Time-dependent high-dimensional partial differential equations (PDEs) with spatially localised and dynamically evolving solutions pose a fundamental challenge for physics-informed neural networks (PINNs), as uniform collocation sampling becomes increasingly ineffective in high-dimensional spatiotemporal domains. In this work, a deep adaptive sampling framework for PINNs is extended to the time-dependent setting by treating space and time as a unified domain without any explicit time marching. A normalising flow neural network model effectively learns the distribution induced by the PDE residual and generates new collocation points concentrated in regions where the solution is most difficult to learn. Unlike conventional adaptive strategies that require explicit time stepping or moving meshes, high-residual regions are automatically identified and tracked across both space and time, driven purely by the PDE residual distribution. The effectiveness of the proposed strategy is assessed on a range of benchmark problems, from sharp and moving features in two spatial dimensions to localised structures in up to eight spatial dimensions.

This paper establishes a theoretical framework for the uniform convergence of smoothly activated deep neural network (DNN) estimators. While standard ReLU networks achieve minimax-optimal rates in the $L^2(P)$ norm for various nonparametric regression tasks, we establish a theoretical lower bound demonstrating that least-squares ReLU estimators can suffer from the curse of dimensionality in their uniform convergence behavior. Motivated by the need for reliable uniform guarantees in downstream tasks requiring worst-case reliability, we address this limitation by analyzing smoothly activated DNNs (smooth DNNs), encompassing both feedforward and residual structures. We establish novel pseudo-dimension bounds, non-asymptotic approximation guarantees, and Hölder-norm bounds for the approximators of these models. Leveraging these results, we derive non-asymptotic uniform convergence rates for smooth DNN estimators across multiple statistical contexts, including Huber, least-squares, quantile, and logistic regression. We prove that smooth DNNs can mitigate the {curse of dimensionality} in uniform convergence by adaptively exploiting the low-dimensional hierarchical composition structure of the target function. Supported by both simulation studies and a real-world application, our results position smooth DNNs as a theoretically grounded and practically viable alternative to ReLU networks for statistical learning tasks requiring uniform guarantees.

E-values have recently emerged as a versatile alternative to p-values for statistical inference (Ramdas and Wang, 2025). They offer several advantages: they remain valid under optional stopping (Grünwald et al., 2024a), combine easily under arbitrary dependence, and exist for irregular problems where no other inferential method is known (Wasserman et al., 2020), among others. Beyond being useful, they have also proven necessary in various problems, such as multiple testing (Wang and Ramdas, 2022; Fischer and Ramdas, 2024; Xu et al., 2025), statistical contract theory (Bates et al., 2022), and post-hoc inference (Grünwald, 2024). Formally, an e-value is a nonnegative test statistic whose expected value is at most one under the null hypothesis. Ideally, analysts want e-values that are large under the alternative--that is, e-values with high power.

Principal component analysis (PCA) is one of the most widely used unsupervised dimension reduction techniques. We study PCA for data from multiple related domains. Since principal components generally differ across domains, one way to obtain a shared low-rank embedding is to perform PCA on the pooled data. However, this approach can focus on spurious directions that exhibit high variation in only a few domains. To find a robust embedding that still explains most variance in unseen but similar domains, we propose instead to focus on shared directions of variation. To this end, we introduce Anchor PCA which trades off overall explained variance with agreement between the shared and domain-specific low-rank embeddings. Anchor PCA amounts to PCA on a modified target matrix and thus can be solved efficiently. Moreover, we show that Anchor PCA recovers a maximal invariant subspace and admits a minimax reconstruction interpretation under bounded domain-specific covariance inflations. On simulated and real-world gas sensor data with temporal drift, we demonstrate, respectively, that Anchor PCA recovers the maximally invariant subspace and yields embeddings that explain more variance on unseen domains than the pooling baseline and a worst-case alternative. Taken together, these findings establish Anchor PCA as a promising approach to robust unsupervised dimension reduction from multi-domain data.

