  In this paper we propose a new model-based unsupervised learning method,
called VarNet, for the solution of partial differential equations (PDEs) using
deep neural networks (NNs). Particularly, we propose a novel loss function that
relies on the variational (integral) form of PDEs as apposed to their
differential form which is commonly used in the literature. Our loss function
is discretization-free, highly parallelizable, and more effective in capturing
the solution of PDEs since it employs lower-order derivatives and trains over
measure non-zero regions of space-time. Given this loss function, we also
propose an approach to optimally select the space-time samples, used to train
the NN, that is based on the feedback provided from the PDE residual. The
models obtained using VarNet are smooth and do not require interpolation. They
are also easily differentiable and can directly be used for control and
optimization of PDEs. Finally, VarNet can straight-forwardly incorporate
parametric PDE models making it a natural tool for model order reduction (MOR)
of PDEs. We demonstrate the performance of our method through extensive
numerical experiments for the advection-diffusion PDE as an important
case-study.
