Solving inverse-PDE problems with physics-aware neural networks

•We present a novel hybrid framework that enables discovery of unknown fields in inverse partial differential problems.•We implement trainable finite discretization solver layers that are composable with pre-existing neural layers.•The network can be pre-trained in a self-supervised fashion and used on unseen data without further training.•This framework enables consideration of domain specific knowledge about the unknown fields.•In contrast to constrained optimization methods, the loss function is simply difference between data and prediction.