A0881
Title: Measuring the risk of solving fluid dynamics by neural nets
Authors: Anirbit Mukherjee - The University of Manchester (United Kingdom)
Dibyakanti Kumar - The University of Manchester (United Kingdom) [presenting]
Abstract: What is the relationship between a machine learning model's error in approximating the PDE solution and its physics-informed neural net (PINN) loss function value? This is critical for scientific-ML, and it is generally quite unclear. The attempt is to prove relationships between these two components for non-viscous pressure-less fluids (Burgers' PDE) in arbitrary dimensions while allowing for flow divergence. This is an interesting edge because it allows for PDE solutions that blow up in finite time while starting from smooth solutions - and hence, it allows for hard tests of theory to be conducted. Recently, some other empirical studies have pointed out that such solutions might be detectable by the PINN method if one regularizes for the gradients of the neural surrogate. Interestingly, this way of doing the population risk analysis reveals that this risk does indeed scale with such functional norms of the surrogate - and hence, it gives theoretical foundations for such penalizers to be added to PINN losses. Can risk bounds be obtained on PINNs that vanish in the limit of a large number of collocation points being used? That remains unclear - and definitely so for these cases as considered. Hence, the neural PDE solving scenario considered motivates exciting new research directions about generalization error bounding.
