Complex mechanical systems often exhibit strongly nonlinear behavior due to the presence of nonlinearities in the energy dissipation mechanisms, material constitutive relationships, or geometric/connectivity mechanics. Numerical modeling of these systems leads to nonlinear full-order models that possess an underlying Lagrangian structure. This work proposes a Lagrangian operator inference method enhanced with structure-preserving machine learning to learn nonlinear reduced-order models (ROMs) of nonlinear mechanical systems. This two-step approach first learns the best-fit linear Lagrangian ROM via Lagrangian operator inference and then presents a structure-preserving machine learning method to learn nonlinearities in the reduced space. The proposed approach can learn a structure-preserving nonlinear ROM purely from data, unlike the existing operator inference approaches that require knowledge about the mathematical form of nonlinear terms. From a machine learning perspective, it accelerates the training of the structure-preserving neural network by providing an informed prior, and it reduces the computational cost of the network training by operating on the reduced space. The method is first demonstrated on two simulated examples: a conservative nonlinear rod model and a two-dimensional nonlinear membrane with nonlinear internal damping. Finally, the method is demonstrated on an experimental dataset consisting of digital image correlation measurements taken from a lap-joint beam structure from which a predictive model is learned that captures amplitude-dependent frequency and damping characteristics accurately. The numerical results demonstrate that the proposed approach yields generalizable nonlinear ROMs that exhibit bounded energy error, capture the nonlinear characteristics reliably, and provide accurate long-time predictions outside the training data regime.
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機器學習（Machine Learning）是一個研究計算學習方法的國際論壇。該雜志發表文章，報告廣泛的學習方法應用于各種學習問題的實質性結果。該雜志的特色論文描述研究的問題和方法，應用研究和研究方法的問題。有關學習問題或方法的論文通過實證研究、理論分析或與心理現象的比較提供了堅實的支持。應用論文展示了如何應用學習方法來解決重要的應用問題。研究方法論文改進了機器學習的研究方法。所有的論文都以其他研究人員可以驗證或復制的方式描述了支持證據。論文還詳細說明了學習的組成部分，并討論了關于知識表示和性能任務的假設。
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We consider stochastic gradient descents on the space of large symmetric matrices of suitable functions that are invariant under permuting the rows and columns using the same permutation. We establish deterministic limits of these random curves as the dimensions of the matrices go to infinity while the entries remain bounded. Under a ``small noise'' assumption the limit is shown to be the gradient flow of functions on graphons whose existence was established in~\cite{oh2021gradient}. We also consider limits of stochastic gradient descents with added properly scaled reflected Brownian noise. The limiting curve of graphons is characterized by a family of stochastic differential equations with reflections and can be thought of as an extension of the classical McKean-Vlasov limit for interacting diffusions to the graphon setting. The proofs introduce a family of infinite-dimensional exchangeable arrays of reflected diffusions and a novel notion of propagation of chaos for large matrices of diffusions converging to such arrays in a suitable sense.
The main objective of this work is to explore the possibility of incorporating radiomic information from multiple lesions into survival models. We hypothesise that when more lesions are present, their inclusion can improve model performance, and we aim to find an optimal strategy for using multiple distinct regions in modelling. The idea of using multiple regions of interest (ROIs) to extract radiomic features for predictive models has been implemented in many recent works. However, in almost all studies, analogous regions were segmented according to particular criteria for all patients -- for example, the primary tumour and peritumoral area, or subregions of the primary tumour. They can be included in a model in a straightforward way as additional features. A more interesting scenario occurs when multiple distinct ROIs are present, such as multiple lesions in a regionally disseminated cancer. Since the number of such regions may differ between patients, their inclusion in a model is non-trivial and requires additional processing steps. We proposed several methods of handling multiple ROIs representing either ROI or risk aggregation strategy, compared them to a published one, and evaluated their performance in different classes of survival models in a Monte Carlo Cross-Validation scheme. We demonstrated the effectiveness of the methods using a cohort of 115 non-small cell lung cancer patients, for whom we predicted the metastasis risk based on features extracted from PET images in original resolution or interpolated to CT image resolution. For both feature sets, incorporating all available lesions, as opposed to a singular ROI representing the primary tumour, allowed for considerable improvement of predictive ability regardless of the model.
