Advancements in materials play a crucial role in technological progress. However, the process of discovering and developing materials with desired properties is often impeded by substantial experimental costs, extensive resource utilization, and lengthy development periods. To address these challenges, modern approaches often employ machine learning (ML) techniques such as Bayesian Optimization (BO), which streamline the search for optimal materials by iteratively selecting experiments that are most likely to yield beneficial results. However, traditional BO methods, while beneficial, often struggle with balancing the trade-off between exploration and exploitation, leading to sub-optimal performance in material discovery processes. This paper introduces a novel Threshold-Driven UCB-EI Bayesian Optimization (TDUE-BO) method, which dynamically integrates the strengths of Upper Confidence Bound (UCB) and Expected Improvement (EI) acquisition functions to optimize the material discovery process. Unlike the classical BO, our method focuses on efficiently navigating the high-dimensional material design space (MDS). TDUE-BO begins with an exploration-focused UCB approach, ensuring a comprehensive initial sweep of the MDS. As the model gains confidence, indicated by reduced uncertainty, it transitions to the more exploitative EI method, focusing on promising areas identified earlier. The UCB-to-EI switching policy dictated guided through continuous monitoring of the model uncertainty during each step of sequential sampling results in navigating through the MDS more efficiently while ensuring rapid convergence. The effectiveness of TDUE-BO is demonstrated through its application on three different material datasets, showing significantly better approximation and optimization performance over the EI and UCB-based BO methods in terms of the RMSE scores and convergence efficiency, respectively.
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Fault detection and isolation in complex systems are critical to ensure reliable and efficient operation. However, traditional fault detection methods often struggle with issues such as nonlinearity and multivariate characteristics of the time series variables. This article proposes a generative adversarial wavelet neural operator (GAWNO) as a novel unsupervised deep learning approach for fault detection and isolation of multivariate time series processes.The GAWNO combines the strengths of wavelet neural operators and generative adversarial networks (GANs) to effectively capture both the temporal distributions and the spatial dependencies among different variables of an underlying system. The approach of fault detection and isolation using GAWNO consists of two main stages. In the first stage, the GAWNO is trained on a dataset of normal operating conditions to learn the underlying data distribution. In the second stage, a reconstruction error-based threshold approach using the trained GAWNO is employed to detect and isolate faults based on the discrepancy values. We validate the proposed approach using the Tennessee Eastman Process (TEP) dataset and Avedore wastewater treatment plant (WWTP) and N2O emissions named as WWTPN2O datasets. Overall, we showcase that the idea of harnessing the power of wavelet analysis, neural operators, and generative models in a single framework to detect and isolate faults has shown promising results compared to various well-established baselines in the literature.
Within Bayesian nonparametrics, dependent Dirichlet process mixture models provide a highly flexible approach for conducting inference about the conditional density function. However, several formulations of this class make either rather restrictive modelling assumptions or involve intricate algorithms for posterior inference, thus preventing their widespread use. In response to these challenges, we present a flexible, versatile, and computationally tractable model for density regression based on a single-weights dependent Dirichlet process mixture of normal distributions model for univariate continuous responses. We assume an additive structure for the mean of each mixture component and incorporate the effects of continuous covariates through smooth nonlinear functions. The key components of our modelling approach are penalised B-splines and their bivariate tensor product extension. Our proposed method also seamlessly accommodates parametric effects of categorical covariates, linear effects of continuous covariates, interactions between categorical and/or continuous covariates, varying coefficient terms, and random effects, which is why we refer our model as a Dirichlet process mixture of normal structured additive regression models. A noteworthy feature of our method is its efficiency in posterior simulation through Gibbs sampling, as closed-form full conditional distributions for all model parameters are available. Results from a simulation study demonstrate that our approach successfully recovers true conditional densities and other regression functionals in various challenging scenarios. Applications to a toxicology, disease diagnosis, and agricultural study are provided and further underpin the broad applicability of our modelling framework. An R package, \texttt{DDPstar}, implementing the proposed method is publicly available at \url{//bitbucket.org/mxrodriguez/ddpstar}.