Enforcing nonlinear inequality constraints in neural networks remains challenging, especially when the output is subject to many coupled constraints. Existing hard constraint methods often impose structural restrictions on the constraint set or introduce substantial computational overhead for large-scale nonlinear problems. Here, we propose DiffSlack, a differentiable projection layer for nonlinear inequality-constrained neural prediction. DiffSlack reformulates inequalities as equalities with learnable slack variables, which are predicted as part of the augmented network output and provide a data-driven warm start for damped Gauss-Newton projection. The projection layer maps raw predictions onto the augmented feasible manifold while preserving end-to-end differentiability. A two-stage curriculum further stabilizes training and improves constraint satisfaction. We evaluate DiffSlack on vehicle path planning with 200 nonlinear inequality constraints from collision avoidance, curvature limits, and waypoint spacing. Compared with existing learning-based baselines, DiffSlack achieves a higher planning success rate and stronger geometric constraint satisfaction under a comparable inference budget. Ablation studies further show that the hard projection layer reduces sensitivity to supervision quality. Closed-loop tracking in CARLA and real-world vehicle experiments confirms the executability of the generated trajectories. These results demonstrate that DiffSlack provides a practical and scalable approach to embedding hard inequality constraints into neural networks for engineering applications.

Diffusion models have demonstrated strong performance in time series modeling due to their ability to progressively capture complex data distributions through iterative denoising. However, existing approaches struggle with frequency-sensitive denoising, high-frequency reconstruction and balancing global trends with local dynamics. To address these limitations, we propose \textbf{HyFAD}, a \textbf{Hy}brid time-frequency \textbf{D}iffusion model with \textbf{F}requency-\textbf{A}ware embedding for time series imputation. Built upon the DDPM paradigm, HyFAD adopts a coupled time-frequency diffusion framework, in which the reverse denoising proceeds sequentially from the time domain to the frequency domain, enabling coarse-to-fine generation. Specifically, the time-domain diffusion process captures low-frequency global trends, while the frequency-domain diffusion process refines high-frequency spectral components. We further introduce a frequency-aware step embedding that exploits the relationship between diffusion steps and spectral components, providing step-dependent spectral guidance and facilitates more accurate band-wise reconstruction. Extensive experiments on multiple benchmark datasets demonstrate that HyFAD achieves state-of-the-art performance. Our source code is available at https://github.com/hongfangao/HyFAD.

Things to Do in L.A. Tap to enable a layout that focuses on the article. O.C. immigration attorneys suspended for filing briefs filled with AI-hallucinated errors The attorneys were fined $2,500 each and suspended from practicing in the U.S. 9th Circuit Court of Appeals for six months. This is read by an automated voice. Please report any issues or inconsistencies here . A pair of Orange County immigration attorneys received temporary suspensions after the court discovered they used generative AI to write briefs that included "multiple nonexistent cases, misattributed quotations, and gross misrepresentations."

Plus: The EU has proposed new legislation to end its Big Tech dependence. Most days in her chambers, Judge Maritza Braswell, a federal magistrate judge in Colorado, sifts through stacks of documents written by people without a lawyer. The number of these filings has more than doubled compared to before 2023. She puts that jump down to AI. But while AI appears to be expanding access to justice, it doesn't seem to be improving people's chances of winning. Judges are starting to question what rights and duties chatbots should have as they stand in for lawyers.

In 2025, the tech journalist invited artificial intelligence to do nearly everything for her, including editing the book she was writing about the experiment. F or a year, Joanna Stern decided to turn herself into a "lab rat" - the object of her own experiment. Throughout 2025, she invited artificial intelligence into "every corner" of her life. She let AI answer her texts, decide what she ate and cooked, mow her lawn, fold her washing, drive her places, parse her mammograms and even, in the darkness of a burner phone, be her lover. The resulting book, I Am Not a Robot: My Year Using AI to Do (Almost) Everything, asks all the big questions, including: what happens when AI can do everything humans can do? And what comes after that?

Non-convex regularizers usually improve the performance of sparse estimation in practice. To prove this fact, we study the conditions of sparse estimations for the sharp concave regularizers which are a general family of non-convex regularizers including many existing regularizers. For the global solutions of the regularized regression, our sparse eigenvalue based conditions are weaker than that of L1-regularization for parameter estimation and sparseness estimation. For the approximate global and approximate stationary (AGAS) solutions, almost the same conditions are also enough. We show that the desired AGAS solutions can be obtained by coordinate descent (CD) based methods. Finally, we perform some experiments to show the performance of CD methods on giving AGAS solutions and the degree of weakness of the estimation conditions required by the sharp concave regularizers. Keywords: Sparse estimation, non-convex regularization, sparse eigenvalue, coordinate descent 1. Introduction High-dimensional estimation concerns the parameter estimation problems in which the dimensions of parameters are comparable to or larger than the sampling size.