Solving parametric PDE problems with artificial neural networks

The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modeled into the equations as random coefficients. However, very often the variability of physical quantities derived from a PDE can be captured by a few features on the space of the coefficient fields. Based on such an observation, we propose using a neural-network (NN) based method to parameterize the physical quantity of interest as a function of input coefficients. The representability of such quantity using a neural-network can be justified by viewing the neural-network as performing time evolution to find the solutions to the PDE. We further demonstrate the simplicity and accuracy of the approach through notable examples of PDEs in engineering and physics.