Further experimental details are described in Ref. [4]. Each spatiotemporal stimulus spanned over 400 ms corresponding to the retinal integration timescale. Figure 1: (a) The log-likelihood of empirical data for each PMF averaged over cells. Black line is the identity line. The central 20 20 arrays are shown.

Sampling equilibrium molecular configurations from the Boltzmann distribution is a longstanding challenge. Boltzmann Generators (BGs) address this by combining exact-likelihood generative models with importance sampling, but their practical scalability is limited. Meanwhile, coarse-grained surrogates enable the modeling of larger systems by reducing effective dimensionality, yet often lack the reweighting process required to ensure asymptotically correct statistics. In this work, we propose Coarse-Grained Boltzmann Generators (CG-BGs), a principled framework that unifies scalable reduced-order modeling with the exactness of importance sampling. CG-BGs act in a coarse-grained coordinate space, using a learned potential of mean force (PMF) to reweight samples generated by a flow-based model. Crucially, we show that this PMF can be efficiently learned from rapidly converged data via force matching. Our results demonstrate that CG-BGs faithfully capture complex interactions mediated by explicit solvent within highly reduced representations, establishing a scalable pathway for the unbiased sampling of larger molecular systems.

Process Model Forecasting (PMF) aims to predict how the control-flow structure of a process evolves over time by mode ling the temporal dynamics of directly-follows (DF) relations, comple menting predictive process monitoring that focuses on single-case prefixe s. Prior benchmarks show that machine learning and deep learning models pr ovide only modest gains over statistical baselines, mainly due to the s parsity and heterogeneity of the DF time series. We investigate Time Ser ies Foundation Models (TSFMs), large pre-trained models for generic t ime series, as an alternative for PMF. Using DF time series derived from rea l-life event logs, we compare zero-shot use of TSFMs, without additional training, with fine-tuned variants adapted on PMF-specific data. TSFMs generally achieve lower forecasting errors (MAE and RMSE) than tradit ional and specialized models trained from scratch on the same logs, in dicating effective transfer of temporal structure from non-process do mains. While fine-tuning can further improve accuracy, the gains are ofte n small and may disappear on smaller or more complex datasets, so zero-s hot use remains a strong default. Our study highlights the generaliza tion capability and data efficiency of TSFMs for process-related time series a nd, to the best of our knowledge, provides the first systematic evaluat ion of temporal foundation models for PMF.

Machine learning (ML) classification models are increasingly being used in a wide range of applications where it is important that predictions are accompanied by uncertainties, including in climate and earth observation, medical diagnosis and bioaerosol monitoring. The output of an ML classification model is a type of categorical variable known as a nominal property in the International Vocabulary of Metrology (VIM). However, concepts related to uncertainty evaluation for nominal properties are not defined in the VIM, nor is such evaluation addressed by the Guide to the Expression of Uncertainty in Measurement (GUM). In this paper we propose a metrological conceptual uncertainty evaluation framework for nominal properties. This framework is based on probability mass functions and summary statistics thereof, and it is applicable to ML classification. We also illustrate its use in the context of two applications that exemplify the issues and have significant societal impact, namely, climate and earth observation and medical diagnosis. Our framework would enable an extension of the GUM to uncertainty for nominal properties, which would make both applicable to ML classification models.

Current evaluations of LLMs for code generation emphasize functional correctness, overlooking the fact that functionally correct solutions can differ significantly in algorithmic complexity. For instance, an $(O(n^2))$ versus $(O(n \log n))$ sorting algorithm may yield similar output but incur vastly different performance costs in production. This discrepancy reveals a critical limitation in current evaluation methods: they fail to capture the behavioral and performance diversity among correct solutions. To address this, we introduce a principled framework for evaluating the dynamic stability of generated code. We propose two metrics derived from opcode distributions: Static Canonical Trace Divergence (SCTD), which captures algorithmic structure diversity across generated solutions, and Dynamic Canonical Trace Divergence (DCTD), which quantifies runtime behavioral variance. Their ratio, the Behavioral Expression Factor (BEF), serves as a diagnostic signal: it indicates critical runtime instability when BEF $\ll$ 1 and functional redundancy when BEF $\gg$ 1. Empirical results on BigOBench and CodeContests show that state-of-the-art LLMs exhibit significant algorithmic variance even among functionally correct outputs. Notably, increasing sampling temperature improves pass@1 rates but degrades stability, revealing an unrecognized trade-off: searching for correct solutions in diverse output spaces introduces a "penalty of instability" between correctness and behavioral consistency. Our findings call for stability-aware objectives in code generation and new benchmarks with asymptotic test cases for robust, real-world LLM evaluation.