We explore Langevin dynamics in the spherical Sherrington-Kirkpatrick model, delving into the asymptotic energy limit. Our approach involves integro-differential equations, incorporating the Crisanti-Horner-Sommers-Cugliandolo-Kurchan equation from spin glass literature, to analyze the system's size and its temperature-dependent phase transition. Additionally, we conduct an average case complexity analysis, establishing hitting time bounds for the bottom eigenvector of a Wigner matrix. Our investigation also includes the power iteration algorithm, examining its average case complexity in identifying the top eigenvector overlap, with comprehensive complexity bounds.
Chiral molecule assignation is crucial for asymmetric catalysis, functional materials, and the drug industry. The conventional approach requires theoretical calculations of electronic circular dichroism (ECD) spectra, which is time-consuming and costly. To speed up this process, we have incorporated deep learning techniques for the ECD prediction. We first set up a large-scale dataset of Chiral Molecular ECD spectra (CMCDS) with calculated ECD spectra. We further develop the ECDFormer model, a Transformer-based model to learn the chiral molecular representations and predict corresponding ECD spectra with improved efficiency and accuracy. Unlike other models for spectrum prediction, our ECDFormer creatively focused on peak properties rather than the whole spectrum sequence for prediction, inspired by the scenario of chiral molecule assignation. Specifically, ECDFormer predicts the peak properties, including number, position, and symbol, then renders the ECD spectra from these peak properties, which significantly outperforms other models in ECD prediction, Our ECDFormer reduces the time of acquiring ECD spectra from 1-100 hours per molecule to 1.5s.
In the present paper, we propose a block variant of the extended Hessenberg process for computing approximations of matrix functions and other problems producing large-scale matrices. Applications to the computation of a matrix function such as f(A)V, where A is an nxn large sparse matrix, V is an nxp block with p<<n, and f is a function are presented. Solving shifted linear systems with multiple right hand sides are also given. Computing approximations of these matrix problems appear in many scientific and engineering applications. Different numerical experiments are provided to show the effectiveness of the proposed method for these problems.
In a topology optimization setting, design-dependent fluidic pressure loads pose several challenges as their direction, magnitude, and location alter with topology evolution. This paper offers a compact 100-line MATLAB code, TOPress, for topology optimization of structures subjected to fluidic pressure loads using the method of moving asymptotes. The code is intended for pedagogical purposes and aims to ease the beginners' and students' learning toward topology optimization with design-dependent fluidic pressure loads. TOPress is developed per the approach first reported in Kumar et al. (Struct Multidisc Optim 61(4):1637-1655, 2020). The Darcy law, in conjunction with the drainage term, is used to model the applied pressure load. The consistent nodal loads are determined from the obtained pressure field. The employed approach facilitates inexpensive computation of the load sensitivities using the adjoint-variable method. Compliance minimization subject to volume constraint optimization problems are solved. The success and efficacy of the code are demonstrated by solving benchmark numerical examples involving pressure loads, wherein the importance of load sensitivities is also demonstrated. TOPress contains six main parts, is described in detail, and is extended to solve different problems. Steps to include a projection filter are provided to achieve loadbearing designs close to~0-1. The code is provided in Appendix~B and can also be downloaded along with its extensions from \url{//github.com/PrabhatIn/TOPress}.
As a pivotal approach in machine learning and data science, manifold learning aims to uncover the intrinsic low-dimensional structure within complex nonlinear manifolds in high-dimensional space. By exploiting the manifold hypothesis, various techniques for nonlinear dimension reduction have been developed to facilitate visualization, classification, clustering, and gaining key insights. Although existing manifold learning methods have achieved remarkable successes, they still suffer from extensive distortions incurred in the global structure, which hinders the understanding of underlying patterns. Scalability issues also limit their applicability for handling large-scale data. Here, we propose a scalable manifold learning (scML) method that can manipulate large-scale and high-dimensional data in an efficient manner. It starts by seeking a set of landmarks to construct the low-dimensional skeleton of the entire data, and then incorporates the non-landmarks into the learned space based on the constrained locally linear embedding (CLLE). We empirically validated the effectiveness of scML on synthetic datasets and real-world benchmarks of different types, and applied it to analyze the single-cell transcriptomics and detect anomalies in electrocardiogram (ECG) signals. scML scales well with increasing data sizes and embedding dimensions, and exhibits promising performance in preserving the global structure. The experiments demonstrate notable robustness in embedding quality as the sample rate decreases.
