Reliability analysis of structures by multiple sparse polynomial chaos expansion based adaptive metamodel

Abstract In high-stakes scientific contexts, explainable AI is crucial for deriving meaningful insights from complex tabular data. A formidable challenge is ensuring both rigorous statistical guarantees and clear interpretability in feature extraction. While traditional methods like PCA are limited by linear assumptions , powerful neural network approaches often lack the transparency required in scientific domains. To address this gap, we introduce \TheName{}, a novel self-supervised learning pipeline that makes nonlinear feature interactions interpretable. \TheName{} marries the power of kernel principal components for capturing complex dependencies with a sparse, principled polynomial representation to achieve clear interpretability with statistical rigor. Our approach bridges data-driven complexity and statistical reliability via three stages. First, it generates self-supervised signals using kernel principal components to model complex patterns. Second, it distills these signals into sparse polynomial functions for interpretability. Third, it applies a multi-objective knockoff selection procedure with significance testing to rigorously identify important features while controlling the false discovery rate (FDR). Extensive experiments on diverse real-world datasets demonstrate the effectiveness of \TheName{}, consistently surpassing other methods in feature selection for regression and classification tasks. In particular, applications on physics datasets highlight the ability of the proposed method to produce scientifically valid and interpretable explanations, reinforcing its practical utility and the critical role of explainability in AI for science.

Probabilistic stability analysis of bolt-reinforced rock slopes considering corrosion based on sparse polynomial chaos expansions

Active sparse polynomial chaos expansion for reliability analysis of underground structures considering the spatial variability of soil properties

Uncertainty quantification of geomagnetically induced current based on improved sparse polynomial chaos expansion

Parallel evaluation of sparse polynomials; based on Estrin’s algorithm: conception and analysis of schedulings

To meet the increasing demands for high thrust-to-weight ratios and low specific fuel consumption in modern aero engines, compressor designs are trending toward higher stage loading to reduce stage counts. However, these advancements introduce challenges related to low Reynolds number (Re) effect and elevated sensitivity to uncertainty. This study proposes a data-driven sparse arbitrary polynomial chaos expansion (DD-SAPCE) method to investigate the impact of inflow uncertainty on the aerodynamic performance of a transonic compressor rotor across different Re conditions. The DD-SAPCE framework effectively mitigates the curse of dimensionality inherent in conventional polynomial chaos approaches, reducing computational cost by nearly 60% while achieving comparable accuracy. Furthermore, the method supports modeling with discrete data, making it suitable for practical engineering applications. Statistical comparisons reveal that low Re conditions intensify the variability in mass flow while reducing the dispersion of isentropic efficiency. These effects are accompanied by enhanced nonlinear sensitivity of aerodynamic performance to inflow uncertainty. Spanwise analyses further show that changes in blade loading and shock structures are the key contributors to performance fluctuations. Furthermore, global and local sensitivity analyses based on the Shapley method identify inlet total pressure, back pressure, and flow angle as the dominant factors influencing the aerodynamic performance sensitivity to inlet flow disturbances. These findings offer critical insights for the performance influence mechanism and robust aerodynamic optimization of transonic compressor systems.

Chebyshev varieties are algebraic varieties parametrized by Chebyshev polynomials or their multivariate generalizations. We determine the dimension, degree, singular locus and defining equations of these varieties. We explain how they play the role of toric varieties in sparse polynomial root finding, when monomials are replaced by Chebyshev polynomials. We present numerical root finding algorithms that exploit our results.

Stochastic extended finite element analysis based on sparse polynomial chaos expansion

Adaptive bayesian sparse polynomial chaos expansion for voltage balance of an isolated microgrid at peak load

On Solving Sparse Polynomial Factorization Related Problems

In renewable energy generation, uncertainties mainly refer to power output fluctuations caused by the intermittency, variability, and forecasting errors of wind and photovoltaic power. These uncertainties have adverse effects on the secure operation of the power systems. Probabilistic load supply capability (LSC) serves as an effective perspective for evaluating power system security under uncertainties. Therefore, this paper studies the influence of renewable energy generation on probabilistic LSC to quantify the impact of these uncertainties on the secure operation of the power systems. Global sensitivity analysis (GSA) is introduced for the first time into probabilistic LSC evaluation. It can quantify the impact of renewable energy generation on the system’s LSC and rank the importance of renewable energy power stations based on GSA indices. GSA necessitates multiple rounds of probabilistic LSC evaluation, which is computationally intensive. To address it, this paper introduces a novel probabilistic repeated power flow (PRPF) algorithm, which employs a basis-adaptive sparse polynomial chaos expansion (BASPCE) model as a surrogate model for the original repeated power flow model, thereby accelerating the probabilistic LSC evaluation. Finally, the effectiveness of the proposed methods is verified through case studies on the IEEE 39-bus system. This study provides a practical approach for analyzing the impact of renewable generation uncertainties on power system security, contributing to more informed planning and operational decisions.

Abstract This paper studies the sparse Moment-SOS hierarchy of relaxations for solving sparse polynomial optimization problems. We show that this sparse hierarchy is tight if and only if the objective can be written as a sum of sparse nonnegative polynomials, each of which belongs to the sum of the ideal and quadratic module generated by the corresponding sparse constraints. Based on this characterization, we give several sufficient conditions for the sparse Moment-SOS hierarchy to be tight. In particular, we show that this sparse hierarchy is tight under some assumptions such as convexity, optimality conditions or finiteness of constraining sets.

