arXiv subjects

Andrew B. Duncan

Publications and source records attributed to Andrew B. Duncan.

At least 19 recordsLinked to original sources

arXiv2026

Walking the Score Manifold: Continuous-time Generative Dynamics on Learned Data Manifolds

Generative modeling of time-dependent data is typically formulated on a discrete temporal grid, restricting supervision to the observed timestamps in the training data. We instead frame generation as continuous-time evolution on a learned data manifold. To this end, we leverage pretrained score-based models as geometric priors and learn a vector field that evolves data along score-induced interpolation paths. Because these dynamics follow transitions that respect the geometry learned by the score model, they support generation at arbitrary timestamps and temporal super-resolution beyond the discretization of the training data. Moreover, this geometric formulation allows us to train the vector field simulation-free through a regression objective. To improve long-horizon rollout robustness, we introduce an objective that promotes path-relative transverse exponential stability. While motivated by stability theory, it admits a practical interpretation as denoising score matching transverse to the interpolation path. Further, we extend the framework to a probabilistic setting that models a distribution over plausible future trajectories. We demonstrate the method on natural video and scientific dynamical data, including temporal super-resolution, PDE-based spatiotemporal fields, and molecular dynamics. Our results show that score-based priors provide a strong foundation for learning stochastic continuous-time generative dynamics.

cs.LG↗

arXiv2026

Physics-Constrained Fine-Tuning of Flow-Matching Models for Generation and Inverse Problems

We present a framework for fine-tuning flow-matching generative models to enforce physical constraints and solve inverse problems in scientific systems. Starting from a model trained on low-fidelity or observational data, we apply a differentiable post-training procedure that minimizes weak-form residuals of governing partial differential equations (PDEs), promoting physical consistency and adherence to boundary conditions without distorting the underlying learned distribution. To infer unknown physical inputs, such as source terms, material parameters, or boundary data, we augment the generative process with a learnable latent parameter predictor and propose a joint optimization strategy. The resulting model produces physically valid field solutions alongside plausible estimates of hidden parameters, effectively addressing ill-posed inverse problems in a data-driven yet physicsaware manner. We validate our method on canonical PDE benchmarks, demonstrating improved satisfaction of PDE constraints and accurate recovery of latent coefficients. Our approach bridges generative modelling and scientific inference, opening new avenues for simulation-augmented discovery and data-efficient modelling of physical systems.

cs.LG↗

arXiv2026

Soft Specialists: $α$-Rényi Ensembles for Uncertainty-Aware LLM Post-Training

Existing training approaches for large language models learn a single set of parameters, based on large volumes of data, which is typically heterogeneous, conflicting and often outright contradictory. As a result, the model is forced to compress conflicting goals, and inherent uncertainties into a single, averaged pattern of behaviour. We propose an $α$-Rényi variational framework for learning distributions over post-training parameters, offering an uncertainty-aware alternative to deep ensemble approaches. The resulting variational objective interpolates between classical variational Bayes and predictively oriented posterior learning, balancing between globally plausible individual models against systems of complementary specialists. We identify local stability criteria, demonstrating how model misspecification can make non-degenerate posterior spread locally favourable, manifesting contradictory or conflicting data as epistemic uncertainty. We apply our framework to LLM post-training, learning an ensemble of LoRA adapters attached to a shared, frozen base model, providing a scalable training procedure for both supervised fine-tuning and preference optimisation. Our approach enables training examples to be softly routed across ensemble members, promoting model specialisation and providing actionable uncertainty estimates across different tasks.

stat.ML↗

arXiv2026

Batch-based Bayesian Optimal Experimental Design in Linear Inverse Problems

Experimental design is central to science and engineering. A ubiquitous challenge is how to maximize the value of information obtained from expensive or constrained experimental settings. Bayesian optimal experimental design (OED) provides a principled framework for addressing such questions. In this paper, we study experimental design problems such as the optimization of sensor locations over a continuous domain in the context of linear Bayesian inverse problems. We focus in particular on batch design, that is, the simultaneous optimization of multiple design variables, which leads to a notoriously difficult non-convex optimization problem. We tackle this challenge using a promising strategy recently proposed in the frequentist setting, which relaxes A-optimal design to the space of finite positive measures. Our main contribution is the rigorous identification of the Bayesian inference problem corresponding to this relaxed A-optimal OED formulation. Moreover, building on recent work, we develop a Wasserstein gradient-flow -based optimization algorithm for the expected utility and introduce novel regularization schemes that guarantee convergence to an empirical measure. These theoretical results are supported by numerical experiments demonstrating both convergence and the effectiveness of the proposed regularization strategy.

