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Predicting the response of nonlinear dynamical systems subject to random, broadband excitation is important across a range of scientific disciplines, such as structural dynamics and neuroscience. Building data-driven models requires experimental measurements of the system input and output, but it can be difficult to determine whether inaccuracies in the model stem from modelling errors or noise. This paper presents a novel method to identify the causal component of the input-output data from measurements of a system in the presence of output noise, as a function of frequency, without needing a high fidelity model. An output prediction, calculated using an available model, is optimally combined with noisy measurements of the output to predict the input to the system. The parameters of the algorithm balance the two output signals and are utilised to calculate a nonlinear coherence metric as a measure of causality. This method is applicable to a broad class of nonlinear dynamical systems. There are currently no solutions to this problem in the absence of a complete benchmark model.

High-fidelity direct numerical simulation of turbulent flows for most real-world applications remains an outstanding computational challenge. Several machine learning approaches have recently been proposed to alleviate the computational cost even though they become unstable or unphysical for long time predictions. We identify that the Fourier neural operator (FNO) based models combined with a partial differential equation (PDE) solver can accelerate fluid dynamic simulations and thus address computational expense of large-scale turbulence simulations. We treat the FNO model on the same footing as a PDE solver and answer important questions about the volume and temporal resolution of data required to build pre-trained models for turbulence. We also discuss the pitfalls of purely data-driven approaches that need to be avoided by the machine learning models to become viable and competitive tools for long time simulations of turbulence.

Inference methods for computing confidence intervals in parametric settings usually rely on consistent estimators of the parameter of interest. However, it may be computationally and/or analytically burdensome to obtain such estimators in various parametric settings, for example when the data exhibit certain features such as censoring, misclassification errors or outliers. To address these challenges, we propose a simulation-based inferential method, called the implicit bootstrap, that remains valid regardless of the potential asymptotic bias of the estimator on which the method is based. We demonstrate that this method allows for the construction of asymptotically valid percentile confidence intervals of the parameter of interest. Additionally, we show that these confidence intervals can also achieve second-order accuracy. We also show that the method is exact in three instances where the standard bootstrap fails. Using simulation studies, we illustrate the coverage accuracy of the method in three examples where standard parametric bootstrap procedures are computationally intensive and less accurate in finite samples.

We consider robust optimal experimental design (ROED) for nonlinear Bayesian inverse problems governed by partial differential equations (PDEs). An optimal design is one that maximizes some utility quantifying the quality of the solution of an inverse problem. However, the optimal design is dependent on elements of the inverse problem such as the simulation model, the prior, or the measurement error model. ROED aims to produce an optimal design that is aware of the additional uncertainties encoded in the inverse problem and remains optimal even after variations in them. We follow a worst-case scenario approach to develop a new framework for robust optimal design of nonlinear Bayesian inverse problems. The proposed framework a) is scalable and designed for infinite-dimensional Bayesian nonlinear inverse problems constrained by PDEs; b) develops efficient approximations of the utility, namely, the expected information gain; c) employs eigenvalue sensitivity techniques to develop analytical forms and efficient evaluation methods of the gradient of the utility with respect to the uncertainties we wish to be robust against; and d) employs a probabilistic optimization paradigm that properly defines and efficiently solves the resulting combinatorial max-min optimization problem. The effectiveness of the proposed approach is illustrated for optimal sensor placement problem in an inverse problem governed by an elliptic PDE.

Faithfully summarizing the knowledge encoded by a deep neural network (DNN) into a few symbolic primitive patterns without losing much information represents a core challenge in explainable AI. To this end, Ren et al. (2024) have derived a series of theorems to prove that the inference score of a DNN can be explained as a small set of interactions between input variables. However, the lack of generalization power makes it still hard to consider such interactions as faithful primitive patterns encoded by the DNN. Therefore, given different DNNs trained for the same task, we develop a new method to extract interactions that are shared by these DNNs. Experiments show that the extracted interactions can better reflect common knowledge shared by different DNNs.

Gaussian random fields (GFs) are fundamental tools in spatial modeling and can be represented flexibly and efficiently as solutions to stochastic partial differential equations (SPDEs). The SPDEs depend on specific parameters, which enforce various field behaviors and can be estimated using Bayesian inference. However, the likelihood typically only provides limited insights into the covariance structure under in-fill asymptotics. In response, it is essential to leverage priors to achieve appropriate, meaningful covariance structures in the posterior. This study introduces a smooth, invertible parameterization of the correlation length and diffusion matrix of an anisotropic GF and constructs penalized complexity (PC) priors for the model when the parameters are constant in space. The formulated prior is weakly informative, effectively penalizing complexity by pushing the correlation range toward infinity and the anisotropy to zero.

We consider nonparametric Bayesian inference in a multidimensional diffusion model with reflecting boundary conditions based on discrete high-frequency observations. We prove a general posterior contraction rate theorem in $L^2$-loss, which is applied to Gaussian priors. The resulting posteriors, as well as their posterior means, are shown to converge to the ground truth at the minimax optimal rate over H\"older smoothness classes in any dimension. Of independent interest and as part of our proofs, we show that certain frequentist penalized least squares estimators are also minimax optimal.

Analogously to de Bruijn sequences, orientable sequences have application in automatic position-location applications and, until recently, studies of these sequences focused on the binary case. In recent work by Alhakim et al., a range of methods of construction were described for orientable sequences over arbitrary finite alphabets; some of these methods involve using negative orientable sequences as a building block. In this paper we describe three techniques for generating such negative orientable sequences, as well as upper bounds on their period. We then go on to show how these negative orientable sequences can be used to generate orientable sequences with period close to the maximum possible for every non-binary alphabet size and for every tuple length. In doing so we use two closely related approaches described by Alhakim et al.

Iterative refinement (IR) is a popular scheme for solving a linear system of equations based on gradually improving the accuracy of an initial approximation. Originally developed to improve upon the accuracy of Gaussian elimination, interest in IR has been revived because of its suitability for execution on fast low-precision hardware such as analog devices and graphics processing units. IR generally converges when the error associated with the solution method is small, but is known to diverge when this error is large. We propose and analyze a novel enhancement to the IR algorithm by adding a line search optimization step that guarantees the algorithm will not diverge. Numerical experiments verify our theoretical results and illustrate the effectiveness of our proposed scheme.

A first proposal of a sparse and cellwise robust PCA method is presented. Robustness to single outlying cells in the data matrix is achieved by substituting the squared loss function for the approximation error by a robust version. The integration of a sparsity-inducing $L_1$ or elastic net penalty offers additional modeling flexibility. For the resulting challenging optimization problem, an algorithm based on Riemannian stochastic gradient descent is developed, with the advantage of being scalable to high-dimensional data, both in terms of many variables as well as observations. The resulting method is called SCRAMBLE (Sparse Cellwise Robust Algorithm for Manifold-based Learning and Estimation). Simulations reveal the superiority of this approach in comparison to established methods, both in the casewise and cellwise robustness paradigms. Two applications from the field of tribology underline the advantages of a cellwise robust and sparse PCA method.