\emph{Continuous-space} attacks find prompts that trigger malicious behaviors, such as overthinking, in black-box LLMs by sampling or optimizing over a low-dimensional Gaussian latent space and decoding it to text through a \emph{random mapping} and nearest-token lookup. Because the random mapping is not learned from language, its prompts can be highly unnatural, and as such easily detectable through input guardrails, and the risk estimates they yield are not grounded in natural prompts. In this paper, we demonstrate that the same continuous latent-space framework can be coupled directly to a learned generative distribution over language, a continuous diffusion language model (dLLM), which removes the artificial random mapping while preserving the standard-Gaussian latent formulation and the downstream sampling or optimization procedure. We evaluate this approach under two settings: sampling-based estimation using \emph{RareTrap} and optimization-based search using \emph{ThinkTrap}. We find that character-level perplexity drops from $10^3$--$10^7$ to 3--5 in RareTrap and from 20--28 to 4.3--5.4 in ThinkTrap. We also find that model risk estimates can differ by an order of magnitude across the two mappings.

Deep learning (DL) methods have substantially improved predictive performance across a wide range of downstream prediction tasks, enabling new discoveries in multiple domains. However, principled DL methods for non-Euclidean random objects remain relatively scarce, despite the growing importance of modern biomedical outcomes taking the form of probability distributions, random graphs, functions, and other random objects in metric spaces. Motivated by a new generation of digital and clinical biomarkers, we propose a deep learning framework for regression with random objects in metric spaces. Our approach operates directly on the metric structure of the response space, avoiding restrictive manifold or Euclidean embedding assumptions imposed by many existing methods. At the same time, it retains modern statistical guarantees under the stated statistical assumptions. We establish statistical consistency and convergence rates for the proposed estimator, and characterize conditions under which it can mitigate the curse of dimensionality for hierarchically compositional regression functions. Extensive simulation studies demonstrate competitive predictive performance, with gains over several competing approaches in multiple Euclidean and non-Euclidean configurations. Finally, we demonstrate the practical utility of the proposed framework in two digital health case studies with distribution-valued outcomes in Wasserstein spaces, using high-resolution accelerometry and continuous glucose monitoring data. Embedding-based deep-learning methods may be less suitable when no informative low-dimensional representation exists or when the learned embedding does not adequately preserve the intrinsic geometry of the response space.

Neural operators learn mappings between function spaces, enabling fast surrogate solutions to partial differential equations. Despite remarkable architectural diversity, these methods often share a common input representation: raw coordinate-value pairs $(x, u(x))$. We ask whether inputs aligned with PDE differential structure can improve learning. Through Picard iteration on mild solutions, we show that derivatives of the input appear explicitly in the solution operator, suggesting that providing derivative features should reduce the network's implicit differentiation burden. We prove this intuition: providing derivative features improves approximation bounds from $O(W^{-(s-m)/d})$ to $O(W^{-s/d})$, where $W$ is parameter count, $s$ is input regularity, $m$ is the PDE order, and $d$ is spatial dimension. Our central finding, however, is a surprise: the optimal derivative order $\beta^*$ is strictly less than the PDE order $m$. This gap arises from a bias-variance tradeoff in spectral space that we characterize in closed form. Learning $\beta$ from data achieves automatic spectral regularization. We introduce $\partial$-NO (del-NO), for derivative-augmented neural operators, an architecture-agnostic augmentation that provides learnable fractional derivative features to any neural operator backbone. Across benchmark problems and architectures, $\partial$-NO consistently improves prediction accuracy, with learned orders that reflect a representation of known physics modulated by noise and finite-sample constraints.

