Surrogate Modeling, Uncertainty Quantification and Machine Learning

Uncertainty quantification relies on the ability to draw representative samples from complex computational models of interest. In order to avoid a large number of full model evaluations, surrogate models provide an efficient approximation to the relationship between the input parameters and the output response. The Graham-Brady research group uses both adaptive sampling and machine learning to build these surrogate models.

Abstracts from a few relevant publications

Stochastic collocation approach with adaptive mesh refinement for parametric uncertainty analysis

The presence of a high-dimensional stochastic input domain with discontinuitiesposes major computational challenges in analyzing and quantifying the effects of the uncertainties in a physical system. In this paper, we propose a stochastic collocation method with adaptive mesh refinement (SCAMR) to deal with high dimensional stochastic systems with discontinuities. Specifically, the proposed approach uses generalized polynomial chaos (gPC) expansion with Legendre polynomial basis and solves for the gPC coefficients using the least squares method. It also implements an adaptive mesh (element) refinement strategy which checks for abrupt variations in the output based on a low-order gPC approximation error to track discontinuities or non-smoothness. In addition, the proposed method involves a criterion for checking possible dimensionality reduction and consequently, the decomposition of the original high-dimensional problem to a number of lower-dimensional subproblems. Specifically, this criterion checks all the existing interactions between input parameters of a specific problem based on the high-dimensional model representation (HDMR) method, and therefore automatically provides the subproblems which only involve interacting input parameters. The efficiency of the approach is demonstrated using examples of both smooth and non-smooth problems with number of input parameters up to 500, and the approach is compared against other existing algorithms.

Bhaduri, A., He, Y., Shields, M., Graham-Brady, L., Kirby, R.M. (2018). “Stochastic collocation approach with adaptive mesh refinement for parametric uncertainty analysis,” Journal of Computational Physics, 371: 732-750.

Perturbation-based surrogate models for dynamic failure of brittle materials in a multi-scale and probabilistic context

Localization of failure in many materials is associated with the heterogeneity in the material microstructure. Multiscale models often address this heterogeneity by passing field variables back and forth between amacroscale model and subscale analyses at each integration point. Although this technique is often effective, it can be extremely costly to perform distinct microscale analyses for every integration point in the domain. The proposed work uses a perturbation-based approach, conceptually similar to in situ adaptive tabulation, which provides a straightforward surrogate model that can be orders of magnitude more efficient than the microscale model. The approach is demonstrated specifically for models of dynamic brittle failure, in which crack populations are tracked from one load step to the next. Furthermore, following an approach similar to that used in perturbation-based stochastic finite elements, this technique streamlines the process of probabilistic characterization of the instantaneous stress and the uniaxial compressive strength. Numerical examples show that the approach is accurate and highly efficient when considering random perturbations in both the underlying flaw population and the strain history in these brittle materials.




Liu, J. & Graham-Brady, L. (2016). “Perturbation-based surrogate models for dynamic failure of brittle materials in a multi-scale and probabilistic context,” International Journal for Multiscale Computational Engineering, 14(3): 273-290.

Free energy calculation using space filled design and weighted reconstruction: A modified single sweep approach

A modified single sweep approach is proposed for generating free energy landscapes. The approach replaces the use of temperature-accelerated molecular dynamics (TAMD) to generate centres in collective variable (CV) space at which mean forces are computed using restrained molecular dynamics (MD) simulations with a sequential space-filling design. This approach also modifies the radial basis function reconstruction step of the traditional single sweep approach and proposes a weighted reconstruction of the free energy surface using the previously generated mean forces. The modified approach is compared to the traditional single sweep (SS) approach on the (φ, ψ) dihedral free-energy map of solvated alanine dipeptide (AD). It is found that the new approach results in a more accurate reconstructed free energy than does the traditional approach when compared to the directly-computed reference free energy landscape. It is shown that the increased accuracy of the overall map stems from the improved 1-dimensional space filling (projective) property of the proposed design compared to that of the TAMD generated centres. A further enhancement in the accuracy of the crucial lower energy regions is enabled by the introduction of weights in the reconstruction step that give more importance to lower energy-valued regions.


Bhaduri, A., Gardner, J., Abrams, C., Graham-Brady, L. (2020). “Free energy calculation using space filled design and weighted reconstruction: A modified single sweep approach,” Molecular Simulations, 46(3):193-206.

Other relevant publications

Bhaduri, A., Brandyberry, D., Shields, M., Geubelle, P., Graham-Brady, L. (2020). “On the usefulness of gradient information in surrogate modeling: Application to uncertainty propagation in composite material models,” Probabilistic Engineering Mechanics, 60,103024.

Bhaduri, A. & Graham-Brady, L. (2018). “An efficient adaptive sparse grid collocation method through derivative estimation,” Probabilistic Engineering Mechanics, 51:11-22.

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