Fitting high-dimensional potential energy surface using active subspace and tensor train (AS+TT) method.

Active subspaces for sensitivity analysis and dimension reduction of an integrated hydrologic model

Groundwater models always involve high‐dimensional parameters, which makes computationally tractable data assimilation using surrogate models very challenging. To address this issue, one common practice is to employ dimension reduction (DR) techniques. Nevertheless, traditional DR methods are usually implemented based on prior parameter statistics, that is, without considering the inherent system dynamics. Here, we show that when significant difference in parameter sensitivity exists, further efficiency can be achieved by adopting a supervised DR method, that is, the active subspace (AS) method. To avoid non‐trivial efforts in calculating the gradient information needed in the standard AS method, a cluster‐based gradient‐free AS (GFAS) method is developed in this study. By combining GFAS with Gaussian process regression, a surrogate model for the CPU‐demanding groundwater model can be adaptively constructed to accelerate data assimilation. Furthermore, a compensation scheme is proposed to cope with uncertainty underestimation caused by DR. The developed approach is tested with numerical experiments and field cases, which illustrated that the new approach is more efficient than the previously developed unsupervised ones by incorporating sensitivity information. Although an iterative ensemble smoother is employed in this study, the proposed method can also be used in other data assimilation approaches, such as Markov chain Monte Carlo and ensemble Kalman filter.

Gradient free active subspace construction using Morris screening elementary effects

We present a distributed active subspace method for training surrogate models of complex physical processes with high-dimensional inputs and function valued outputs. Specifically, we represent the model output with a truncated Karhunen–Loeve (KL) expansion, screen the structure of the input space with respect to each KL mode via the active subspace method, and finally form an overall surrogate model of the output by combining surrogates of individual output KL modes. To ensure scalable computation of the gradients of the output KL modes, needed in active subspace discovery, we rely on adjoint-based gradient computation. The proposed method combines benefits of active subspace methods for input dimension reduction and KL expansions used for spectral representation of the output field. We provide a mathematical framework for the proposed method and conduct an error analysis of the mixed KL active subspace approach. Specifically, we provide an error estimate that quantifies errors due to active subspace projection and truncated KL expansion of the output. We demonstrate the numerical performance of the surrogate modeling approach with an application example from biotransport.

Shared low-dimensional subspaces for propagating kinetic uncertainty to multiple outputs

We present a distributed active subspace method for training surrogate models of complex physical processes with high-dimensional inputs and function valued outputs. Specifically, we represent the model output with a truncated Karhunen-Lo\`eve (KL) expansion, screen the structure of the input space with respect to each KL mode via the active subspace method, and finally form an overall surrogate model of the output by combining surrogates of individual output KL modes. To ensure scalable computation of the gradients of the output KL modes, needed in active subspace discovery, we rely on adjoint-based gradient computation. The proposed method combines benefits of active subspace methods for input dimension reduction and KL expansions used for spectral representation of the output field. We provide a mathematical framework for the proposed method and conduct an error analysis of the mixed KL active subspace approach. Specifically, we provide an error estimate that quantifies errors due to active subspace projection and truncated KL expansion of the output. We demonstrate the numerical performance of the surrogate modeling approach with an application example from biotransport.

Global sensitivity analysis and uncertainty quantification of soot formation in an n-dodecane spray flame

Fuel octane number prediction based on topological indices and active subspace method

Reliability analysis and design of a single diode solar cell model using polynomial chaos and active subspace

Cetane number prediction for hydrocarbons from molecular structural descriptors based on active subspace methodology

Aerodynamic shape optimization plays a fundamental role in aircraft design. However, useful parameterizations of shapes for engineering models often result in high-dimensional design spaces which can create challenges for both local and global optimizers. In this paper, we employ an active subspace method (ASM) to discover and exploit low-dimensional, monotonic trends in the quantity of interest as a function of the design variables. The trend enables us to eciently and eectively nd an optimal design in appropriate areas of the design space. We apply this approach to the ONERA-M6 transonic wing, parameterized with 50 Free-Form Deformation (FFD) design variables. Given an initial set of 300 designs, the ASM discovered a low-dimensional linear subspace of the input space that explained the majority of the variability in the drag and lift coecients. This revealed a global trend that we exploited to nd an optimal design with reduced computational cost.

