Over the past few decades, modeling, simulation, and optimization tools have received
attention for their ability to represent and improve complex systems. The use of …

Despite tremendous progress seen in the computational fluid dynamics community for the
past few decades, numerical tools are still too slow for the simulation of practical flow …

Neural networks (NNs) are currently changing the computational paradigm on how to
combine data with mathematical laws in physics and engineering in a profound way …

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 …

Deep generative models (DGM) are neural networks with many hidden layers trained to
approximate complicated, high‐dimensional probability distributions using samples. When …

We describe the new field of the mathematical analysis of deep learning. This field emerged
around a list of research questions that were not answered within the classical framework of …

We use a physics-informed neural network (PINN) to simultaneously model and optimize the
flow around an airfoil to maximize its lift to drag ratio. The parameters of the airfoil shape are …

Deep learning (DL) has had unprecedented success and is now entering scientific
computing with full force. However, current DL methods typically suffer from instability, even …

Machine learning methods trained on raw numerical time series data exhibit fundamental
limitations such as a high sensitivity to the hyper parameters and even to the initialization of …

We present PFNN, a penalty-free neural network method, to efficiently solve a class of
second-order boundary-value problems on complex geometries. To reduce the smoothness …