Towards information optimal simulation of partial differential equations

TL;DR

This paper develops an information-theoretic framework for simulating nonlinear PDEs, aiming for optimal information preservation, and demonstrates its advantages over traditional methods through numerical tests on the Burgers equation.

Contribution

It formulates an informationally optimal simulation scheme for nonlinear PDEs using information field dynamics in a Gaussian approximation, improving accuracy over finite-difference methods.

Findings

The IFD scheme outperforms finite-difference schemes in accuracy.

The scheme recovers spectral methods in certain limits.

Proper subgrid correlation information is crucial for accuracy.

Abstract

Most simulation schemes for partial differential equations (PDEs) focus on minimizing a simple error norm of a discretized version of a field. This paper takes a fundamentally different approach; the discretized field is interpreted as data providing information about a real physical field that is unknown. This information is sought to be conserved by the scheme as the field evolves in time. Such an information theoretic approach to simulation was pursued before by information field dynamics (IFD). In this paper we work out the theory of IFD for nonlinear PDEs in a noiseless Gaussian approximation. The result is an action that can be minimized to obtain an informationally optimal simulation scheme. It can be brought into a closed form using field operators to calculate the appearing Gaussian integrals. The resulting simulation schemes are tested numerically in two instances for the Burgers equation. Their accuracy surpasses finite-difference schemes on the same resolution. The IFD scheme, however, has to be correctly informed on the subgrid correlation structure. In certain limiting cases we recover well-known simulation schemes like spectral Fourier Galerkin methods. We discuss implications of the approximations made.