Measurements of systems taken along a continuous functional dimension, such as time or space, are ubiquitous in many fields, from the physical and biological sciences to economics and engineering.Such measurements can be viewed as realisations of an underlying smooth process sampled over the continuum. However, traditional methods for independence testing and causal learning are not directly applicable to such data, as they do not take into account the dependence along the functional dimension. By using specifically designed kernels, we introduce statistical tests for bivariate, joint, and conditional independence for functional variables. Our method not only extends the applicability to functional data of the HSIC and its d-variate version (d-HSIC), but also allows us to introduce a test for conditional independence by defining a novel statistic for the CPT based on the HSCIC, with optimised regularisation strength estimated through an evaluation rejection rate. Our empirical results of the size and power of these tests on synthetic functional data show good performance, and we then exemplify their application to several constraint- and regression-based causal structure learning problems, including both synthetic examples and real socio-economic data.
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In this paper, the surface of revolution discrete element method (SR-DEM) is introduced to simulate systems of particles with closed surfaces of revolution. Due to the cylindrical symmetry of a surface of revolution, the geometry of any cross-section about the axis of rotation remains the same. Taking advantage of this geometric feature, a node-to-cross-section contact algorithm is proposed for efficient contact detection between particles with a surface of revolution. In our SR-DEM framework, the contact algorithm is realized in a master-slave fashion: the master particle is approximated by its surface nodes, while the slave particle is represented by a signed distance field (SDF) of the cross-section about the axis of rotation. This hybrid formulation in both 2D and 3D space allows a very efficient contact calculation yet relatively simple code implementation. We then apply SR-DEM to simulate particle-particle, particle-wall impact, granular packing in a cylindrical container, and tablets in a rotating drum, to demonstrate SR-DEM's ability to predict the post-impact velocities, packing porosity, and dynamic angle of repose, respectively. Finally, we suggest a simple approach to find an optimal surface resolution, by increasing the number of surface nodes until some of the bulk properties that could characterize the system converge.
This work deals with an inverse source problem for the biharmonic wave equation. A two-stage numerical method is proposed to identify the unknown source from the multi-frequency phaseless data. In the first stage, we introduce some artificially auxiliary point sources to the inverse source system and establish a phase retrieval formula. Theoretically, we point out that the phase can be uniquely determined and estimate the stability of this phase retrieval approach. Once the phase information is retrieved, the Fourier method is adopted to reconstruct the source function from the phased multi-frequency data. The proposed method is easy-to-implement and there is no forward solver involved in the reconstruction. Numerical experiments are conducted to verify the performance of the proposed method.
This paper explores an iterative coupling approach to solve linear thermo-poroelasticity problems, with its application as a high-fidelity discretization utilizing finite elements during the training of projection-based reduced order models. One of the main challenges in addressing coupled multi-physics problems is the complexity and computational expenses involved. In this study, we introduce a decoupled iterative solution approach, integrated with reduced order modeling, aimed at augmenting the efficiency of the computational algorithm. The iterative coupling technique we employ builds upon the established fixed-stress splitting scheme that has been extensively investigated for Biot's poroelasticity. By leveraging solutions derived from this coupled iterative scheme, the reduced order model employs an additional Galerkin projection onto a reduced basis space formed by a small number of modes obtained through proper orthogonal decomposition. The effectiveness of the proposed algorithm is demonstrated through numerical experiments, showcasing its computational prowess.
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.
We present a method that employs physics-informed deep learning techniques for parametrically solving partial differential equations. The focus is on the steady-state heat equations within heterogeneous solids exhibiting significant phase contrast. Similar equations manifest in diverse applications like chemical diffusion, electrostatics, and Darcy flow. The neural network aims to establish the link between the complex thermal conductivity profiles and temperature distributions, as well as heat flux components within the microstructure, under fixed boundary conditions. A distinctive aspect is our independence from classical solvers like finite element methods for data. A noteworthy contribution lies in our novel approach to defining the loss function, based on the discretized weak form of the governing equation. This not only reduces the required order of derivatives but also eliminates the need for automatic differentiation in the construction of loss terms, accepting potential numerical errors from the chosen discretization method. As a result, the loss function in this work is an algebraic equation that significantly enhances training efficiency. We benchmark our methodology against the standard finite element method, demonstrating accurate yet faster predictions using the trained neural network for temperature and flux profiles. We also show higher accuracy by using the proposed method compared to purely data-driven approaches for unforeseen scenarios.