More than three decades ago, after a series of results, Kaltofen and Trager in 1990 designed a randomized polynomial time algorithm for factorization of multivariate circuits. Derandomizing this algorithm, even for restricted circuit classes, is an important open problem. In particular, the case of s -sparse polynomials, having individual degree d =O(1), has been very well studied (e.g., by Shpilka and Volkovich in 2010, Vokovich in 2017, and Bhargava et al. in 2018). We give a complete derandomization for this class assuming that the input is a symmetric polynomial over rationals. Generally, we prove an s poly( d ) -sparsity bound for the factors of symmetric polynomials over any field. To factor f , we use techniques from convex geometry and exploit symmetry (only) in the Newton polytope of f . We prove a crucial result about convex polytopes, by introducing the concept of “low min-entropy,” which might also be of independent interest.

The random fluctuations in inlet flow represent a common uncertainty in aero-engine compressors, necessitating the control of its effects through blade optimization design. To account for the impact of inlet flow fluctuations on performance in blade design optimization, an efficient multi-objective adaptive robust aerodynamic design optimization (ARADO) method is proposed. The optimization method employs a novel sparse polynomial chaos expansion (PCE) and the advanced noisy Gaussian process regression (NGPR) technique is used to establish an initial stochastic surrogate model (SSM) containing statistical moments of aerodynamic performance. By introducing advanced sparse signal processing concepts, the sparce PCE significantly enhances the efficiency of acquiring each training sample for SSM. During the optimization process, the initial SSM autonomously updates based on historical optimization data, without requiring high precision across the entire design space. Compared to traditional model-based aerodynamic robust optimizations, the proposed ARADO method exhibits a faster convergence speed and achieves a superior average level of the optimal solution set. It also better balances various optimization objectives, concentrating the space distribution of optimal solutions closer to the average level. Ultimately, the ARADO is applied to the aerodynamic robust design of a high-load compressor airfoil across all operating incidences. The optimization results enhance aerodynamic performance while reducing performance diversity, thus aligning more closely with practical engineering requirements. Through data analysis of the optimal solutions, robust design guidelines for blade aerodynamic shapes are obtained, along with insights into the flow mechanisms that enhance aerodynamic robustness.

Sparse Polynomial Optimization with Unbounded Sets

A new Bayesian sparse polynomial chaos expansion based on Gaussian scale mixture prior model

Abstract This study introduces a novel approach for the prediction and experimental validation of microhardness in High Entropy Alloy (HEA) reinforced Aluminium composites using advanced Machine Learning (ML) techniques. For the first time, five ML models, including XGBoost, Stacking Regressor, Quantile Random Forest, Gaussian Process, and Sparse Polynomial Regression were employed to predict the microhardness of Al/HEA composites. The models were trained using an 80:20 train-test split, with hyperparameters optimized via random search and cross-validation. Among the models, XGBoost delivered the highest accuracy, with an R² value of 0.91, and predicted a microhardness of 177 HV. To validate these predictions experimentally, LM13 aluminium alloy was selected as the matrix material, reinforced with HEA (AlTiVZrCrMo) synthesized through stir casting route. Scanning Electron Microscopy (SEM) analysis confirmed uniform reinforcement distribution, contributing to an experimental microhardness of 168 HV, a 45% improvement over the base alloy. The minimal 5.8% deviation between predicted and experimental values demonstrates the effectiveness of ML in predicting material properties, significantly reducing the need for extensive experimental work.

We present a C implementation of Kaltofen and Yagati's fast algorithm for solving a transposed Vandermonde system of equations modulo a prime p. Such systems arise in sparse polynomial interpolation algorithms. We compare the speed of our C code with a Maple implementation of Kaltofen and Yagati's algorithm and also with a C implementation of Zippel's O(n2) algorithm. The paper describes Zippel's algorithm and derives Kaltofen and Yagati's algorithm from Zippel's algorithm. A version of this paper was presented at the LALO60 Conference in London, Ontario in July 2024.

In this work, an algorithm is introduced based on polynomial chaos expansions (PCEs) to tackle uncertainty quantification problems related to grating filters. Our approach adaptively constructs anisotropic PC models for the quantities of interest, accommodating varying polynomial orders. It exploits the sparsity of the PCE coefficients, which are computed using the least angles regression (LARS) sparse solver, leading to a highly efficient process. In addition, optimal experiments are designed that take advantage of the local variance of the samples, further improving the reliability of the computations. The method is applied to the uncertainty quantification of a typical resonant grating filter, demonstrating its superior efficiency, which is more than 2 orders of magnitude less usage of time demanding full-wave solvers, compared to reference techniques like Monte Carlo (MC). Specifically, the proposed method required approximately 25 calls to a full-wave solver, compared to the 20,000 calls needed by the MC approach. In addition, the constructed PCE model can very efficiently generate samples of the grating filter's quantities of interest, compared to generation by full-wave solvers, which can be used alongside a stochastic optimizer to optimize the grating filter's performance with respect to its design variables. Furthermore, improved optimization results are observed when the presented PCE algorithm is combined with Kriging interpolation.