Consistency and convergence of simulation schemes in Information field dynamics

TL;DR

This paper introduces a new simulation scheme called Information Field Dynamics (IFD) for PDEs, proves its consistency under certain conditions, and provides a simple way to estimate its accuracy, advancing the theoretical understanding of IFD.

Contribution

The paper analytically proves the consistency of a subset of IFD schemes and offers a practical rule for assessing their accuracy, which was not previously established.

Findings

A restricted subset of IFD schemes are consistent and produce valid high-resolution predictions.

An easy rule-of-thumb for estimating the spatial and temporal accuracy of IFD schemes.

The results serve as a general indicator of IFD validity and may extend to more complex systems.

Abstract

We explore a new simulation scheme for partial differential equations (PDE's) called Information Field Dynamics (IFD). Information field dynamics attempts to improve on existing simulation schemes by incorporating Bayesian field inference, which seeks to preserve the maximum amount of information about the field being simulated. The field inference is truly Bayesian and thus depends on a notion of prior belief. 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 modelling 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.