Uncertainty Quantification

• Limitations in the popular MCMC
  • Computational expense due to serial model evaluations
  • Gets stuck in a single mode of multi-modal distributions
• Proposed Method: IISGA – iterative importance sample with genetic algorithm [1]
  • Computational speedup through parallel computing à iterative importance sampling
  • Infuse robustness: avoid sample degeneracy and explore multi-model distributions à genetic algorithm
  • Address curse of dimensionality due to many calibrated parameters à inflated likelihood

• Transform high-dimensional spatio-temporal outputs to low-dimensional, uncorrelated latent space using Singular value decomposition (SVD) [1] or Principal components analysis (PCA) [2]
• Transform high-dimensional inputs using the active subspace (AS) method [2,3], which identifies a subspace of important directions within the gradient space of the model
• The PC-AS method combines these mappings from the physics model to a low-dimensional surrogate model as shown below (first diagram)
• The measured outputs are also transformed into the latent space for comparison to the dimension-reduced surrogate model during Bayesian inference/calibration (second diagram)

1. Model discrepancy (output space)
2. Model form error (governing equation)

3.1. Model discrepancy [4]

• Air cycle machine
  • Cooling on airplanes
  • Complex system to model
    • Thermodynamics
    • Fluid dynamics (compressible fluid)
• Predicted the model discrepancy for a new mission (flight)
  • Use deep learning and available test data to build a model for simulation discrepancy
  • Probabilistic assessment of model reliability

3.2. Model form error [5]

• Model form error (MFE) is the difference between the simulation’s governing equations and the underlying physics
• Model discrepancy is at the level of model output, which is a consequence of model form error
• Prediction requires an extrapolation across one or more of the following: input loadings, system configurations, or output quantities/locations
  • An estimated model discrepancy can be used to locally correct the simulation near the training points where the discrepancy was learned (interpolation only)
  • Model discrepancy cannot be extrapolated, but model form error can be, since it is at the level of governing equations
  • An estimated MFE can correct simulation predictions in extrapolated cases as long as the governing equations still hold

• Aims to provide a quantitative assessment of the relative contribution of each uncertainty source to the uncertainty in the model response
• Expanding GSA to consider both aleatory and epistemic uncertainty sources is beneficial in supporting resource allocation decisions
• If the contribution of epistemic uncertainty is found to be significant, then it may be valuable to collect more data to reduce the epistemic uncertainty and thus its contribution to the output uncertainty
• Developed methodologies for information fusion and machine learning for sensitivity analysis using both physics knowledge and experimental data, while accounting for model uncertainty [6]
  • Several physics-informed machine learning (ML) models were developed by leveraging two strategies for incorporating physics knowledge into ML models:
    • Incorporating physics constraints within the loss functions used in training the ML models, and
    • Pre-training an ML model with simulation data and then updating it with experimental data
• Quantification of the contribution of each uncertainty source to the uncertainty in the quantity of interest; modularized global sensitivity analysis [7] and efficient Gaussian mixture copula (GMC) approximation of the joint distribution for fast computation of sensitivity indices [3, 8].
• Time dependent GSA is performed to quantify the contributions of model parameters at various stages of the physical process [9]

Model validation is the process of determining the degree to which a model is an accurate representation of the real world from the perspective of the intended use of the model. [10,11]

Quantitative validation metrics for model credibility and decision-making:

• High-dimensional model responses and corresponding measurements
  • Time-dependent [12]
  • Multi-dimensional [13]
  • Correlated responses, e.g., field response
• Uncertainty in models and measurements
• Limited measurements