Parametrization of stochastic inputs using generative adversarial networks with application in geology

We investigate artificial neural networks as a parametrization tool for stochastic inputs in numerical simulations. We address parametrization from the point of view of emulating the data generating process, instead of explicitly constructing a parametric form to preserve predefined statistics of the data. This is done by training a neural network to generate samples from the data distribution using a recent deep learning technique called generative adversarial networks. By emulating the data generating process, the relevant statistics of the data are replicated. The method is assessed in subsurface flow problems, where effective parametrization of underground properties such as permeability is important due to the high dimensionality and presence of high spatial correlations. We experiment with realizations of binary channelized subsurface permeability and perform uncertainty quantification and parameter estimation. Results show that the parametrization using generative adversarial networks is very effective in preserving visual realism as well as high order statistics of the flow responses, while achieving a dimensionality reduction of two orders of magnitude

Volumetric Particle Tracking Velocimetry (PTV) Uncertainty Quantification

We introduce the first comprehensive approach to determine the uncertainty in volumetric Particle Tracking Velocimetry (PTV) measurements. Volumetric PTV is a state-of-the-art non-invasive flow measurement technique, which measures the velocity field by recording successive snapshots of the tracer particle motion using a multi-camera set-up. The measurement chain involves reconstructing the three-dimensional particle positions by a triangulation process using the calibrated camera mapping functions. The non-linear combination of the elemental error sources during the iterative self-calibration correction and particle reconstruction steps increases the complexity of the task. Here, we first estimate the uncertainty in the particle image location, which we model as a combination of the particle position estimation uncertainty and the reprojection error uncertainty. The latter is obtained by a gaussian fit to the histogram of disparity estimates within a sub-volume. Next, we determine the uncertainty in the camera calibration coefficients. As a final step the previous two uncertainties are combined using an uncertainty propagation through the volumetric reconstruction process. The uncertainty in the velocity vector is directly obtained as a function of the reconstructed particle position uncertainty. The framework is tested with synthetic vortex ring images. The results show good agreement between the predicted and the expected RMS uncertainty values. The prediction is consistent for seeding densities tested in the range of 0.01 to 0.1 particles per pixel. Finally, the methodology is also successfully validated for an experimental test case of laminar pipe flow velocity profile measurement where the predicted uncertainty is within 17% of the RMS error value

An Alarm System For Segmentation Algorithm Based On Shape Model

It is usually hard for a learning system to predict correctly on rare events that never occur in the training data, and there is no exception for segmentation algorithms. Meanwhile, manual inspection of each case to locate the failures becomes infeasible due to the trend of large data scale and limited human resource. Therefore, we build an alarm system that will set off alerts when the segmentation result is possibly unsatisfactory, assuming no corresponding ground truth mask is provided. One plausible solution is to project the segmentation results into a low dimensional feature space; then learn classifiers/regressors to predict their qualities. Motivated by this, in this paper, we learn a feature space using the shape information which is a strong prior shared among different datasets and robust to the appearance variation of input data.The shape feature is captured using a Variational Auto-Encoder (VAE) network that trained with only the ground truth masks. During testing, the segmentation results with bad shapes shall not fit the shape prior well, resulting in large loss values. Thus, the VAE is able to evaluate the quality of segmentation result on unseen data, without using ground truth. Finally, we learn a regressor in the one-dimensional feature space to predict the qualities of segmentation results. Our alarm system is evaluated on several recent state-of-art segmentation algorithms for 3D medical segmentation tasks. Compared with other standard quality assessment methods, our system consistently provides more reliable prediction on the qualities of segmentation results.Comment: Accepted to ICCV 2019 (10 pages, 4 figures

Bayes' Rays: Uncertainty Quantification for Neural Radiance Fields

Neural Radiance Fields (NeRFs) have shown promise in applications like view synthesis and depth estimation, but learning from multiview images faces inherent uncertainties. Current methods to quantify them are either heuristic or computationally demanding. We introduce BayesRays, a post-hoc framework to evaluate uncertainty in any pre-trained NeRF without modifying the training process. Our method establishes a volumetric uncertainty field using spatial perturbations and a Bayesian Laplace approximation. We derive our algorithm statistically and show its superior performance in key metrics and applications. Additional results available at: https://bayesrays.github.io

Nonparametric Uncertainty Quantification for Stochastic Gradient Flows

This paper presents a nonparametric statistical modeling method for quantifying uncertainty in stochastic gradient systems with isotropic diffusion. The central idea is to apply the diffusion maps algorithm to a training data set to produce a stochastic matrix whose generator is a discrete approximation to the backward Kolmogorov operator of the underlying dynamics. The eigenvectors of this stochastic matrix, which we will refer to as the diffusion coordinates, are discrete approximations to the eigenfunctions of the Kolmogorov operator and form an orthonormal basis for functions defined on the data set. Using this basis, we consider the projection of three uncertainty quantification (UQ) problems (prediction, filtering, and response) into the diffusion coordinates. In these coordinates, the nonlinear prediction and response problems reduce to solving systems of infinite-dimensional linear ordinary differential equations. Similarly, the continuous-time nonlinear filtering problem reduces to solving a system of infinite-dimensional linear stochastic differential equations. Solving the UQ problems then reduces to solving the corresponding truncated linear systems in finitely many diffusion coordinates. By solving these systems we give a model-free algorithm for UQ on gradient flow systems with isotropic diffusion. We numerically verify these algorithms on a 1-dimensional linear gradient flow system where the analytic solutions of the UQ problems are known. We also apply the algorithm to a chaotically forced nonlinear gradient flow system which is known to be well approximated as a stochastically forced gradient flow.Comment: Find the associated videos at: http://personal.psu.edu/thb11