From Displacements to Distributions: A Machine-Learning Enabled Framework for Quantifying Uncertainties in Parameters of Computational Models

TL;DR

This work presents novel extensions for combining two frameworks for quantifying both aleatoric and epistemic sources of uncertainties in the modeling of engineered systems with a robust filtering step in LUQ that can learn the most useful quantitative information present in spatio-temporal datasets.

Abstract

This work presents novel extensions for combining two frameworks for quantifying both aleatoric (i.e., irreducible) and epistemic (i.e., reducible) sources of uncertainties in the modeling of engineered systems. The data-consistent (DC) framework poses an inverse problem and solution for quantifying aleatoric uncertainties in terms of pullback and push-forward measures for a given Quantity of Interest (QoI) map. Unfortunately, a pre-specified QoI map is not always available a priori to the collection of data associated with system outputs. The data themselves are often polluted with measurement errors (i.e., epistemic uncertainties), which complicates the process of specifying a useful QoI. The Learning Uncertain Quantities (LUQ) framework defines a formal three-step machine-learning enabled process for transforming noisy datasets into samples of a learned QoI map to enable DC-based inversion. We develop a robust filtering step in LUQ that can learn the most useful quantitative information present in spatio-temporal datasets. The learned QoI map transforms simulated and observed datasets into distributions to perform DC-based inversion. We also develop a DC-based inversion scheme that iterates over time as new spatial datasets are obtained and utilizes quantitative diagnostics to identify both the quality and impact of inversion at each iteration. Reproducing Kernel Hilbert Space theory is leveraged to mathematically analyze the learned QoI map and develop a quantitative sufficiency test for evaluating the filtered data. An illustrative example is utilized throughout while the final two examples involve the manufacturing of shells of revolution to demonstrate various aspects of the presented frameworks.

Figures (18)

• Figure 1: Trommel configuration, hole coverage 35%. (a) Three different sections are indicated with different graylevels. (b) The sensor locations are indicated with red colour. The boundaries at $x = \pm \pi$ are clamped and $y=0$ and $y=2\pi$ are periodic.
• Figure 2: Contour plot of the QoI map $Q$ over $[0,5]^2$ for illustrative example (part I) with the specified QoI map $Q(a,b)=u\left( 4.0,1.0,2.5;a,b\right)$ which represents the wave height at spatial location $(4.0,1.0)$ 2.5 time units after initial water droplet at $(a,b)$ occurs.
• Figure 3: Histogram of TV values between KDEs and exact joint DG densities using 1000 random sets of 200 random iid samples. On the left is the resulting histogram. The middle and right plots represent the resulting histograms for the horizontal and vertical marginals, respectively. The vertical lines in each plot represent the TV metric between the KDEs of 200 i.i.d. samples used for the illustrative example throughout the paper.
• Figure 4: (Left) Color plot of the DCI solution using $10E3$ uniform samples colored by the updated density for the illustrative example part I with data-generating samples overlaid in black. (Middle and Right) Updated density marginals of $a$ and $b$ for the illustrative wave equation example part I along with the original data-generating marginals. The TV metric between the initial predicted distribution and the data-generating distribution is 0.808 while the TV between the updated solution and the data-generating distribution is 0.718 showing minor improvements in learning the data-generating distribution.
• Figure 5: (Left) Color plot of the DCI solution using $10E3$ uniform samples colored by the updated density for the illustrative example part II with data-generating samples overlaid in black. (Middle and Right) Updated density marginals of $a$ and $b$ for the illustrative wave equation example part II using purely temporal data along with the original data-generating marginals. The TV metric between the updated solution and the data-generating distribution is 0.731. For part I of this example, the resulting TV metric was 0.718. While there is an increase in TV metrics, the increase is within expected error tolerances with the quadrature used to compute the TV metric integral suggesting that there are minor changes in the results from parts I and II.

Theorems & Definitions (2)

• Remark 1
• Remark 2