Bayesian Inverse Problems with Conditional Sinkhorn Generative Adversarial Networks in Least Volume Latent Spaces

TL;DR

This paper tackles Bayesian inverse problems in high dimensions by marrying Least Volume Autoencoders (LVAE) with Conditional Sinkhorn GANs (CSGANs). LVAE discovers a minimal, nonlinear latent manifold, enabling robust dimension reduction, while CSGANs learn conditional posteriors in that latent space via the stable Sinkhorn divergence. The approach is demonstrated on a nonlinear ODE source-identification problem and a high-dimensional subsurface permeability inversion, revealing a clear link between the intrinsic dimension of the input condition and posterior uncertainty. The results show efficient, accurate posterior inferences in reduced latent spaces, with LVAE dimensions serving as a diagnostic of problem complexity and posterior spread, offering practical guidance for experimental design and sensing strategies in complex inverse problems.

Abstract

Solving inverse problems in scientific and engineering fields has long been intriguing and holds great potential for many applications, yet most techniques still struggle to address issues such as high dimensionality, nonlinearity and model uncertainty inherent in these problems. Recently, generative models such as Generative Adversarial Networks (GANs) have shown great potential in approximating complex high dimensional conditional distributions and have paved the way for characterizing posterior densities in Bayesian inverse problems, yet the problems' high dimensionality and high nonlinearity often impedes the model's training. In this paper we show how to tackle these issues with Least Volume--a novel unsupervised nonlinear dimension reduction method--that can learn to represent the given datasets with the minimum number of latent variables while estimating their intrinsic dimensions. Once the low dimensional latent spaces are identified, efficient and accurate training of conditional generative models becomes feasible, resulting in a latent conditional GAN framework for posterior inference. We demonstrate the power of the proposed methodology on a variety of applications including inversion of parameters in systems of ODEs and high dimensional hydraulic conductivities in subsurface flow problems, and reveal the impact of the observables' and unobservables' intrinsic dimensions on inverse problems.

Figures (12)

• Figure 1: Flattening the latent set via Least Volume.
• Figure 2: FFT result of $y_1(t)$ samples
• Figure 3: Dimension Reduction Result of KO Dataset
• Figure 4: Heatmap of predictions made by latent CSGAN on 25 test samples. Here the red '$\times$' signs denote the groundtruth initial conditions $\boldsymbol{\xi}$ (or $y_2(0)$ and $y_3(0)$) and the scattered translucent blue dots are the posterior samples. For each case, a Gaussian filter of $\sigma=5$ is applied on the 2D histogram (on a $100\times 100$ grid over $[-0.1, 0.1]\times[-1,1]$) of these 100 predictions to produce the bluish heatmap, in order to better highlight the high density area of predictions.
• Figure 5: Distributions of Minimum $\ell_2$ Prediction Errors of Test Samples