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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. 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 on 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 goal of this PhD project is to develop new AL algorithms tailored to the unique challenges and opportunities presented by PDEs, such as the extremely high-dimensional output space and temporal evolution. As a first step, we investigated the behavior of existing AL algorithms by creating the AL4PDE benchmark. In contrast to the standard problems considered in the AL literature, the numerical simulator can be queried at any input point, allowing us to investigate the so-called query synthesis scenario. Finally, we want to find new ways to incorporate physical inductive biases into both model architecture and the data generation phase.
Keywords: Active Learning, Partial Differential Equations, Neural PDE Solvers