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On uncertainty-penalized Bayesian information criterion

Pongpisit Thanasutives, Ken-ichi Fukui

TL;DR

This work analyzes uncertainty-penalized information criteria for data-driven PDE discovery. It proves that UBIC is equivalent to applying the conventional BIC to an overparameterized model set, implying their asymptotic properties coincide as the sample size grows. Building on this, the authors discuss limitations when the uncertainty term grows with model size and propose an alternative, physics-informed criterion using ICOMP with the inverse Fisher information, showing it can recover true Burgers' PDE form with an appropriately scaled complexity term. The results clarify the theoretical relationship between UBIC and BIC and motivate developing physics-guided model-selection criteria to improve PDE identification in practice.

Abstract

The uncertainty-penalized information criterion (UBIC) has been proposed as a new model-selection criterion for data-driven partial differential equation (PDE) discovery. In this paper, we show that using the UBIC is equivalent to employing the conventional BIC to a set of overparameterized models derived from the potential regression models of different complexity measures. The result indicates that the asymptotic property of the UBIC and BIC holds indifferently.

On uncertainty-penalized Bayesian information criterion

TL;DR

This work analyzes uncertainty-penalized information criteria for data-driven PDE discovery. It proves that UBIC is equivalent to applying the conventional BIC to an overparameterized model set, implying their asymptotic properties coincide as the sample size grows. Building on this, the authors discuss limitations when the uncertainty term grows with model size and propose an alternative, physics-informed criterion using ICOMP with the inverse Fisher information, showing it can recover true Burgers' PDE form with an appropriately scaled complexity term. The results clarify the theoretical relationship between UBIC and BIC and motivate developing physics-guided model-selection criteria to improve PDE identification in practice.

Abstract

The uncertainty-penalized information criterion (UBIC) has been proposed as a new model-selection criterion for data-driven partial differential equation (PDE) discovery. In this paper, we show that using the UBIC is equivalent to employing the conventional BIC to a set of overparameterized models derived from the potential regression models of different complexity measures. The result indicates that the asymptotic property of the UBIC and BIC holds indifferently.
Paper Structure (3 sections, 7 equations, 2 figures)

This paper contains 3 sections, 7 equations, 2 figures.

Table of Contents

  1. Definitions
  2. Main proof (Connecting the BIC and UBIC)
  3. Necessity of a new physics-inspired model-selection criterion for identifying governing equations

Figures (2)

  • Figure 1: Plot of the relative ICOMP with respect to $\norm{\xi}_{0}$. The relative ICOMP is defined as $\textrm{ICOMP}_{\Phi}(\xi) - \min_{\xi} \textrm{ICOMP}_{\Phi}(\xi)$. The code implementation of the ICOMP is provided in https://github.com/Pongpisit-Thanasutives/ICOMP.
  • Figure 2: Plot of $C_{\xi}(\hat{\mathfrak{F}}^{-1})$ (the maximal information complexity of the estimated IFIM) with respect to $\norm{\xi}_{0}$.