  Simulations of complex physical systems are typically realized by
discretizing partial differential equations (PDEs) on unstructured meshes.
While neural networks have recently been explored for surrogate and reduced
order modeling of PDE solutions, they often ignore interactions or hierarchical
relations between input features, and process them as concatenated mixtures. We
generalize the idea of conditional parameterization -- using trainable
functions of input parameters to generate the weights of a neural network, and
extend them in a flexible way to encode critical information. Inspired by
discretized numerical methods, choices of the parameters include physical
quantities and mesh topology features. The functional relation between the
modeled features and the parameters is built into the network architecture. The
method is implemented on different networks and applied to frontier scientific
machine learning tasks including the discovery of unmodeled physics,
super-resolution of coarse fields, and the simulation of unsteady flows with
chemical reactions. The results show that the conditionally-parameterized
networks provide superior performance compared to their traditional
counterparts. The CP-GNet - an architecture that can be trained on very few
data snapshots - is proposed as the first deep learning model capable of
standalone prediction of reacting flows on irregular meshes.