In the Bayes paradigm and for a given loss function, we propose the construction of a new type of posterior distributions, that extends the classical Bayes one, for estimating the law of an $n$-sample. The loss functions we have in mind are based on the total variation and Hellinger distances as well as some $\mathbb{L}_{j}$-ones. We prove that, with a probability close to one, this new posterior distribution concentrates its mass in a neighbourhood of the law of the data, for the chosen loss function, provided that this law belongs to the support of the prior or, at least, lies close enough to it. We therefore establish that the new posterior distribution enjoys some robustness properties with respect to a possible misspecification of the prior, or more precisely, its support. For the total variation and squared Hellinger losses, we also show that the posterior distribution keeps its concentration properties when the data are only independent, hence not necessarily i.i.d., provided that most of their marginals or the average of these are close enough to some probability distribution around which the prior puts enough mass. The posterior distribution is therefore also stable with respect to the equidistribution assumption. We illustrate these results by several applications. We consider the problems of estimating a location parameter or both the location and the scale of a density in a nonparametric framework. Finally, we also tackle the problem of estimating a density, with the squared Hellinger loss, in a high-dimensional parametric model under some sparsity conditions. The results established in this paper are non-asymptotic and provide, as much as possible, explicit constants.
In the analysis of cluster randomized trials, two typical features are that individuals within a cluster are correlated and that the total number of clusters can sometimes be limited. While model-robust treatment effect estimators have been recently developed, their asymptotic theory requires the number of clusters to approach infinity, and one often has to empirically assess the applicability of those methods in finite samples. To address this challenge, we propose a conformal causal inference framework that achieves the target coverage probability of treatment effects in finite samples without the need for asymptotic approximations. Meanwhile, we prove that this framework is compatible with arbitrary working models, including machine learning algorithms leveraging baseline covariates, possesses robustness against arbitrary misspecification of working models, and accommodates a variety of within-cluster correlations. Under this framework, we offer efficient algorithms to make inferences on treatment effects at both the cluster and individual levels, applicable to user-specified covariate subgroups and two types of test data. Finally, we demonstrate our methods via simulations and a real data application based on a cluster randomized trial for treating chronic pain.
Network meta-analysis (NMA) combines evidence from multiple trials to compare the effectiveness of a set of interventions. In public health research, interventions are often complex, made up of multiple components or features. This makes it difficult to define a common set of interventions on which to perform the analysis. One approach to this problem is component network meta-analysis (CNMA) which uses a meta-regression framework to define each intervention as a subset of components whose individual effects combine additively. In this paper, we are motivated by a systematic review of complex interventions to prevent obesity in children. Due to considerable heterogeneity across the trials, these interventions cannot be expressed as a subset of components but instead are coded against a framework of characteristic features. To analyse these data, we develop a bespoke CNMA-inspired model that allows us to identify the most important features of interventions. We define a meta-regression model with covariates on three levels: intervention, study, and follow-up time, as well as flexible interaction terms. By specifying different regression structures for trials with and without a control arm, we relax the assumption from previous CNMA models that a control arm is the absence of intervention components. Furthermore, we derive a correlation structure that accounts for trials with multiple intervention arms and multiple follow-up times. Although our model was developed for the specifics of the obesity data set, it has wider applicability to any set of complex interventions that can be coded according to a set of shared features.
Defining a successful notion of a multivariate quantile has been an open problem for more than half a century, motivating a plethora of possible solutions. Of these, the approach of [8] and [25] leading to M-quantiles, is very appealing for its mathematical elegance combining elements of convex analysis and probability theory. The key idea is the description of a convex function (the K-function) whose gradient (the K-transform) is in one-to-one correspondence between all of R^d and the unit ball in R^d. By analogy with the d=1 case where the K-transform is a cumulative distribution function-like object (an M-distribution), the fact that its inverse is guaranteed to exist lends itself naturally to providing the basis for the definition of a quantile function for all d>=1. Over the past twenty years the resulting M-quantiles have seen applications in a variety of fields, primarily for the purpose of detecting outliers in multidimensional spaces. In this article we prove that for odd d>=3, it is not the gradient but a poly-Laplacian of the K-function that is (almost everywhere) proportional to the density function. For d even one cannot establish a differential equation connecting the K-function with the density. These results show that usage of the K-transform for outlier detection in higher odd-dimensions is in principle flawed, as the K-transform does not originate from inversion of a true M-distribution. We demonstrate these conclusions in two dimensions through examples from non-standard asymmetric distributions. Our examples illustrate a feature of the K-transform whereby regions in the domain with higher density map to larger volumes in the co-domain, thereby producing a magnification effect that moves inliers closer to the boundary of the co-domain than outliers. This feature obviously disrupts any outlier detection mechanism that relies on the inverse K-transform.
Most state-of-the-art machine learning techniques revolve around the optimisation of loss functions. Defining appropriate loss functions is therefore critical to successfully solving problems in this field. We present a survey of the most commonly used loss functions for a wide range of different applications, divided into classification, regression, ranking, sample generation and energy based modelling. Overall, we introduce 33 different loss functions and we organise them into an intuitive taxonomy. Each loss function is given a theoretical backing and we describe where it is best used. This survey aims to provide a reference of the most essential loss functions for both beginner and advanced machine learning practitioners.
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