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Uncertainty quantification (UQ) is a field of study that focuses on understanding, modeling, and reducing uncertainties in computational models and real-world systems. It is widely used in engineering, physics, finance, climate modeling, and many other areas where decisions are made based on models that are not entirely certain due to inherent complexities and limitations in data or assumptions. UQ seeks to quantify the impact of these uncertainties to make predictions and decisions more robust and reliable.Ìý
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Failure Probability Estimation

Uncertainty quantification (UQ) plays a critical role in predicting the behavior of systems under uncertain conditions, especially when assessing the probability of failure. In engineering, uncertainties arise from two types of errors: epistemic and aleatoric. Epistemic uncertainty stems from incomplete knowledge or lack of data, while aleatoric uncertainty originates from inherent variability in the system or environment. Both must be modeled accurately to ensure robust predictions, particularly in high-stakes scenarios like aerospace and structural engineering, where minimizing the probability of failure is essential.

The ACCESS project sponsored by NASA focuses on estimating the probability of failure for a forebody heatshield during atmospheric re-entry in Titan atmosphere, where uncertainties exist in both input and output data. Input variables include chemical reaction rates and the structural properties of the thermal protection system, while output is tied to the bondline temperature. Exceeding a bondline temperature threshold leads to failure of the system. This creates a high-dimensional challenge due to numerous uncertain inputs and very low failure probabilities analysis. The "curse of dimensionality" exacerbates the problem, as the required number of simulations increases exponentially with the number of input variables using traditional UQ methods. Additionally, obtaining output data from high-fidelity simulations is costly and time-consuming, necessitating a more efficient approach to reduce the number of simulations while ensuring reliability.

To address these challenges, we employ a hybrid approach that combines global and local surrogate models, improving the estimation of failure probabilities. A key aspect of this approach is Christoffel adaptive sampling, which strategically selects sample points within the "buffer zone"—the area surrounding the limit state function. The limit state function represents the boundary between successful and failed system performance. By accurately approximating this critical transition zone, the method enables precise failure probability predictions with fewer high-fidelity simulations, ultimately enhancing the system's reliability analysis.

Related:
Audrey Gaymann

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Bi-fidelity Failure Probability Estimation

Estimating failure probability is one of the key tasks in the field of uncertainty quantification. In this domain, the importance sampling method emerges as an effective estimation strategy but struggles to efficiently determine the biasing distribution. One way to solve this problem is to leverage a less expensive low-fidelity surrogate. Building on the accessibility to such a low-fidelity model and its derivative, we introduce an importance-sampling-based estimator, termed the Langevin bi-fidelity importance sampling (L-BF-IS) estimator, which uses score-function-based sampling algorithms to generate new samples and substantially reduces the mean square error (MSE). It demonstrates strong performance, especially in high-dimensional (>100) input space and limited high-fidelity evaluations. The L-BF-IS estimator's effectiveness is validated through experiments with two synthetic functions and two real-world applications. These real-world applications involve a composite beam, which is represented using a simplified Euler-Bernoulli equation as a low-fidelity surrogate, and a steady-state stochastic heat equation, for which a pre-trained neural operator serves as the low-fidelity surrogate.Ìý

Source: Langevin Bi-fidelity Importance Sampling for Failure Probability Estimation, Nuojin Cheng, Alireza Doostan, ongoing work

Bi-fidelity Variational Auto-encoder

Quantifying the uncertainty of quantities of interest (QoIs) from physical systems is a primary objective in model validation. However, achieving this goal entails balancing the need for computational efficiency with the requirement for numerical accuracy. To address this trade-off, we propose a novel bi-fidelity formulation of variational auto-encoders (BF-VAE) designed to estimate the uncertainty associated with a QoI from low-fidelity (LF) and high-fidelity (HF) samples of the QoI. This model allows for the approximation of the statistics of the HF QoI by leveraging information derived from its LF counterpart. Specifically, we design a bi-fidelity auto-regressive model in the latent space that is integrated within the VAE's probabilistic encoder-decoder structure. An effective algorithm is proposed to maximize the variational lower bound of the HF log-likelihood in the presence of limited HF data, resulting in the synthesis of HF realizations with a reduced computational cost. Additionally, we introduce the concept of the bi-fidelity information bottleneck (BF-IB) to provide an information-theoretic interpretation of the proposed BF-VAE model. Our numerical results demonstrate that the BF-VAE leads to considerably improved accuracy, as compared to a VAE trained using only HF data, when limited HF data is available.

