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Chaotic systems make long-horizon forecasts difficult because small perturbations in initial conditions cause trajectories to diverge at an exponential rate. In this setting, neural operators trained to minimize squared error losses, while capable of accurate short-term forecasts, often fail to reproduce statistical or structural properties of the dynamics over longer time horizons and can yield degenerate results. In this paper, we propose an alternative framework designed to preserve invariant measures of chaotic attractors that characterize the time-invariant statistical properties of the dynamics. Specifically, in the multi-environment setting (where each sample trajectory is governed by slightly different dynamics), we consider two novel approaches to training with noisy data. First, we propose a loss based on the optimal transport distance between the observed dynamics and the neural operator outputs. This approach requires expert knowledge of the underlying physics to determine what statistical features should be included in the optimal transport loss. Second, we show that a contrastive learning framework, which does not require any specialized prior knowledge, can preserve statistical properties of the dynamics nearly as well as the optimal transport approach. On a variety of chaotic systems, our method is shown empirically to preserve invariant measures of chaotic attractors.

Construction of a universal detector poses a crucial question: How can we most effectively train a model on a large mixture of datasets? The answer lies in learning dataset-specific features and ensembling their knowledge but do all this in a single model. Previous methods achieve this by having separate detection heads on a common backbone but that results in a significant increase in parameters. In this work, we present Mixture-of-Experts as a solution, highlighting that MoEs are much more than a scalability tool. We propose Dataset-Aware Mixture-of-Experts, DAMEX where we train the experts to become an `expert' of a dataset by learning to route each dataset tokens to its mapped expert. Experiments on Universal Object-Detection Benchmark show that we outperform the existing state-of-the-art by average +10.2 AP score and improve over our non-MoE baseline by average +2.0 AP score. We also observe consistent gains while mixing datasets with (1) limited availability, (2) disparate domains and (3) divergent label sets. Further, we qualitatively show that DAMEX is robust against expert representation collapse.

When modeling a vector of risk variables, extreme scenarios are often of special interest. The peaks-over-thresholds method hinges on the notion that, asymptotically, the excesses over a vector of high thresholds follow a multivariate generalized Pareto distribution. However, existing literature has primarily concentrated on the setting when all risk variables are always large simultaneously. In reality, this assumption is often not met, especially in high dimensions. In response to this limitation, we study scenarios where distinct groups of risk variables may exhibit joint extremes while others do not. These discernible groups are derived from the angular measure inherent in the corresponding max-stable distribution, whence the term extreme direction. We explore such extreme directions within the framework of multivariate generalized Pareto distributions, with a focus on their probability density functions in relation to an appropriate dominating measure. Furthermore, we provide a stochastic construction that allows any prespecified set of risk groups to constitute the distribution's extreme directions. This construction takes the form of a smoothed max-linear model and accommodates the full spectrum of conceivable max-stable dependence structures. Additionally, we introduce a generic simulation algorithm tailored for multivariate generalized Pareto distributions, offering specific implementations for extensions of the logistic and H\"usler-Reiss families capable of carrying arbitrary extreme directions.

Complete observation of event histories is often impossible due to sampling effects such as right-censoring and left-truncation, but also due to reporting delays and incomplete event adjudication. This is for example the case during interim stages of clinical trials and for health insurance claims. In this paper, we develop a parametric method that takes the aforementioned effects into account, treating the latter two as partially exogenous. The method, which takes the form of a two-step M-estimation procedure, is applicable to multistate models in general, including competing risks and recurrent event models. The effect of reporting delays is derived via thinning, extending existing results for Poisson models. To address incomplete event adjudication, we propose an imputed likelihood approach which, compared to existing methods, has the advantage of allowing for dependencies between the event history and adjudication processes as well as allowing for unreported events and multiple event types. We establish consistency and asymptotic normality under standard identifiability, integrability, and smoothness conditions, and we demonstrate the validity of the percentile bootstrap. Finally, a simulation study shows favorable finite sample performance of our method compared to other alternatives, while an application to disability insurance data illustrates its practical potential.