In the early stages of aerospace design, reduced-order models (ROMs) are crucial for minimizing computational costs associated with using physics-rich field information in many-query scenarios requiring multiple evaluations. The intricacy of aerospace design demands the use of high-dimensional design spaces to capture detailed features and design variability accurately. However, these spaces introduce significant challenges, including the curse of dimensionality, which stems from both high-dimensional inputs and outputs necessitating substantial training data and computational effort. To address these complexities, this study introduces a novel multifidelity, parametric, and nonintrusive ROM framework designed for high-dimensional contexts. It integrates machine learning techniques for manifold alignment and dimension reduction—employing proper orthogonal decomposition and model-based active subspace—with multifidelity regression for ROM construction. Our approach is validated through two test cases: the 2D RAE 2822 airfoil and the 3D NASA CRM wing, assessing various fidelity levels, training data ratios, and sample sizes. Compared to the single-fidelity principal component–active subspace (PCAS) method, our multifidelity solution offers improved cost-accuracy benefits and achieves better predictive accuracy with reduced computational demands. Moreover, our methodology outperforms the manifold-aligned ROM method by 50% in handling scenarios with large input dimensions, underscoring its efficacy in addressing the complex challenges of aerospace design.

The fluid dynamic preliminary design of unconventional turbomachinery is customary done with meanline design procedures coupled with gradient-free optimizers. This method features various drawbacks, since it might become computationally expensive, and it does not provide design insights or guidelines to the designer. This work proposes a strategy to abate this disadvantages, namely, the construction of a reduced-order model by means of active sub-spaces, and the use of the surrogate combined with a gradient-based optimizer. The case study is the design optimization of a Organic Rankine Cycle radial inflow turbine. The results show that active subspaces exist for this application, and that it is possible to construct a surrogate with an approximate error of ±1% for the total-to-static efficiency. Additionally, the optimization using the surrogate leads to accurate results and a computational cost at least four times faster. Furthermore, the results reveal that the models for unconventional turbomachinery feature multiple regions containing constrained optima. Active subspace methods thus prove to be a promising alternative for optimization of unconventional turbomachinery.

Modern internal combustion (IC) engines have complex configurations with many parameters to be tuned, and this system complication usually leads to a large amount of engine calibration experiments. In this study, the active subspace method was used to build predictive models for the gas exchange-related parameters, including volumetric efficiency, intake mass flow and pumping loss, and the power-related parameters, including engine torque and engine power. The results show that the predicted outputs fit well with the experimental data, with satisfactory coefficients of determination and average absolute errors (AAE). Further, the contributions and influence directions of the input parameters to the outputs were provided based on a sensitivity analysis, which is consistent with the existing knowledge, and therefore, verifies the reliability of the predictive model built based on the active subspace method. Finally, the relation between the training group size and the prediction performance was explored. It is shown that a reduction, up to 66%, in the training group size is still able to maintain good predictive performances of the models, indicating the substantial capability of the active subspace method to reduce the experimental efforts.

This paper presents a regularization framework that aims to improve the fidelity of Tikhonov inverse solutions. At the heart of the framework is the data-informed regularization idea that only data-uninformed parameters need to be regularized, while the data-informed parameters, on which data and forward model are integrated, should remain untouched. We propose to employ the active subspace method to determine the data-informativeness of a parameter. The resulting framework is thus called a data-informed (DI) active subspace (DIAS) regularization. Four proposed DIAS variants are rigorously analyzed, shown to be robust with the regularization parameter and capable of avoiding polluting solution features informed by the data. They are thus well suited for problems with small or reasonably small noise corruptions in the data. Furthermore, the DIAS approaches can effectively reuse any Tikhonov regularization codes/libraries. Though they are readily applicable for nonlinear inverse problems, we focus on linear problems in this paper in order to gain insights into the framework. Various numerical results for linear inverse problems are presented to verify theoretical findings and to demonstrate advantages of the DIAS framework over the Tikhonov, truncated SVD, and the TSVD-based DI approaches.