Whilst the vulnerability of graph neural networks (GNNs) to adversarial attacks poses a critical threat to graph representation learning, the understanding of the robust generalization behavior remains a fundamental challenge in the adversarial setting. Recently, PAC-Bayesian margin-based generalization analysis substantially advances this line of research by providing a flexible and data-dependent analytical framework. However, existing robust analyses often rely on isotropic Gaussian posteriors and control weight perturbations in the full parameter space, which limits the ability to capture heterogeneous parameter sensitivity yet hinges on hidden-width-dependent complexity terms, resulting in not-tight-enough generalization bounds. In this paper, we extend a recently proposed sensitivity-aware PAC-Bayesian framework from deep neural networks to message passing GNNs (MPGNNs) and derive a tighter robust generalization bound in the adversarial setting. Specifically, we first quantify how sensitive the perturbations across different parameter blocks are to the network outputs by deriving the output Jacobians with respect to the weight parameters. Exploiting the fact that these Jacobian matrices have rank at most $K$ in $K$-class graph classification, we then construct Jacobian-aligned sensitivity matrices and use anisotropic Gaussian posteriors with optimized covariances to upper bound the KL divergence in a tight way. Notably, by refining the spectral-norm dependence on the learned weights and reducing the leading dimension factor from hidden-width-dependent terms to the number of classes $K$, our analysis yields much tighter robust generalization guarantees for MPGNNs, thereby guiding their designs to enhance adversarial robustness.

Subgraph detection seeks to identify whether and where instances of query patterns occur within a larger graph. This problem is fundamental across scientific domains and is closely related to subgraph isomorphism, which is NP-complete, limiting combinatorial approaches to small patterns or moderately sized graphs. We introduce GraphDETR, a deep learning framework that formulates subgraph detection as a set prediction problem, analogous to DETR in object detection. GraphDETR encodes the target graph with a graph neural network, and employs a fixed set of learnable query vectors, decoded via a transformer decoder, to predict all pattern occurrences jointly in a single forward pass. This is enabled by training the model end-to-end with bipartite matching. Unlike traditional combinatorial methods that only solve exact structural matching, GraphDETR naturally extends to approximate matching, enabling detection beyond exact pattern correspondence. Empirically, we show that GraphDETR can detect diverse patterns, such as molecular structures, cycles, cliques, and fuzzy patterns of up to 50 nodes, in target graphs with up to 1000 nodes. We further evaluate on molecular functional group detection over the ChEMBL dataset, where GraphDETR predicts the complete set of functional groups per molecule, achieving a strong performance of $\text{AP}_{100} = 91.2$.

Per-ticker forecasting models dominate financial time-series work yet remain blind to cross-company propagation: a foundry disruption in Taiwan does not register in a single-asset model until Apple's own price has already moved. To address this limitation, we introduce a heterogeneous Rust-Python streaming architecture that maps cross-company attention as a continuous-time graph driven directly from text. We show that on the ingestion side, a zero-copy Rust edge parses news records in $\sim$100 ns and scans the target equity universe in $\sim$1.2 $μ$s. On the inference end, a multivariate Neural Hawkes Process featuring per-node continuous-time LSTM states and a bilinear latent projection propagates directed excitation, while an adaptive pruning rule bounds the computational cost of dynamic neighborhood updates. Combining these stages, we demonstrate an end-to-end processing latency of $\sim$13 ms per incoming news record on a single commodity CPU. Evaluated on a one-month temporal holdout of the FNSPID corpus (638 articles across 47 tickers), the system delivers a $1.70\times$ precision lift over random at the 90th-percentile next-day return threshold, and $3.36\times$ over a same-sector baseline. Crucially, removing the graph topology collapses precision to zero, confirming that the dynamic attention network is the sole driver of cross-company signal in this architecture.

Topological Data Analysis (TDA) offers a principled, intrinsic lens for comparing neural representations. However, existing paired topological divergences (e.g., RTD) are limited by heuristic asymmetry and, more critically, unbounded scores that depend on sample size, hindering reliable cross-scenario benchmarking. To address these challenges, we develop a unified topological toolkit serving two complementary needs: fine-grained structural diagnosis and robust, standardized evaluation. First, we complete the RTD framework by introducing Symmetric Representation Topology Divergence (SRTD) and its efficient variant SRTD-lite. Beyond resolving the theoretical asymmetry of prior variants, SRTD consolidates diagnostic information into a single, comprehensive cross-barcode signature. This allows for precise localization of structural discrepancies and serves as an effective optimization objective without the overhead of dual directional computations. Second, to enable reliable benchmarking across heterogeneous settings, we propose Normalized Topological Similarity (NTS). By measuring the rank correlation of hierarchical merge orders, NTS yields a scale-invariant metric bounded between -1 and 1, effectively overcoming the scale and sample-dependence of unnormalized divergences. Experiments across synthetic and real-world deep learning settings demonstrate that our toolkit captures functional shifts in CNNs missed by geometric measures and robustly maps LLM genealogy even under distance saturation, offering a rigorous, topology-aware perspective that complements measures like CKA.

Physics-Informed Neural Networks inherently suffer from task interference because they rely on a shared parameter space to satisfy both governing differential equations and boundary conditions. We analyze this structural conflict using the Fisher Information Matrix to quantify the effective degrees of freedom ($d_{eff}$) in a physics-constrained model. Unlike the classical $d_{eff}$ which measures how many parameter directions are informed by data against a statistical prior, our $d_{eff}$ measures the dimension of the parameter directions unconstrained by the differential operator. For operators with finite-dimensional kernel, we show that $d_{eff}$ converges to the kernel dimension exactly, independent of network width, depth, or activation function, recasting it from a fit diagnostic into a structural invariant of the underlying continuous operator. For operators with infinite-dimensional kernel, $d_{eff}$ instead measures the network's finite-dimensional representational bandwidth for that kernel rather than recovering an integer invariant. Importantly, $d_{eff}$ also serves as an a priori structural diagnostic. Driving $d_{eff}$ of a well-posed problem to zero certifies that the physics and boundary constraints have absorbed the network's free directions. Building on this characterization, we introduce subspace projection strategies for boundary adaptation. Rather than retraining from scratch, we project parameter updates into the null space of the pre-trained physics operator so that new boundary conditions are satisfied without disturbing the learned physics. Gradient-based fine-tuning can match or exceed this but needs more wall-clock time and tuning, whereas subspace projection delivers near-equivalent quality in seconds to minutes. We validate on linear and nonlinear operators, demonstrating accurate adaptation to initial and boundary shifts and unencountered constraint types.

Reduced-order modeling of high-dimensional dynamical systems is often hindered by the non-Markovian closure term that represents the effect of unresolved variables on the resolved dynamics. Inspired by the Mori--Zwanzig formalism, in which the closure takes the form of a memory functional of the resolved trajectory, we recast closure modeling as a sequence modeling problem and propose the Mamba-Assisted Closure (MAC) framework: a Mamba-based sequence model, trained to predict the closure from the resolved trajectory, is coupled with the reduced-order governing equations through a numerical integrator to advance the resolved variables in time. A key feature of the framework is its exploitation of the dual representation of state-space models -- the model is trained in a sequence-to-sequence fashion via the convolutional form, and deployed for step-by-step autoregressive rollout via the recurrent form, yielding both efficient long-trajectory training and constant per-step inference cost. On the viscous Burgers' equation and the chaotic two-scale Lorenz '96 system, the MAC model substantially outperforms the Markovian reduced-order model, the GRU-based sequence model, and the Wilks method in predictive accuracy and long-time rollout stability.

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.

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."