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Geometric Autoencoder Priors for Bayesian Inversion: Learn First Observe Later

Arnaud Vadeboncoeur, Gregory Duthé, Mark Girolami, Eleni Chatzi

TL;DR

This work introduces Geometric Autoencoders for Bayesian Inversion (GABI), a framework for learning geometry-aware generative models of physical responses that serve as highly informative geometry-conditioned priors for Bayesian inversion.

Abstract

Uncertainty Quantification (UQ) is paramount for inference in engineering. A common inference task is to recover full-field information of physical systems from a small number of noisy observations, a usually highly ill-posed problem. Sharing information from multiple distinct yet related physical systems can alleviate this ill-possendess. Critically, engineering systems often have complicated variable geometries prohibiting the use of standard multi-system Bayesian UQ. In this work, we introduce Geometric Autoencoders for Bayesian Inversion (GABI), a framework for learning geometry-aware generative models of physical responses that serve as highly informative geometry-conditioned priors for Bayesian inversion. Following a ''learn first, observe later'' paradigm, GABI distills information from large datasets of systems with varying geometries, without requiring knowledge of governing PDEs, boundary conditions, or observation processes, into a rich latent prior. At inference time, this prior is seamlessly combined with the likelihood of a specific observation process, yielding a geometry-adapted posterior distribution. Our proposed framework is architecture agnostic. A creative use of Approximate Bayesian Computation (ABC) sampling yields an efficient implementation that utilizes modern GPU hardware. We test our method on: steady-state heat over rectangular domains; Reynold-Averaged Navier-Stokes (RANS) flow around airfoils; Helmholtz resonance and source localization on 3D car bodies; RANS airflow over terrain. We find: the predictive accuracy to be comparable to deterministic supervised learning approaches in the restricted setting where supervised learning is applicable; UQ to be well calibrated and robust on challenging problems with complex geometries.

Geometric Autoencoder Priors for Bayesian Inversion: Learn First Observe Later

TL;DR

This work introduces Geometric Autoencoders for Bayesian Inversion (GABI), a framework for learning geometry-aware generative models of physical responses that serve as highly informative geometry-conditioned priors for Bayesian inversion.

Abstract

Uncertainty Quantification (UQ) is paramount for inference in engineering. A common inference task is to recover full-field information of physical systems from a small number of noisy observations, a usually highly ill-posed problem. Sharing information from multiple distinct yet related physical systems can alleviate this ill-possendess. Critically, engineering systems often have complicated variable geometries prohibiting the use of standard multi-system Bayesian UQ. In this work, we introduce Geometric Autoencoders for Bayesian Inversion (GABI), a framework for learning geometry-aware generative models of physical responses that serve as highly informative geometry-conditioned priors for Bayesian inversion. Following a ''learn first, observe later'' paradigm, GABI distills information from large datasets of systems with varying geometries, without requiring knowledge of governing PDEs, boundary conditions, or observation processes, into a rich latent prior. At inference time, this prior is seamlessly combined with the likelihood of a specific observation process, yielding a geometry-adapted posterior distribution. Our proposed framework is architecture agnostic. A creative use of Approximate Bayesian Computation (ABC) sampling yields an efficient implementation that utilizes modern GPU hardware. We test our method on: steady-state heat over rectangular domains; Reynold-Averaged Navier-Stokes (RANS) flow around airfoils; Helmholtz resonance and source localization on 3D car bodies; RANS airflow over terrain. We find: the predictive accuracy to be comparable to deterministic supervised learning approaches in the restricted setting where supervised learning is applicable; UQ to be well calibrated and robust on challenging problems with complex geometries.

Paper Structure

This paper contains 40 sections, 3 theorems, 16 equations, 17 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setup
  3. Related Works
  4. Direct Map Inversion
  5. Forward and Inverse Problems on Graphs
  6. Statistical Autoencoders
  7. Notation
  8. Methodology
  9. Posterior from a Pushforward Prior
  10. Geometric Autoencoder Priors for Bayesian Inversion (GABI)
  11. Observational Noise Estimation
  12. Implementation
  13. Numerics
  14. Rectangles -- Steady-State Heat
  15. Airfoils -- Flow Field
  16. ...and 25 more sections

Key Result

Lemma 2.1

Let $(\mathcal{U}, \mathcal{F}, \mathbb{P}_{u})$, and $(\mathcal{Z}, \mathcal{S}, \mathbb{P}_z)$ be measure spaces and $g:\mathcal{Z}\rightarrow\mathcal{U}$ be $(\mathcal{S, F})$-measurable. Furthermore, let $\mathbb{P}_{u|y}$ be the Bayesian posterior on $\mathcal{U}$ with likelihood proportional t where $\mathrm{d} \mathbb{P}_{z|y} (z) = 1/Z(y) \exp(-{\Phi}(g (z);y)) \mathrm{d} \mathbb{P}_z(z)$

Figures (17)

  • Figure 1: Four out of 1k geometries and solutions in dataset for the steady-state heat problem.
  • Figure 2: (a) Four selected query locations for the sampled predictive solutions given data, the full field estimations are in Figure \ref{['fig:data_heat_rect_pred']}. (b) The corresponding histograms for the predictive posterior at specified query locations; the dashed black line indicates the ground truth for each histogram.
  • Figure 3: Comparison of ground truth (GT), inferred mean, error, and standard deviation for pressure ($p$). The red dots correspond to the observation location for the pressure. Full results including reconstruction of vertical and horizontal velocity fields are in Figure \ref{['fig:airfoil_results_full']}
  • Figure 4: Ground truth, inferred mean, stddev., and error for ampltitude $u$, and forcing $f$. In magenta are the observation locations.
  • Figure 5: Ground truth, inferred mean, error, and standard deviation for pressure and the magnitude of the velocity vector, ($\|{v}\|$) across two terrains. The red dots correspond to the observation locations.
  • ...and 12 more figures

Theorems & Definitions (6)

  • Lemma 2.1
  • Theorem 2.2
  • Corollary 2.3
  • proof
  • proof
  • proof