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Common techniques for the spatial discretisation of PDEs on a macroscale gridinclude finite difference, finite elements and finite volume methods. Suchmethods typically impose assumed microscale structures on the subgrid fields,so without further tailored analysis are not suitable for systems withsubgrid-scale heterogeneity or nonlinearities. We provide a new algebraic routeto systematically approximate, in principle exactly, the macroscale closure ofthe spatially-discrete dynamics of a general class of heterogeneousnon-autonomous reaction-advection-diffusion PDEs. This holistic discretisationapproach, developed through rigorous theory and verified with computer algebra,systematically constructs discrete macroscale models through physics informedby the PDE out-of-equilibrium dynamics, thus relaxing many assumptionsregarding the subgrid structure. The construction is analogous to recentgray-box machine learning techniques in that predictions are directed byiterative layers (as in neural networks), but informed by the subgrid physics(or 'data') as expressed in the PDEs. A major development of the holisticmethodology, presented herein, is novel inter-element coupling between subgridfields which preserve self-adjointness of the PDE after macroscalediscretisation, thereby maintaining the spectral structure of the originalsystem. This holistic methodology also encompasses homogenisation of microscaleheterogeneous systems, as shown here with the canonical examples ofheterogeneous 1D waves and diffusion.

Learning high-order spatial discretisations of PDEs with symmetry-preserving iterative algorithms