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Gaussian processes for Bayesian inverse problems associated with linear partial differential equations

Tianming Bai, Aretha L. Teckentrup, Konstantinos C. Zygalakis

TL;DR

This work extends the framework of Raissi et.

Abstract

This work is concerned with the use of Gaussian surrogate models for Bayesian inverse problems associated with linear partial differential equations. A particular focus is on the regime where only a small amount of training data is available. In this regime the type of Gaussian prior used is of critical importance with respect to how well the surrogate model will perform in terms of Bayesian inversion. We extend the framework of Raissi et. al. (2017) to construct PDE-informed Gaussian priors that we then use to construct different approximate posteriors. A number of different numerical experiments illustrate the superiority of the PDE-informed Gaussian priors over more traditional priors.

Gaussian processes for Bayesian inverse problems associated with linear partial differential equations

TL;DR

This work extends the framework of Raissi et.

Abstract

This work is concerned with the use of Gaussian surrogate models for Bayesian inverse problems associated with linear partial differential equations. A particular focus is on the regime where only a small amount of training data is available. In this regime the type of Gaussian prior used is of critical importance with respect to how well the surrogate model will perform in terms of Bayesian inversion. We extend the framework of Raissi et. al. (2017) to construct PDE-informed Gaussian priors that we then use to construct different approximate posteriors. A number of different numerical experiments illustrate the superiority of the PDE-informed Gaussian priors over more traditional priors.
Paper Structure (28 sections, 2 theorems, 63 equations, 15 figures, 6 tables, 1 algorithm)

This paper contains 28 sections, 2 theorems, 63 equations, 15 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. PDE Inverse problems
  4. Gaussian processes
  5. Gaussian emulators and approximate posteriors
  6. Emulating the forward map
  7. Emulating the log-likelihood
  8. Methodology
  9. Correlated and PDE-informed priors
  10. Computational implementation
  11. MCMC algorithms
  12. Numerical experiments
  13. Examples in one spatial dimension
  14. Constant diffusion coefficient
  15. Baseline model:
  16. ...and 13 more sections

Key Result

Theorem 1

Consider two Gaussian processes $\mathbf{g}_{0} (\boldsymbol{\theta}) \sim \textrm{GP}(\mathbf{m}(\boldsymbol{\theta}),k_p(\boldsymbol{\theta},\boldsymbol{\theta}')I_{{d}_{\mathbf{y}}})$ and $\mathbf{g}_{0,s} (\boldsymbol{\theta}) \sim \textrm{GP}(\mathbf{m}(\boldsymbol{\theta}),k_p(\boldsymbol{\the where $k_{N,p}(\boldsymbol{\theta},\boldsymbol{\theta}')$ is scalar-valued.

Figures (15)

  • Figure 1: True posterior with different ${d}_{\mathbf{y}}$
  • Figure 2: (\ref{['fig:exp0_basic_mean']}) Baseline model mean-based posterior with different $N$. (\ref{['fig:exp0_basic_infl']}) Baseline model marginal posterior with different $N$. (\ref{['fig:exp0_basic_hell']}) Hellinger distance between approximated posteriors and true posteriors when N increases. \ref{['fig:exp0_basic_var']} Average predictive variance of the Gaussian process emulator as $N$ increases. $\mathcal{G}_X$ is the discretised solution $u$ in \ref{['eq:example0']}.
  • Figure 3: (\ref{['fig:exp0_spc_infl']}) Baseline and spatially correlated model marginal posterior for $N=2$. (\ref{['fig:exp0_spc_hell']}) Hellinger distance between approximated posteriors and true posterior as $N$ increases. \ref{['fig:exp0_spc_var']} Average predictive variance of the Gaussian process emulator as $N$ increases. $\mathcal{G}_X$ is the discretised solution $u$ in \ref{['eq:example0']}.
  • Figure 4: Comparison of different models when $N=2$, for PDE model $d_f = 5$. (\ref{['fig:exp0_pde_mean']}) Mean-based posteriors (\ref{['fig:exp0_pde_infl']}) Marginal posteriors (\ref{['fig:exp0_pde_hell']}) Hellinger distance between approximated posteriors and true posterior as $d_f$ increases. (\ref{['fig:exp0_pde_var']}) Average predictive variance of emulator as $d_f$ increases. $\mathcal{G}_X$ is the discretised solution $u$ in \ref{['eq:example0']}.
  • Figure 5: Error between the predictive mean of PDE constrained emulators and the ground truth at observation points ($\boldsymbol{\theta} = \boldsymbol{\theta}^{\dagger}$) for different (a) $d_{f}$ ($\Bar{N} = 10$) (b) $\Bar{N}$ ($d_{f} = 20$)
  • ...and 10 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Lemma 2
  • proof