We propose a coefficient that measures dependence in paired samples of functions. It has properties similar to the Pearson correlation, but differs in significant ways: 1) it is designed to measure dependence between curves, 2) it focuses only on extreme curves. The new coefficient is derived within the framework of regular variation in Banach spaces. A consistent estimator is proposed and justified by an asymptotic analysis and a simulation study. The usefulness of the new coefficient is illustrated on financial and and climate functional data.
For boundary value problem of an elliptic equation with variable coefficients describing the physical field distribution in inhomogeneous media, the Levi function can represent the solution in terms of volume and surface potentials, with the drawback that the volume potential involving in the solution expression requires heavy computational costs as well as the solvability of the integral equations with respect to the density pair. We introduce an modified integral expression for the solution to an elliptic equation in divergence form under the Levi function framework. The well-posedness of the linear integral system with respect to the density functions to be determined is rigorously proved. Based on the singularity decomposition for the Levi function, we propose two schemes to deal with the volume integrals so that the density functions can be solved efficiently. One method is an adaptive discretization scheme (ADS) for computing the integrals with continuous integrands, leading to the uniform accuracy of the integrals in the whole domain, and consequently the efficient computations for the density functions. The other method is the dual reciprocity method (DRM) which is a meshless approach converting the volume integrals into boundary integrals equivalently by expressing the volume density as the combination of the radial basis functions determined by the interior grids. The proposed schemes are justified numerically to be of satisfactory computation costs. Numerical examples in 2-dimensional and 3-dimensional cases are presented to show the validity of the proposed schemes.
Solid-state dewetting (SSD), a widespread phenomenon in solid-solid-vapor system, could be used to describe the accumulation of solid thin films on the substrate. In this work, we consider the sharp interface model for axisymmetric SSD with anisotropic surface energy. By introducing two types of surface energy matrices from the anisotropy functions,we aim to design two structure-preserving algorithms for the axisymmetric SSD. The newly designed schemes are applicable to a broader range of anisotropy functions, and we can theoretically prove their volume conservation and energy stability. In addition, based on a novel weak formulation for the axisymmetric SSD, we further build another two numerical schemes that have good mesh properties. Finally, numerous numerical tests are reported to showcase the accuracy and efficiency of the numerical methods.
Continuous p-dispersion problems with and without boundary constraints are NP-hard optimization problems with numerous real-world applications, notably in facility location and circle packing, which are widely studied in mathematics and operations research. In this work, we concentrate on general cases with a non-convex multiply-connected region that are rarely studied in the literature due to their intractability and the absence of an efficient optimization model. Using the penalty function approach, we design a unified and almost everywhere differentiable optimization model for these complex problems and propose a tabu search-based global optimization (TSGO) algorithm for solving them. Computational results over a variety of benchmark instances show that the proposed model works very well, allowing popular local optimization methods (e.g., the quasi-Newton methods and the conjugate gradient methods) to reach high-precision solutions due to the differentiability of the model. These results further demonstrate that the proposed TSGO algorithm is very efficient and significantly outperforms several popular global optimization algorithms in the literature, improving the best-known solutions for several existing instances in a short computational time. Experimental analyses are conducted to show the influence of several key ingredients of the algorithm on computational performance.
Detecting differences in gene expression is an important part of single-cell RNA sequencing experiments, and many statistical methods have been developed for this aim. Most differential expression analyses focus on comparing expression between two groups (e.g., treatment vs. control). But there is increasing interest in multi-condition differential expression analyses in which expression is measured in many conditions, and the aim is to accurately detect and estimate expression differences in all conditions. We show that directly modeling single-cell RNA-seq counts in all conditions simultaneously, while also inferring how expression differences are shared across conditions, leads to greatly improved performance for detecting and estimating expression differences compared to existing methods. We illustrate the potential of this new approach by analyzing data from a single-cell experiment studying the effects of cytokine stimulation on gene expression. We call our new method "Poisson multivariate adaptive shrinkage", and it is implemented in an R package available online at //github.com/stephenslab/poisson.mash.alpha.