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Bi-fidelity Boosting: Efficient Sub-sampling Algorithm

Least squares regression is a ubiquitous tool for building emulators (a.k.a. surrogate models) of problems across science and engineering for purposes such as design space exploration and uncertainty quantification. When the regression data are generated using an experimental design process (e.g., a quadrature grid), sketching techniques have shown promise to reduce the cost of the construction of the regression model while ensuring accuracy comparable to that of the full data. However, random sketching strategies, such as those based on leverage scores, lead to regression errors that are random and may exhibit large variability. To mitigate this issue, we present a novel boosting approach that leverages cheaper, lower-fidelity data of the problem at hand to identify the best sketch among a set of candidate sketches. This in turn specifies the sketch of the intended high-fidelity model and the associated data. We provide theoretical analyses of this bi-fidelity boosting (BFB) approach and discuss the conditions the low- and high-fidelity data must satisfy for a successful boosting. In doing so, we derive a bound on the residual norm of the BFB sketched solution relating it to its ideal, but computationally expensive, high-fidelity boosted counterpart. Empirical results on both manufactured and PDE data corroborate the theoretical analyses and illustrate the efficacy of the BFB solution in reducing the regression error, as compared to the non-boosted solution.Ìý

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Polynomial Chaos Expansion: Modified Orthogonal Matching Pursuit

Uncertainty Quantification for Laser-Ignition Modeling

Consider a physical phenomenon described by two models: experimental and simulation. The former measures a QoI or the solution's model, but it is expensive to evaluate. The latter approximates the ground truth model but is cheaper to compute. The first model depends on known and unknown variables, and the second depends on the known variables. Modeling the discrepancy between these two models using the same input parameters is the main goal in this scenario. Recent works have used Bayesian modeling and invertible neural networks but have struggled to learn a map between the models. We aim to develop a general framework that models the discrepancy between these two models by adding unknown variables drawn from a known prior distribution to construct a probabilistic map between experiments and simulation. We learn a non-linear probabilistic map via Neural Networks. The learning strategy considers multiple sets of data from the simulation and random variables that minimize the mean-squared error and the Maximum Mean Discrepancy between the distribution on the experimental and simulation models. This map models the uncertainty presented by the hidden variables in the simulation model approximating the experimental model. There are two future avenues. The uncertainty can improve the simulation model. Next, we can use the EigenVI framework inspired by spectral theory that measures the uncertainty via Hermite polynomials on scarce data.
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Uncertainty Quantification in Space Weather

Equatorial plasma bubbles (EPB) are electron density irregularities in the equatorial and low latitude ionosphere F region (~200 km–1000km) covering a wide range of scales from centimeters to thousand kilometers.Ìý EPB is a common space weather phenomenon that can impact the quality of satellite communication, GPS position, and navigation through abrupt changes in amplitudes and phases of the radio wave. Better forecasting the occurrence of EPB is a critical step to decrease or avoid such impacts on human activities. Due to the complexity of the geospace system (i.e., drivers from the solar activity and the lower atmosphere and internal processes), our EPB forecast capability is still far from the operational requirement. Uncertainty quantification (UQ) plays a crucial role in this field, both in providing forecasts with quantified uncertainty and in exploring the sensitivity of forecasts to input drivers. Polynomial chaos expansion (PCE)-based uncertainty quantification and sensitivity analysis, with essential dimensionality reduction for the input drivers, have been applied to the ionosphere-thermosphere system using the Whole Atmosphere Model, WAM-IPE.