Minisymposium Presentation

Model reduction for hyperbolic PDEs using classical techniques is difficult due to the slow decay in the Kolmogorov n-width, making it necessary to explore new forms of approximation. We will discuss a new approach using deep neural networks endowed with a particular low-rank structure, which we call low-rank Physics-Informed Neural Networks (LR-PINNs). LR-PINNs are a form of implicit neural representation in which the weights and biases belong to linear spaces of small dimensions. We will show that entropy solutions to scalar conservation laws can be represented efficiently by such a representation. Numerical examples illustrating the efficacy of the neural network will be shown, and we will also discuss applications of LR-PINNs regarding the so-called failure modes of PINNs.