In sectors such as energy, aquaculture and chemical processing, the effectiveness of both mechanistic and data-driven models is frequently limited by poor-quality training data, which obscures underlying physical laws and degrades predictive accuracy. Such quality issues commonly manifest as anomalies in the dataset, ranging from sensor faults to process upsets and incorrect operator inputs. While traditional rule-based filtering can remove a subset of the anomalies, it is unable to leverage contextual information stored as unstructured text, such as engineer comments and operational logs that explain under what conditions the data was collected. We propose a text-informed data filtering method that leverages large language models (LLMs) to integrate unstructured operational data with tabular sensor data. The method consists of two steps: (1) contextual anomaly detection by reasoning over both data modalities, and (2) generation of a binary mask that is applied to filter the training dataset for model development. To investigate the potential benefit, a steady-state soft-sensor regression model, namely a virtual flow meter that estimates well flow rates, is used as the downstream evaluation target. Experiments on a synthetic oil and gas dataset show improved model performance compared to traditional rule-based filtering. Ablation studies show that textual operator comments can provide filtering-relevant signals beyond sensor data alone, with the multimodal approach reducing downstream MAPE from 7.1% to 5.4% relative to sensor-only filtering. This work demonstrates that integrating unstructured operational knowledge with sensor data through LLMs can improve data quality for industrial machine learning applications, offering a scalable alternative to manual data curation.

Inference-time scaling adapts pretrained diffusion models to new sampling tasks without additional training. Existing methods rely primarily on Monte Carlo sampling with more particles, yet are premised on the pretrained model being exact. In practice, data and training limitations make the model imperfect, and these methods inherit its error. More particles reduce Monte Carlo error but cannot remove the mismatch between the endpoint and the desired target or the error in tracking the prescribed probability path. We introduce the Energy-based Feynman-Kac Corrector (EBFKC), a framework for energy-based diffusion models that corrects these errors on the fly given a reference energy. We first derive Feynman-Kac dynamics that track a prescribed path exactly in the continuous-time population limit even when the model is imperfect, and approximate these dynamics using sequential Monte Carlo with variance-controlling guidance. To remove the endpoint mismatch, we use the pretrained energy as a surrogate along the diffusion path and progressively incorporate the discrepancy between the learned and target terminal energies. Experiments on Gaussian mixture models, particle systems, alanine dipeptide, and alanine tetrapeptide show that our method closely matches target distributions and molecular free-energy profiles under annealing and reward tilting, whereas standard inference-time scaling baselines retain substantial sampling errors.

We consider estimating the conditional distribution of a multivariate outcome given covariates when its coordinates may be continuous, binary, categorical, ordinal or rankings, and are conditionally dependent on one another. Different statistical methods have been developed for each outcome type, and most of them target a summary of the conditional distribution, such as the mean of each coordinate, rather than the joint distribution of the outcome vector. We develop generalized engression models, a unified nonparametric distributional regression framework for outcomes of any type. The proposed method builds upon engression, a scoring-rule-based deep generative model, and introduces a data-type-specific link function and a stochastic perturbation that smooths the loss, enabling gradient-based training even with discontinuous links. We establish universal representation results for continuous, discrete and mixed outcomes. In simulations and in two applications, 242 species in a community ecology benchmark and a 17-dimensional mixed-type health outcome, the method matches type-specific models on marginal scores, improves on them on the joint distribution, and matches or exceeds purpose-built state-of-the-art joint species distribution models. Software is available in Python.

Neural activity often exhibits multiple timescales that can vary with behavioral states and task conditions. Identifying these timescales from neural recordings is important for better understanding neural computation and function. However, traditional approaches based on autocorrelation fitting are difficult to scale to high-dimensional population recordings and can become unreliable when neural dynamics change with behavior. State-space models have been a powerful framework for modeling high-dimensional neural population activity through latent dynamical systems, but standard formulations and inference methods do not explicitly account for multiple timescales and therefore do not guarantee accurate recovery of the underlying temporal structure. Motivated by these questions, we introduce the Multi-Timescale Switching Linear Dynamical System (MTS-SLDS), a framework for identifying regime-specific latent timescales from continuous or spiking neural observations. MTS-SLDS combines a multi-lag moment initialization, which captures temporal structure across multiple observation lags, with \textit{regime-conditioned} Laplace-EM inference, which reduces mixing of dynamical statistics across uncertain regimes. Characteristic timescales can then be extracted directly from the eigenvalues of the learned latent transition matrices. In synthetic and neural experiments with Gaussian and Poisson spike observations, MTS-SLDS accurately recovers timescales and switching structure over multiple datasets.