[Paper Review] Least-biased correction of extended dynamical systems using observational data

Keith W. Myerscough, Jason Frank+1 more|arXiv (Cornell University)|Nov 21, 2014

Fluid Dynamics and Turbulent Flows48 references3 citations

TL;DR

This paper proposes a least-biased correction method for extended dynamical systems using observational data to minimize statistical bias in equilibrium distributions. By combining entropy-optimized estimation with on-the-fly thermostatting, the approach corrects model biases using incomplete or noisy data while preserving key dynamical features such as autocorrelation functions and transport coefficients.

ABSTRACT

We consider dynamical systems evolving near an equilibrium statistical state where the interest is in modelling long term behavior that is consistent with thermodynamic constraints. We adjust the distribution using an entropy-optimizing formulation that can be computed on-the- fly, making possible partial corrections using incomplete information, for example measured data or data computed from a different model (or the same model at a different scale). We employ a thermostatting technique to sample the target distribution with the aim of capturing relavant statistical features while introducing mild dynamical perturbation (thermostats). The method is tested for a point vortex fluid model on the sphere, and we demonstrate both convergence of equilibrium quantities and the ability of the formulation to balance stationary and transient- regime errors.

Motivation & Objective

• Address statistical bias in long-time simulations of chaotic dynamical systems due to time and spatial discretization.
• Correct model bias when the true invariant measure is unknown but partial expectations of observables are available from data or simulations.
• Develop a computationally efficient, on-the-fly method that enables partial corrections using short-time ensemble bursts.
• Balance stationary (equilibrium) and transient (dynamical) errors in statistical modeling of complex systems.
• Enable accurate sampling of target probability distributions without requiring full knowledge of the underlying invariant measure.

Proposed method

• Use entropy maximization to construct a least-biased probability density consistent with observed moment constraints, starting from a prior distribution.
• Apply a thermostatting technique to sample the target distribution by perturbing dynamics with a stochastic damping term that maintains detailed balance.
• Integrate the perturbed dynamics using symmetric composition (Strang splitting) to preserve geometric structure and ensure stability.
• Compute Lagrange multipliers iteratively via ensemble averages of observables, enabling adaptation with incomplete or noisy data.
• Use short-time ensemble bursts to tune parameters dynamically, reducing computational cost while maintaining accuracy.
• Model vortex interactions via symmetric and antisymmetric decomposition of the velocity field, enabling analytical integration of the thermostated dynamics.

Experimental results

Research questions

• RQ1 How can statistical bias in long-time simulations of dynamical systems be minimized when the true invariant measure is unknown?
• RQ2 To what extent can a least-biased probability density be constructed from partial observational data using entropy optimization?
• RQ3 Can thermostatting techniques be adapted to correct bias in extended dynamical systems without degrading dynamical properties like autocorrelation functions?
• RQ4 How does the proposed method balance errors in equilibrium statistics and transient dynamics during simulation?
• RQ5 What is the impact of partial or noisy data on the convergence and accuracy of the corrected statistical distribution?

Key findings

• The method successfully converges to the correct equilibrium distribution even when only partial moment information is available.
• Autocorrelation functions are only modestly perturbed, indicating that transport coefficients and dynamical mixing properties are preserved.
• The iterative Lagrange multiplier computation converges efficiently using short-time ensemble bursts, enabling real-time adaptation.
• The thermostatting mechanism introduces minimal dynamical perturbation while ensuring exact adherence to the least-biased distribution.
• Numerical experiments on the point vortex model on the sphere confirm that both stationary and transient statistical features are accurately captured.
• The symmetric composition of dynamics preserves geometric structure and ensures robust integration of the thermostated system.