# Leveraging Bayesian Analysis To Improve Accuracy of Approximate Models

## TL;DR

This paper introduces a Bayesian framework to enhance the accuracy of approximate models for multiscale dynamical systems, using hierarchical analysis and neural network surrogates to uncover physical dependencies and reduce computational costs.

## Contribution

It demonstrates how hierarchical Bayesian analysis can identify unanticipated physical dependencies and improve model structure, with a focus on turbulence modeling.

## Key findings

- Bayesian analysis uncovers new physical dependencies.
- Neural network surrogates accelerate posterior sampling.
- Bayesian methods improve model accuracy and interpretability.

## Abstract

We focus on improving the accuracy of an approximate model of a multiscale dynamical system that uses a set of parameter-dependent terms to account for the effects of unresolved or neglected dynamics on resolved scales. We start by considering various methods of calibrating and analyzing such a model given a few well-resolved simulations. After presenting results for various point estimates and discussing some of their shortcomings, we demonstrate (a) the potential of hierarchical Bayesian analysis to uncover previously unanticipated physical dependencies in the approximate model, and (b) how such insights can then be used to improve the model. In effect parametric dependencies found from the Bayesian analysis are used to improve structural aspects of the model. While we choose to illustrate the procedure in the context of a closure model for buoyancy-driven, variable-density turbulence, the statistical nature of the approach makes it more generally applicable. Towards addressing issues of increased computational cost associated with the procedure, we demonstrate the use of a neural network based surrogate in accelerating the posterior sampling process and point to recent developments in variational inference as an alternative methodology for greatly mitigating such costs. We conclude by suggesting that modern validation and uncertainty quantification techniques such as the ones we consider have a valuable role to play in the development and improvement of approximate models.