Ordinal categorical data are routinely encountered in a wide range of practical applications. When the primary goal is to construct a regression model for ordinal outcomes, cumulative link models represent one of the most popular choices to link the cumulative probabilities of the response with a set of covariates through a parsimonious linear predictor, shared across response categories. When the number of observations grows, standard sampling algorithms for Bayesian inference scale poorly, making posterior computation increasingly challenging in large datasets. In this article, we propose three scalable algorithms for approximating the posterior distribution of the regression coefficients in cumulative probit models relying on Variational Bayes and Expectation Propagation. We compare the proposed approaches with inference based on Markov Chain Monte Carlo, demonstrating superior computational performance and remarkable accuracy; finally, we illustrate the utility of the proposed algorithms on a challenging case study to investigate the structure of a criminal network.

We study nonparametric estimation of a probability mass function (PMF) on a large discrete support, where the PMF is multi-modal and heavy-tailed. The core idea is to treat the empirical PMF as a signal on a line graph and apply a data-dependent low-pass filter. Concretely, we form a symmetric tri-diagonal operator, the path graph Laplacian perturbed with a diagonal matrix built from the empirical PMF, then compute the eigenvectors, corresponding to the smallest feq eigenvalues. Projecting the empirical PMF onto this low dimensional subspace produces a smooth, multi-modal estimate that preserves coarse structure while suppressing noise. A light post-processing step of clipping and re-normalizing yields a valid PMF. Because we compute the eigenpairs of a symmetric tridiagonal matrix, the computation is reliable and runs time and memory proportional to the support times the dimension of the desired low-dimensional supspace. We also provide a practical, data-driven rule for selecting the dimension based on an orthogonal-series risk estimate, so the method "just works" with minimal tuning. On synthetic and real heavy-tailed examples, the approach preserves coarse structure while suppressing sampling noise, compares favorably to logspline and Gaussian-KDE baselines in the intended regimes. However, it has known failure modes (e.g., abrupt discontinuities). The method is short to implement, robust across sample sizes, and suitable for automated pipelines and exploratory analysis at scale because of its reliability and speed.

Diffusion-based samplers learn to sample complex, high-dimensional distributions using energies or log densities alone, without training data. Yet, they remain impractical for molecular sampling because they are often slower than molecular dynamics and miss thermodynamically relevant modes. Inspired by enhanced sampling, we encourage exploration by introducing a sequential bias along bespoke, information-rich, low-dimensional projections of atomic coordinates known as collective variables (CVs). We introduce a repulsive potential centered on the CVs from recent samples, which pushes future samples towards novel CV regions and effectively increases the temperature in the projected space. Our resulting method improves efficiency, mode discovery, enables the estimation of free energy differences, and retains independent sampling from the approximate Boltzmann distribution via reweighting by the bias. On standard peptide conformational sampling benchmarks, the method recovers diverse conformational states and accurate free energy profiles. We are the first to demonstrate reactive sampling using a diffusion-based sampler, capturing bond breaking and formation with universal interatomic potentials at near-first-principles accuracy. The approach resolves reactive energy landscapes at a fraction of the wall-clock time of standard sampling methods, advancing diffusion-based sampling towards practical use in molecular sciences.

Further experimental details are described in Ref. [4]. Each spatiotemporal stimulus spanned over 400 ms corresponding to the retinal integration timescale. Figure 1: (a) The log-likelihood of empirical data for each PMF averaged over cells. Black line is the identity line. The central 20 20 arrays are shown.