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Towards information-optimal simulation of partial differential equations.

Reimar H Leike1, Torsten A Enßlin1

  • 1Max-Planck-Institut für Astrophysik, Karl-Schwarzschildstrasse 1, 85748 Garching, Germany and Ludwig-Maximilians-Universität München, Geschwister-Scholl-Platz 1, 80539 Munich, Germany.

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This study introduces an information-theoretic approach for simulating partial differential equations (PDEs), treating discretized fields as data. This novel method optimizes simulations by conserving information, outperforming traditional schemes.

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Area of Science:

  • Computational Physics
  • Information Theory
  • Numerical Analysis

Background:

  • Traditional partial differential equation (PDE) simulations focus on minimizing error norms of discretized fields.
  • A novel approach interprets discretized fields as data, aiming to conserve information about the underlying physical field during simulation.

Purpose of the Study:

  • To develop an information-theoretic framework for simulating nonlinear PDEs using Information Field Dynamics (IFD).
  • To derive an information-optimal simulation scheme through action minimization.
  • To validate the scheme's accuracy and compare it with existing methods.

Main Methods:

  • Developed the theory of IFD for nonlinear PDEs under a noiseless Gaussian approximation.
  • Derived a simulation scheme by minimizing a derived action.
  • Utilized field operators to compute Gaussian integrals for a closed-form solution.
  • Numerically tested the scheme using the Burgers equation.

Main Results:

  • The derived IFD scheme demonstrates superior accuracy compared to finite-difference schemes at the same resolution.
  • The scheme's performance is contingent on accurate subgrid correlation structure information.
  • In limiting cases, the scheme recovers established methods like spectral Fourier-Galerkin.

Conclusions:

  • The information-theoretic approach offers a powerful alternative for PDE simulation, prioritizing information conservation.
  • The derived IFD scheme provides an information-optimal simulation method with demonstrated advantages.
  • Further research should explore the implications of approximations and refine subgrid correlation modeling.