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Simulations of complex physical systems are typically realized bydiscretizing partial differential equations (PDEs) on unstructured meshes.While neural networks have recently been explored for surrogate and reducedorder modeling of PDE solutions, they often ignore interactions or hierarchicalrelations between input features, and process them as concatenated mixtures. Wegeneralize the idea of conditional parameterization -- using trainablefunctions of input parameters to generate the weights of a neural network, andextend them in a flexible way to encode critical information. Inspired bydiscretized numerical methods, choices of the parameters include physicalquantities and mesh topology features. The functional relation between themodeled features and the parameters is built into the network architecture. Themethod is implemented on different networks and applied to frontier scientificmachine learning tasks including the discovery of unmodeled physics,super-resolution of coarse fields, and the simulation of unsteady flows withchemical reactions. The results show that the conditionally-parameterizednetworks provide superior performance compared to their traditionalcounterparts. The CP-GNet - an architecture that can be trained on very fewdata snapshots - is proposed as the first deep learning model capable ofstandalone prediction of reacting flows on irregular meshes.

Conditionally Parameterized, Discretization-Aware Neural Networks for Mesh-Based Modeling of Physical Systems