent illnesses

MEMEX: Detecting Explanatory Evidence for Memes via Knowledge-Enriched Contextualization

Memes are a powerful tool for communication over social media. Their affinity for evolving across politics, history, and sociocultural phenomena makes them an ideal communication vehicle. To comprehend the subtle message conveyed within a meme, one must understand the background that facilitates its holistic assimilation. Besides digital archiving of memes and their metadata by a few websites like knowyourmeme.com, currently, there is no efficient way to deduce a meme's context dynamically. In this work, we propose a novel task, MEMEX - given a meme and a related document, the aim is to mine the context that succinctly explains the background of the meme. At first, we develop MCC (Meme Context Corpus), a novel dataset for MEMEX. Further, to benchmark MCC, we propose MIME (MultImodal Meme Explainer), a multimodal neural framework that uses common sense enriched meme representation and a layered approach to capture the cross-modal semantic dependencies between the meme and the context. MIME surpasses several unimodal and multimodal systems and yields an absolute improvement of ~ 4% F1-score over the best baseline. Lastly, we conduct detailed analyses of MIME's performance, highlighting the aspects that could lead to optimal modeling of cross-modal contextual associations.Comment: 9 pages main + 1 ethics + 3 pages ref. + 4 pages app (Total: 17 pages

Toward Refactoring of DMARF and GIPSY Case Studies – a Team 3 SOEN6471-S14 Project Report

The software architecture of a system is an illustration of the system which supports the understanding of the behaviour of the system. The architecture aids as the blueprint of the system, defining the work obligations which must be conceded by design and implementation teams. It is an artifact for early enquiry to make sure that a design methodology will produce a standard system. This paper depicts the software architecture and design of two frameworks DMARF and GIPSY. Primarily it inaugurates a comprehensive understanding of the frameworks and their applications. DMARF is high-volume processing of recorded audio, textual, or imagery data for pattern recognition and biometric forensic analysis, whereas GIPSY system provides a platform for a distributed multi-tier demand driven evaluation of heterogeneous programs. Secondly, the paper illustrates the use of several tools for the code analysis for both platforms and provides the outcome of the analysis. Thirdly, it establishes the architecture and design of the systems. Fourthly, it fuses the architecture for both the systems into one. The paper ends with depicting properties like code smells and refactoring to improve code quality for the frameworks

On compactness properties of subgroups

The class of locally compact groups has been widely studied in group theory, representation theory, and harmonic analysis. There is a current program of extending geometric techniques used in the study of discrete groups to this larger class [Wil94, KM08,CCMT15,CdlH16]. This thesis is part of that program. We use geometric methods to study the compactness properties of subgroups in the class of topological groups containing a compact open subgroup. This class includes discrete groups, profinite groups, and totally disconnected locally compact groups as subclasses. In the first project, we study discrete hyperbolic groups. Finitely presented subgroups of hyperbolic groups are not necessarily hyperbolic; the first examples of this phenomenon were constructed by Brady [Bra99]. In contrast, for hyperbolic groups of integral cohomological dimension at most two, finitely presented subgroups are hyperbolic; this is a result of Gersten [Ger96b]. We extend this result to hyperbolic groups with rational cohomological dimension bounded by two. This applies to examples of groups constructed by Bestvina and Mess, fully describing the nature of their finitely presented subgroups, which was previously unknown. In the second project, we extend Gersten’s result for totally disconnected locally compact (TDLC) groups. In particular, we prove that closed compactly presented subgroups of hyperbolic TDLC groups of discrete rational cohomological dimension bounded by two are hyperbolic. We also characterize hyperbolic TDLC groups in terms of isoperimetric inequalities and study small cancellation quotients of amalgamated free products of profinite groups over open subgroups. In the last project, we study coherence of topological groups. A group is coherent if every compactly generated subgroup is compactly presented. We prove that amalgamated free products of coherent groups over compact open subgroups are coherent. We also show that certain small cancellation quotients of these groups are also coherent, generalizing a result of McCammond and Wise [MW05]. In order to prove the main results, we study relative hyperbolicity for topological groups containing compact open subgroups with respect to finite collection of open subgroups, and extend some results of Osin [Osi06]