Partial Differential Equations (PDEs) are notoriously difficult to solve. In general, closed-form solutions are not available and numerical approximation schemes are computationally expensive. In this paper, we propose to approach the solution of PDE
s based on a novel technique that combines the advantages of two recently emerging machine learning based approaches. First, physics-informed neural networks (PINNs) learn continuous solutions of PDEs and can be trained with little to no ground truth data. However, PINNs do not generalize well to unseen domains. Second, convolutional neural networks provide fast inference and generalize but either require large amounts of training data or a physics-constrained loss based on finite differences that can lead to inaccuracies and discretization artifacts. We leverage the advantages of both of these approaches by using Hermite spline kernels in order to continuously interpolate a grid-based state representation that can be handled by a CNN. This allows for training without any precomputed training data using a physics-informed loss function only and provides fast, continuous solutions that generalize to unseen domains. We demonstrate the potential of our method at the examples of the incompressible Navier-Stokes equation and the damped wave equation. Our models are able to learn several intriguing phenomena such as Karman vortex streets, the Magnus effect, Doppler effect, interference patterns and wave reflections. Our quantitative assessment and an interactive real-time demo show that we are narrowing the gap in accuracy of unsupervised ML based methods to industrial CFD solvers while being orders of magnitude faster.
Parquetry is the art and craft of decorating a surface with a pattern of differently colored veneers of wood, stone or other materials. Traditionally, the process of designing and making parquetry has been driven by color, using the texture found in
real wood only for stylization or as a decorative effect. Here, we introduce a computational pipeline that draws from the rich natural structure of strongly textured real-world veneers as a source of detail in order to approximate a target image as faithfully as possible using a manageable number of parts. This challenge is closely related to the established problems of patch-based image synthesis and stylization in some ways, but fundamentally different in others. Most importantly, the limited availability of resources (any piece of wood can only be used once) turns the relatively simple problem of finding the right piece for the target location into the combinatorial problem of finding optimal parts while avoiding resource collisions. We introduce an algorithm that allows to efficiently solve an approximation to the problem. It further addresses challenges like gamut mapping, feature characterization and the search for fabricable cuts. We demonstrate the effectiveness of the system by fabricating a selection of photo-realistic pieces of parquetry from different kinds of unstained wood veneer.