Quasiperiodic systems, related to irrational numbers, are space-filling structures without decay nor translation invariance. How to accurately recover these systems, especially for non-smooth cases, presents a big challenge in numerical computation. In this paper, we propose a new algorithm, finite points recovery (FPR) method, which is available for both smooth and non-smooth cases, to address this challenge. The FPR method first establishes a homomorphism between the lower-dimensional definition domain of the quasiperiodic function and the higher-dimensional torus, then recovers the global quasiperiodic system by employing interpolation technique with finite points in the definition domain without dimensional lifting. Furthermore, we develop accurate and efficient strategies of selecting finite points according to the arithmetic properties of irrational numbers. The corresponding mathematical theory, convergence analysis, and computational complexity analysis on choosing finite points are presented. Numerical experiments demonstrate the effectiveness and superiority of FPR approach in recovering both smooth quasiperiodic functions and piecewise constant Fibonacci quasicrystals. While existing spectral methods encounter difficulties in accurately recovering non-smooth quasiperiodic functions.
Gradient-enhanced Kriging (GE-Kriging) is a well-established surrogate modelling technique for approximating expensive computational models. However, it tends to get impractical for high-dimensional problems due to the size of the inherent correlation matrix and the associated high-dimensional hyper-parameter tuning problem. To address these issues, a new method, called sliced GE-Kriging (SGE-Kriging), is developed in this paper for reducing both the size of the correlation matrix and the number of hyper-parameters. We first split the training sample set into multiple slices, and invoke Bayes' theorem to approximate the full likelihood function via a sliced likelihood function, in which multiple small correlation matrices are utilized to describe the correlation of the sample set rather than one large one. Then, we replace the original high-dimensional hyper-parameter tuning problem with a low-dimensional counterpart by learning the relationship between the hyper-parameters and the derivative-based global sensitivity indices. The performance of SGE-Kriging is finally validated by means of numerical experiments with several benchmarks and a high-dimensional aerodynamic modeling problem. The results show that the SGE-Kriging model features an accuracy and robustness that is comparable to the standard one but comes at much less training costs. The benefits are most evident for high-dimensional problems with tens of variables.
The multigrid V-cycle method is a popular method for solving systems of linear equations. It computes an approximate solution by using smoothing on fine levels and solving a system of linear equations on the coarsest level. Solving on the coarsest level depends on the size and difficulty of the problem. If the size permits, it is typical to use a direct method based on LU or Cholesky decomposition. In settings with large coarsest-level problems, approximate solvers such as iterative Krylov subspace methods, or direct methods based on low-rank approximation, are often used. The accuracy of the coarsest-level solver is typically determined based on the experience of the users with the concrete problems and methods. In this paper we present an approach to analyzing the effects of approximate coarsest-level solves on the convergence of the V-cycle method for symmetric positive definite problems. Using these results, we derive coarsest-level stopping criterion through which we may control the difference between the approximation computed by a V-cycle method with approximate coarsest-level solver and the approximation which would be computed if the coarsest-level problems were solved exactly. The coarsest-level stopping criterion may thus be set up such that the V-cycle method converges to a chosen finest-level accuracy in (nearly) the same number of V-cycle iterations as the V-cycle method with exact coarsest-level solver. We also utilize the theoretical results to discuss how the convergence of the V-cycle method may be affected by the choice of a tolerance in a coarsest-level stopping criterion based on the relative residual norm.
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