Efficient Surrogate Modeling with Dimensionality Reduction for Robust Design Optimization

Centrala begrepp

The authors propose a reduced dimension variational Gaussian process (RDVGP) surrogate model that efficiently approximates complex computational models with high-dimensional and uncertain inputs. The RDVGP surrogate incorporates dimensionality reduction and Bayesian inference to capture both epistemic and aleatoric uncertainties.

Sammanfattning

The content presents a statistical surrogate modeling approach called the reduced dimension variational Gaussian process (RDVGP) surrogate. The key highlights are:

The RDVGP surrogate is designed to handle high-dimensional and uncertain inputs, which is critical for robust design optimization (RDO) problems.
It employs a latent variable formulation to achieve dimensionality reduction, where the high-dimensional input variables are mapped to a low-dimensional latent space using an orthogonal projection matrix.
Bayesian inference is used to determine the posterior probability density of the surrogate model, which captures both epistemic and aleatoric uncertainties.
Variational Bayes is used to approximate the intractable posterior, by minimizing the Kullback-Leibler divergence between the true posterior and a simpler trial density.
A sparse formulation is introduced to reduce the computational complexity of the RDVGP surrogate, by augmenting the training data with pseudo output variables.
The RDVGP surrogate is shown to outperform standard Gaussian process regression in accurately approximating the true marginal posterior probability density, especially for problems with high input uncertainty.
The RDVGP surrogate is demonstrated on illustrative examples and robust design optimization problems, showcasing its accuracy and versatility.

Statistik

The objective function is J(s) = (s2s1 - 2)^2 sin(12s1 - 4) + 8s1 + s3.
The constraint function is H(s) = -cos(2πs1) - s3 - 0.7.
The input variables s = (s1 s2 s3)^T, where s1 is a design variable with mean μ_s1 ∈ [0, 1] and standard deviation σ_1 ∈ {0.025, 0.05, 0.075}, and s2 and s3 are immutable variables with means 6.0 and 0.0, and standard deviations σ_2 ∈ {0.25, 0.5, 0.75} and 0.1, respectively.

Citat

"The key challenge in Bayesian inference with an uncertain input s is that the posterior density is not a GP with a closed-form solution (even when both the prior probability density and the likelihood density are Gaussians)."
"Variational Bayes provides an optimisation-based formulation for approximating the posterior density and parameters of statistical models."

Viktiga insikter från

Variational Bayesian surrogate modelling with application to robust design optimisation

by Thomas A. Ar... på arxiv.org 04-24-2024

https://arxiv.org/pdf/2404.14857.pdf

Djupare frågor

How can the RDVGP surrogate be extended to handle non-Gaussian input variable distributions

To extend the RDVGP surrogate to handle non-Gaussian input variable distributions, we can incorporate techniques such as copulas or transformation methods. Copulas can be used to model the dependence structure between variables, allowing for the generation of samples from arbitrary marginal distributions. By using copulas to model the joint distribution of the input variables, we can then transform them into Gaussian space for processing by the RDVGP surrogate. Additionally, transformation methods like the Box-Cox transformation or quantile mapping can be applied to normalize the input variables to a Gaussian distribution before feeding them into the surrogate model. These approaches enable the RDVGP surrogate to handle a wider range of input variable distributions beyond Gaussian.

What are the potential limitations of the dimensionality reduction approach used in the RDVGP surrogate, and how could it be further improved

One potential limitation of the dimensionality reduction approach in the RDVGP surrogate is the assumption of linearity in the mapping from the high-dimensional input space to the low-dimensional latent space. If the relationship between the input and output variables is highly non-linear, the linear projection matrix W may not capture the complex interactions effectively. To address this limitation, non-linear dimensionality reduction techniques such as autoencoders or kernel PCA could be explored to capture the non-linear relationships in the data more accurately. Additionally, incorporating adaptive methods to adjust the dimensionality reduction during training based on the data characteristics could enhance the flexibility and performance of the surrogate model.

What other applications beyond robust design optimization could benefit from the RDVGP surrogate modeling approach

The RDVGP surrogate modeling approach has a wide range of applications beyond robust design optimization. Some potential applications include uncertainty quantification in complex systems, inverse problems, model calibration, and surrogate-based optimization in various engineering and scientific domains. For example, in uncertainty quantification, the RDVGP surrogate can efficiently handle input uncertainties and provide accurate predictions of system responses under varying conditions. In inverse problems, the surrogate model can aid in estimating model parameters or inputs from observed data. Moreover, in surrogate-based optimization tasks, the RDVGP surrogate can enable efficient exploration of design spaces and identification of optimal solutions in multi-query scenarios. The versatility and accuracy of the RDVGP approach make it valuable for a broad range of applications requiring fast and reliable surrogate modeling.