High-order homogenisation by learning spatial discretisations of PDEs that provably preserve self-adjointness

Common techniques for the spatial discretisation of pdes on a macroscale grid include finite difference, finite elements and finite volume methods. Such methods typically impose assumed microscale structures on the subgrid fields, so without further tailored analysis are not suitable for systems with subgrid-scale heterogeneity or nonlinearities. We provide a new algebraic route to systematically approximate, in principle exactly, the macroscale closure of the spatially-discrete dynamics of a general class of heterogeneous non-autonomous reaction–advection–diffusion pdes. This holistic discretisation approach, developed through rigorous theory and verified with computer algebra, systematically constructs discrete macroscale models by direct reference to the pde out-of-equilibrium dynamics, thus relaxing many assumptions regarding the subgrid structure. The construction is analogous to recent grey-box machine learning techniques in that predictions are directed by iterative layers (as in neural networks), but informed by the subgrid physics (or ‘data’) as expressed in the pde. A major development of the holistic methodology, presented herein, is novel inter-element coupling between subgrid fields which preserve self-adjointness of the pde after macroscale discretisation, thereby maintaining the spectral structure of the original system. This holistic methodology also encompasses homogenisation of microscale heterogeneous systems, as shown here with the canonical examples of heterogeneous 1D waves and diffusion.