We consider the problem of obtaining effective representations for the solutions of linear, vector-valued stochastic differential equations (SDEs) driven by non-Gaussian pure-jump L\'evy processes, and we show how such representations lead to efficient simulation methods. The processes considered constitute a broad class of models that find application across the physical and biological sciences, mathematics, finance and engineering. Motivated by important relevant problems in statistical inference, we derive new, generalised shot-noise simulation methods whenever a normal variance-mean (NVM) mixture representation exists for the driving L\'evy process, including the generalised hyperbolic, normal-Gamma, and normal tempered stable cases. Simple, explicit conditions are identified for the convergence of the residual of a truncated shot-noise representation to a Brownian motion in the case of the pure L\'evy process, and to a Brownian-driven SDE in the case of the L\'evy-driven SDE. These results provide Gaussian approximations to the small jumps of the process under the NVM representation. The resulting representations are of particular importance in state inference and parameter estimation for L\'evy-driven SDE models, since the resulting conditionally Gaussian structures can be readily incorporated into latent variable inference methods such as Markov chain Monte Carlo (MCMC), Expectation-Maximisation (EM), and sequential Monte Carlo.

We introduce a semi-explicit time-stepping scheme of second order for linear poroelasticity satisfying a weak coupling condition. Here, semi-explicit means that the system, which needs to be solved in each step, decouples and hence improves the computational efficiency. The construction and the convergence proof are based on the connection to a differential equation with two time delays, namely one and two times the step size. Numerical experiments confirm the theoretical results and indicate the applicability to higher-order schemes.

Modern high-throughput sequencing assays efficiently capture not only gene expression and different levels of gene regulation but also a multitude of genome variants. Focused analysis of alternative alleles of variable sites at homologous chromosomes of the human genome reveals allele-specific gene expression and allele-specific gene regulation by assessing allelic imbalance of read counts at individual sites. Here we formally describe an advanced statistical framework for detecting the allelic imbalance in allelic read counts at single-nucleotide variants detected in diverse omics studies (ChIP-Seq, ATAC-Seq, DNase-Seq, CAGE-Seq, and others). MIXALIME accounts for copy-number variants and aneuploidy, reference read mapping bias, and provides several scoring models to balance between sensitivity and specificity when scoring data with varying levels of experimental noise-caused overdispersion.

We combine the recent relaxation approach with multiderivative Runge-Kutta methods to preserve conservation or dissipation of entropy functionals for ordinary and partial differential equations. Relaxation methods are minor modifications of explicit and implicit schemes, requiring only the solution of a single scalar equation per time step in addition to the baseline scheme. We demonstrate the robustness of the resulting methods for a range of test problems including the 3D compressible Euler equations. In particular, we point out improved error growth rates for certain entropy-conservative problems including nonlinear dispersive wave equations.

Sparse polynomial approximation has become indispensable for approximating smooth, high- or infinite-dimensional functions from limited samples. This is a key task in computational science and engineering, e.g., surrogate modelling in uncertainty quantification where the function is the solution map of a parametric or stochastic differential equation (DE). Yet, sparse polynomial approximation lacks a complete theory. On the one hand, there is a well-developed theory of best $s$-term polynomial approximation, which asserts exponential or algebraic rates of convergence for holomorphic functions. On the other, there are increasingly mature methods such as (weighted) $\ell^1$-minimization for computing such approximations. While the sample complexity of these methods has been analyzed with compressed sensing, whether they achieve best $s$-term approximation rates is not fully understood. Furthermore, these methods are not algorithms per se, as they involve exact minimizers of nonlinear optimization problems. This paper closes these gaps. Specifically, we consider the following question: are there robust, efficient algorithms for computing approximations to finite- or infinite-dimensional, holomorphic and Hilbert-valued functions from limited samples that achieve best $s$-term rates? We answer this affirmatively by introducing algorithms and theoretical guarantees that assert exponential or algebraic rates of convergence, along with robustness to sampling, algorithmic, and physical discretization errors. We tackle both scalar- and Hilbert-valued functions, this being key to parametric or stochastic DEs. Our results involve significant developments of existing techniques, including a novel restarted primal-dual iteration for solving weighted $\ell^1$-minimization problems in Hilbert spaces. Our theory is supplemented by numerical experiments demonstrating the efficacy of these algorithms.