Physics-informed Hyper-networks

There is a growing trend towards the development of parsimonious surrogate models for studying physical phenomena. While they typically offer less accuracy, these models bypass the computational costs of numerical methods, usually by multiple orders of magnitude, allowing statistical applications such as sensitivity analysis, stochastic treatments, parametric problems, and uncertainty quantification. Researchers have explored generalized surrogate frameworks leveraging Gaussian processes, various basis function expansions, support vector machines, and neural networks. Dynamical fields, represented through time-dependent partial differential equation, pose a particular hardship for existing frameworks due to their high dimensional representation, and possibly multi-scale solutions.

In this work, we present a novel architecture for solving time-dependent partial differential equations using co-ordinate neural networks and time-marching updates through hyper-networks. We show that it provides a temporally meshed and spatially mesh-free solution which are causally coherent as justified through a theoretical treatment of Lie groups. We showcase results on some benchmark problems in computational physics while discussing their performance against similar physics-informed approaches like physics-informed DeepOnets and Physics informed neural networks.