Daniel Musekamp

Partial differential equations (PDE) play an essential role in many areas of science but may need an enormous amount of time and computational resources to be solved using numerical solvers. In order to overcome these challenges, neural PDE Solvers can be trained to replace the numerical simulator. Not only provide these surrogates significant speed-ups but are end-to-end differentiable and can be finetuned to experimental data. The problem of neural PDE solvers is that generating the training data using the numerical solver presents a high initial cost. Additionally, it can be challenging to select the simulation inputs such that important edge cases or desired scenarios are sufficiently covered with training data points. A principled solution to these issues is active learning (AL), which iteratively selects the most informative data points based on the previous model behavior. Thus, the focus of this PhD project is to improve the data generation for neural PDE Solvers using AL. Based on current benchmarks for neural PDE Solvers, the initial step will be to evaluate existing AL algorithms on the problem. Afterward, the goal is to develop new AL algorithms tailored to the unique challenges and opportunities presented by PDEs, such as the extremely high-dimensional output space. In contrast to the standard problems considered in the AL literature, the numerical simulator can be queried at any input point, allowing to investigate the query-synthesis scenario. Finally, we want to find new ways to incorporate physical inductive biases into both model architecture and the data generation phase.