Abstract
We explore a new simulation scheme for partial differential equations (PDE's) called information field dynamics (IFD). Information field dynamics is a probabilistic numerics method that seeks to preserve the maximum amount of information about the field being simulated. It rests on Bayesian field inference and therefore allows the incorporation of prior knowledge on the field. This makes IFD attractive to address the closure problem of simulations—how to incorporate knowledge about subgrid dynamics into a scheme on a grid with limited resolution. Here we analytically prove that a restricted subset of simulation schemes in IFD are consistent and thus deliver valid predictions in the limit of high resolutions. This has not previously been done for any IFD schemes. This restricted subset is roughly analogous to traditional fixed-grid numerical PDE solvers, given the additional restriction of translational symmetry. Furthermore, given an arbitrary IFD scheme modeling a PDE, it is a priori not obvious to what order the scheme is accurate in space and time. For this subset of models, we also derive an easy rule of thumb for determining the order of accuracy of the simulation. As with all analytic consistency analysis, an analysis for nontrivial systems is intractable; thus these results are intended as a general indicator of the validity of the approach, and it is hoped that the results will generalize.
- Received 26 June 2018
DOI:https://doi.org/10.1103/PhysRevE.98.043307
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