This paper addresses the numerical implementation of the transparent boundary condition (TBC) and its various approximations for the free Schr\"odinger equation on a rectangular computational domain. In particular, we consider the exact TBC and its spatially local approximation under high frequency assumption along with an appropriate corner condition. For the spatial discretization, we use a Legendre-Galerkin spectral method where Lobatto polynomials serve as the basis. Within variational formalism, we first arrive at the time-continuous dynamical system using spatially discrete form of the initial boundary-value problem incorporating the boundary conditions. This dynamical system is then discretized using various time-stepping methods, namely, the backward-differentiation formula of order 1 and 2 (i.e., BDF1 and BDF2, respectively) and the trapezoidal rule (TR) to obtain a fully discrete system. Next, we extend this approach to the novel Pad\'e based implementation of the TBC presented by Yadav and Vaibhav [arXiv:2403.07787(2024)]. Finally, several numerical tests are presented to demonstrate the effectiveness of the boundary maps (incorporating the corner conditions) where we study the stability and convergence behavior empirically.
The transparent boundary condition for the free Schr\"{o}dinger equation on a rectangular computational domain requires implementation of an operator of the form $\sqrt{\partial_t-i\triangle_{\Gamma}}$ where $\triangle_{\Gamma}$ is the Laplace-Beltrami operator. It is known that this operator is nonlocal in time as well as space which poses a significant challenge in developing an efficient numerical method of solution. The computational complexity of the existing methods scale with the number of time-steps which can be attributed to the nonlocal nature of the boundary operator. In this work, we report an effectively local approximation for the boundary operator such that the resulting complexity remains independent of number of time-steps. At the heart of this algorithm is a Pad\'e approximant based rational approximation of certain fractional operators that handles corners of the domain adequately. For the spatial discretization, we use a Legendre-Galerkin spectral method with a new boundary adapted basis which ensures that the resulting linear system is banded. A compatible boundary-lifting procedure is also presented which accommodates the segments as well as the corners on the boundary. The proposed novel scheme can be implemented within the framework of any one-step time marching schemes. In particular, we demonstrate these ideas for two one-step methods, namely, the backward-differentiation formula of order 1 (BDF1) and the trapezoidal rule (TR). For the sake of comparison, we also present a convolution quadrature based scheme conforming to the one-step methods which is computationally expensive but serves as a golden standard. Finally, several numerical tests are presented to demonstrate the effectiveness of our novel method as well as to verify the order of convergence empirically.
Inference for functional linear models in the presence of heteroscedastic errors has received insufficient attention given its practical importance; in fact, even a central limit theorem has not been studied in this case. At issue, conditional mean estimates have complicated sampling distributions due to the infinite dimensional regressors, where truncation bias and scaling issues are compounded by non-constant variance under heteroscedasticity. As a foundation for distributional inference, we establish a central limit theorem for the estimated conditional mean under general dependent errors, and subsequently we develop a paired bootstrap method to provide better approximations of sampling distributions. The proposed paired bootstrap does not follow the standard bootstrap algorithm for finite dimensional regressors, as this version fails outside of a narrow window for implementation with functional regressors. The reason owes to a bias with functional regressors in a naive bootstrap construction. Our bootstrap proposal incorporates debiasing and thereby attains much broader validity and flexibility with truncation parameters for inference under heteroscedasticity; even when the naive approach may be valid, the proposed bootstrap method performs better numerically. The bootstrap is applied to construct confidence intervals for centered projections and for conducting hypothesis tests for the multiple conditional means. Our theoretical results on bootstrap consistency are demonstrated through simulation studies and also illustrated with a real data example.
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