On the Practical Applications of Information Field Dynamics

TL;DR

This paper introduces and analyzes Information Field Dynamics (IFD), a Bayesian-based simulation scheme for PDEs, demonstrating its convergence, error scaling, and relation to traditional numerical methods through theoretical and prototype developments.

Contribution

It provides the first proof of convergence for IFD schemes, analyzes their error scaling, and develops prototype translation-invariant and SPH-like IFD schemes.

Findings

Translation-invariant schemes converge to true field behavior at high resolution.

Error scaling of IFD schemes is analogous to high-order finite differences.

Prototypes demonstrate practical implementation and stability of IFD methods.

Abstract

In this study we explore a new simulation scheme for partial differential equations known as Information Field Dynamics (IFD). Information field dynamics attempts to improve on existing simulation schemes by incorporating Bayesian field inference into the simulation scheme. The field inference is truly Bayesian and thus depends on a notion of prior belief. A number of results are presented, both theoretical and practical. Many small fixes and results on the general theory are presented, before exploring two general classes of simulation schemes that are possible in the IFD framework. For both, we present a set of theoretical results alongside the development of a prototype scheme. The first class of models corresponds roughly to traditional fixed-grid numerical PDE solvers. The prior Bayesian assumption in these models is that the fields are smooth, and their correlation structure does not vary between locations. For these reasons we call them translation-invariant schemes. We show the requirements for stability of these schemes, but most importantly we prove that these schemes indeed converge to the true behaviour of the field in the limit of high resolutions. Convergence had never been shown for any previous IFD scheme. We also find the error scaling of these codes and show that they implement something very analogous to a high-order finite-difference derivative approximation, which are the most elementary and well-studied numerical schemes. This is an important result, which proves the validity of the IFD approach. The second class of schemes, called the SPH-like schemes are similar to existing Smooth Particle Hydrodynamics codes, in which the simulation grid moves with the flow of the field being modelled.