Geometric NeuralPDE (GNPnet) Models for Learning Dynamics

Real-world phenomena, such as cellular dynamics, electromagnetic wave propagation,and heat diffusion vary with respect to space and time and therefore candescribed by partial differential equations (PDEs). We focus on the problem offinding a dynamic model and parameterization that can generate and match observedtime-series data. For this purpose, we introduce Geometric Neural PDEnetwork (GNPnet), a neural network that learns to match and interpolate measuredphenomenon using an autoregressive framework. GPNnet has several novelfeatures including a geometric scattering network that leverages spatial problemstructure, and an FEM solver that is incorporated within the network. GPNnetlearns parameters of a PDE via an FEM solver that generates solution values thatare compared with measured phenomenon. By using the adjoint sensitivity methodto differentiate the output loss function, we can train the model end-to-end. Wedemonstrate GPNnet by learning the parameters of a simulated wave equation.