Can physics-informed neural networks beat the finite element method?

Partial differential equations (PDEs) play a fundamental role in the mathematical modelling of many processes and systems in physical, biological and other sciences. To simulate such processes and systems, the solutions of PDEs often need to be approximated numerically. The finite element method, for instance, is a usual standard methodology to do so. The recent success of deep neural networks at various approximation tasks has motivated their use in the numerical solution of PDEs. These so-called physics-informed neural networks and their variants have shown to be able to successfully approximate a large range of PDEs. So far, physics-informed neural networks and the finite element method have mainly been studied in isolation of each other. In this work, we compare the methodologies in a systematic computational study. Indeed, we employ both methods to numerically solve various linear and nonlinear PDEs: Poisson in 1D, 2D and 3D, Allen-Cahn in 1D, semilinear Schrödinger in 1D and 2D. We then compare computational costs and approximation accuracies. In terms of solution time and accuracy, physics-informed neural networks have not been able to outperform the finite element method in our study. In some experiments, they were faster at evaluating the solved PDE.

Extracting chain lines and laid lines from digital images of medieval paper using spectral total variation decomposition.

Medieval paper, a handmade product, is made with a mould which leaves an indelible imprint on the sheet of paper. This imprint includes chain lines, laid lines and watermarks which are often visible on the sheet. Extracting these features allows the identification of the paper stock and gives information about the chronology, localisation and movement of manuscripts and people. Most computational work for feature extraction of paper analysis has so far focused on radiography or transmitted light images. While these imaging methods provide clear visualisation of the features of interest, they are expensive and time consuming in their acquisition and not feasible for smaller institutions. However, reflected light images of medieval paper manuscripts are abundant and possibly cheaper in their acquisition. In this paper, we propose algorithms to detect and extract the laid and chain lines from reflected light images. We tackle the main drawback of reflected light images, that is, the low contrast attenuation of chain and laid lines and intensity jumps due to noise and degradation, by employing the spectral total variation decomposition and develop methods for subsequent chain and laid line extraction. Our results clearly demonstrate the feasibility of using reflected light images in paper analysis. This work enables feature extraction for paper manuscripts that have otherwise not been analysed due to a lack of appropriate images. We also open the door for paper stock identification at scale.