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We study the computational scalability of a Gaussian process (GP) framework for solving general nonlinear partial differential equations (PDEs). This framework transforms solving PDEs to solving quadratic optimization problem with nonlinear constraints. Its complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel of the GP and its partial derivatives at collocation points. We present a sparse Cholesky factorization algorithm for such kernel matrices based on the near-sparsity of the Cholesky factor under a new ordering of Diracs and derivative measurements. We rigorously identify the sparsity pattern and quantify the exponentially convergent accuracy of the corresponding Vecchia approximation of the GP, which is optimal in the Kullback-Leibler divergence. This enables us to compute $\epsilon$-approximate inverse Cholesky factors of the kernel matrices with complexity $O(N\log^d(N/\epsilon))$ in space and $O(N\log^{2d}(N/\epsilon))$ in time. With the sparse factors, gradient-based optimization methods become scalable. Furthermore, we can use the oftentimes more efficient Gauss-Newton method, for which we apply the conjugate gradient algorithm with the sparse factor of a reduced kernel matrix as a preconditioner to solve the linear system. We numerically illustrate our algorithm's near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Amp\`ere equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs.

In this article, a copula-based method for mixed regression models is proposed, where the conditional distribution of the response variable, given covariates, is modelled by a parametric family of continuous or discrete distributions, and the effect of a common latent variable pertaining to a cluster is modelled with a factor copula. We show how to estimate the parameters of the copula and the parameters of the margins, and we find the asymptotic behaviour of the estimation errors. Numerical experiments are performed to assess the precision of the estimators for finite samples. An example of an application is given using COVID-19 vaccination hesitancy from several countries. Computations are based on R package CopulaGAMM.

Discovering nonlinear differential equations that describe system dynamics from empirical data is a fundamental challenge in contemporary science. Here, we propose a methodology to identify dynamical laws by integrating denoising techniques to smooth the signal, sparse regression to identify the relevant parameters, and bootstrap confidence intervals to quantify the uncertainty of the estimates. We evaluate our method on well-known ordinary differential equations with an ensemble of random initial conditions, time series of increasing length, and varying signal-to-noise ratios. Our algorithm consistently identifies three-dimensional systems, given moderately-sized time series and high levels of signal quality relative to background noise. By accurately discovering dynamical systems automatically, our methodology has the potential to impact the understanding of complex systems, especially in fields where data are abundant, but developing mathematical models demands considerable effort.

SARSA, a classical on-policy control algorithm for reinforcement learning, is known to chatter when combined with linear function approximation: SARSA does not diverge but oscillates in a bounded region. However, little is known about how fast SARSA converges to that region and how large the region is. In this paper, we make progress towards this open problem by showing the convergence rate of projected SARSA to a bounded region. Importantly, the region is much smaller than the region that we project into, provided that the magnitude of the reward is not too large. Existing works regarding the convergence of linear SARSA to a fixed point all require the Lipschitz constant of SARSA's policy improvement operator to be sufficiently small; our analysis instead applies to arbitrary Lipschitz constants and thus characterizes the behavior of linear SARSA for a new regime.

An old problem in multivariate statistics is that linear Gaussian models are often unidentifiable, i.e. some parameters cannot be uniquely estimated. In factor (component) analysis, an orthogonal rotation of the factors is unidentifiable, while in linear regression, the direction of effect cannot be identified. For such linear models, non-Gaussianity of the (latent) variables has been shown to provide identifiability. In the case of factor analysis, this leads to independent component analysis, while in the case of the direction of effect, non-Gaussian versions of structural equation modelling solve the problem. More recently, we have shown how even general nonparametric nonlinear versions of such models can be estimated. Non-Gaussianity is not enough in this case, but assuming we have time series, or that the distributions are suitably modulated by some observed auxiliary variables, the models are identifiable. This paper reviews the identifiability theory for the linear and nonlinear cases, considering both factor analytic models and structural equation models.

This paper considers a downlink satellite communication system where a satellite cluster, i.e., a satellite swarm consisting of one leader and multiple follower satellites, serves a ground terminal. The satellites in the cluster form either a linear or circular formation moving in a group and cooperatively send their signals by maximum ratio transmission precoding. We first conduct a coordinate transformation to effectively capture the relative positions of satellites in the cluster. Next, we derive an exact expression for the orbital configuration-dependent outage probability under the Nakagami fading by using the distribution of the sum of independent Gamma random variables. In addition, we obtain a simpler approximated expression for the outage probability with the help of second-order moment-matching. We also analyze asymptotic behavior in the high signal-to-noise ratio regime and the diversity order of the outage performance. Finally, we verify the analytical results through Monte Carlo simulations. Our analytical results provide the performance of satellite cluster-based communication systems based on specific orbital configurations, which can be used to design reliable satellite clusters in terms of cluster size, formation, and orbits.

In many fields, including environmental epidemiology, researchers strive to understand the joint impact of a mixture of exposures. This involves analyzing a vector of exposures rather than a single exposure, with the most significant exposure sets being unknown. Examining every possible interaction or effect modification in a high-dimensional vector of candidates can be challenging or even impossible. To address this challenge, we propose a method for the automatic identification and estimation of exposure sets in a mixture with explanatory power, baseline covariates that modify the impact of an exposure and sets of exposures that have synergistic non-additive relationships. We define these parameters in a realistic nonparametric statistical model and use machine learning methods to identify variables sets and estimate nuisance parameters for our target parameters to avoid model misspecification. We establish a prespecified target parameter applied to variable sets when identified and use cross-validation to train efficient estimators employing targeted maximum likelihood estimation for our target parameter. Our approach applies a shift intervention targeting individual variable importance, interaction, and effect modification based on the data-adaptively determined sets of variables. Our methodology is implemented in the open-source SuperNOVA package in R. We demonstrate the utility of our method through simulations, showing that our estimator is efficient and asymptotically linear under conditions requiring fast convergence of certain regression functions. We apply our method to the National Institute of Environmental Health Science mixtures workshop data, revealing correct identification of antagonistic and agonistic interactions built into the data. Additionally, we investigate the association between exposure to persistent organic pollutants and longer leukocyte telomere length.

Rapid evolution of sensor technology, advances in instrumentation, and progress in devising data-acquisition softwares/hardwares are providing vast amounts of data for various complex phenomena, ranging from those in atomospheric environment, to large-scale porous formations, and biological systems. The tremendous increase in the speed of scientific computing has also made it possible to emulate diverse high-dimensional, multiscale and multiphysics phenomena that contain elements of stochasticity, and to generate large volumes of numerical data for them in heterogeneous systems. The difficulty is, however, that often the governing equations for such phenomena are not known. A prime example is flow, transport, and deformation processes in macroscopically-heterogeneous materials and geomedia. In other cases, the governing equations are only partially known, in the sense that they either contain various coefficients that must be evaluated based on data, or that they require constitutive relations, such as the relationship between the stress tensor and the velocity gradients for non-Newtonian fluids in the momentum conservation equation, in order for them to be useful to the modeling. Several classes of approaches are emerging to address such problems that are based on machine learning, symbolic regression, the Mori-Zwanzig projection operator formulation, sparse identification of nonlinear dynamics, data assimilation, and stochastic optimization and analysis, or a combination of two or more of such approaches. This Perspective describes the latest developments in this highly important area, and discusses possible future directions.

The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.