On The Expressive Power of Knowledge Graph Embedding Methods

Knowledge Graph Embedding (KGE) is a popular approach, which aims to represent entities and relations of a knowledge graph in latent spaces. Their representations are known as embeddings. To measure the plausibility of triplets, score functions are defined over embedding spaces. Despite wide dissemination of KGE in various tasks, KGE methods have limitations in reasoning abilities. In this paper we propose a mathematical framework to compare reasoning abilities of KGE methods. We show that STransE has a higher capability than TransComplEx, and then present new STransCoRe method, which improves the STransE by combining it with the TransCoRe insights, which can reduce the STransE space complexity.

PINNs error estimates for nonlinear equations in $\mathbb{R}$-smooth Banach spaces

In the paper, we describe in operator form classes of PDEs that admit PINN's error estimation. Also, for $L^p$ spaces, we obtain a Bramble-Hilbert type lemma that is a tool for PINN's residuals bounding.

Scaling transformation of the multimode nonlinear Schrödinger equation for physics-informed neural networks

Single-mode optical fibers (SMFs) have become the backbone of modern communication systems. However, their throughput is expected to reach its theoretical limit in the nearest future. Utilization of multimode fibers (MMFs) is considered as one of the most promising solutions rectifying this capacity crunch. Nevertheless, differential equations describing light propagation in MMFs are a way more sophisticated than those for SMFs, which makes numerical modelling of MMF-based systems computationally demanding and impractical for the most part of realistic scenarios. Physics-informed neural networks (PINNs) are known to outperform conventional numerical approaches in various domains and have been successfully applied to the nonlinear Schr\"odinger equation (NLSE) describing light propagation in SMFs. A comprehensive study on application of PINN to the multimode NLSE (MMNLSE) is still lacking though. To the best of our knowledge, this paper is the first to deploy the paradigm of PINN for MMNLSE and to demonstrate that a straightforward implementation of PINNs by analogy with NLSE does not work out. We pinpoint all issues hindering PINN convergence and introduce a novel scaling transformation for the zero-order dispersion coefficient that makes PINN capture all relevant physical effects. Our simulations reveal good agreement with the split-step Fourier (SSF) method and extend numerically attainable propagation lengths up to several hundred meters. All major limitations are also highlighted.