stat.ME↗

arXiv2025

Learning Density Functionals to Bridge Particle and Continuum Scales

Predicting interfacial thermodynamics across molecular and continuum scales remains a central challenge in computational science. Classical density functional theory (cDFT) provides a first-principles route to connect microscopic interactions with macroscopic observables, but its predictive accuracy depends on approximate free-energy functionals that are difficult to generalize. Here we introduce a physics-informed learning framework that augments cDFT with neural corrections trained directly against molecular-dynamics data through adjoint optimization. Rather than replacing the theory with a black-box surrogate, we embed compact neural networks within the Helmholtz free-energy functional, learning local and nonlocal corrections that preserve thermodynamic consistency while capturing missing correlations. Applied to Lennard-Jones fluids, the resulting augmented excess free-energy functional quantitatively reproduces equilibrium density profiles, coexistence curves, and surface tensions across a broad temperature range, and accurately predicts contact angles and droplet shapes far beyond the training regime. This approach combines the interpretability of statistical mechanics with the adaptability of modern machine learning, establishing a general route to learned thermodynamic functionals that bridge molecular simulations and continuum-scale models.

physics.comp-ph↗

arXiv2025

Sampling by averaging: A multiscale approach to score estimation

We introduce a novel framework for efficient sampling from complex, unnormalised target distributions by exploiting multiscale dynamics. Traditional score-based sampling methods either rely on learned approximations of the score function or involve computationally expensive nested Markov chain Monte Carlo (MCMC) loops. In contrast, the proposed approach leverages stochastic averaging within a slow-fast system of stochastic differential equations (SDEs) to estimate intermediate scores along a diffusion path without training or inner-loop MCMC. Two algorithms are developed under this framework: MultALMC, which uses multiscale annealed Langevin dynamics, and MultCDiff, based on multiscale controlled diffusions for the reverse-time Ornstein-Uhlenbeck process. Both overdamped and underdamped variants are considered, with theoretical guarantees of convergence to the desired diffusion path. The framework is extended to handle heavy-tailed target distributions using Student's t-based noise models and tailored fast-process dynamics. Empirical results across synthetic and real-world benchmarks, including multimodal and high-dimensional distributions, demonstrate that the proposed methods are competitive with existing samplers in terms of accuracy and efficiency, without the need for learned models.

stat.CO↗

arXiv2025

Certified Self-Consistency: Statistical Guarantees and Test-Time Training for Reliable Reasoning in LLMs

Recent advances such as self-consistency and test-time reinforcement learning (TTRL) improve the reliability of large language models (LLMs) without additional supervision, yet their underlying mechanisms and statistical guarantees remain poorly understood. We present a unified framework for certifiable inference in LLMs, showing that majority voting provides a statistical certificate of self-consistency: under mild assumptions, the aggregated answer coincides with the mode of the model's terminal distribution with high probability. We derive finite-sample and anytime-valid concentration bounds that quantify this confidence, and introduce the Martingale Majority Certificate (MMC), a sequential stopping rule that adaptively determines when sufficient samples have been drawn. We further prove that label-free post-training methods such as TTRL implicitly sharpen the answer distribution by exponentially tilting it toward its mode, thereby reducing the number of samples required for certification. Building on this insight, we propose new post-training objectives that explicitly optimise this trade-off between sharpness and bias. Together, these results explain and connect two central test-time scaling strategies, self-consistency and TTRL, within a single statistical framework for label-free, certifiable reliability in reasoning LLMs.

stat.ML↗

arXiv2025

Non-asymptotic Analysis of Diffusion Annealed Langevin Monte Carlo for Generative Modelling

We investigate the theoretical properties of general diffusion (interpolation) paths and their Langevin Monte Carlo implementation, referred to as diffusion annealed Langevin Monte Carlo (DALMC), under weak conditions on the data distribution. Specifically, we analyse and provide non-asymptotic error bounds for the annealed Langevin dynamics where the path of distributions is defined as Gaussian convolutions of the data distribution as in diffusion models. We then extend our results to recently proposed heavy-tailed (Student's t) diffusion paths, demonstrating their theoretical properties for heavy-tailed data distributions for the first time. Our analysis provides theoretical guarantees for a class of score-based generative models that interpolate between a simple distribution (Gaussian or Student's t) and the data distribution in finite time. This approach offers a broader perspective compared to standard score-based diffusion approaches, which are typically based on a forward Ornstein-Uhlenbeck (OU) noising process.

stat.ML↗

arXiv2024

Energy-Based Modelling for Discrete and Mixed Data via Heat Equations on Structured Spaces

Energy-based models (EBMs) offer a flexible framework for probabilistic modelling across various data domains. However, training EBMs on data in discrete or mixed state spaces poses significant challenges due to the lack of robust and fast sampling methods. In this work, we propose to train discrete EBMs with Energy Discrepancy, a loss function which only requires the evaluation of the energy function at data points and their perturbed counterparts, thus eliminating the need for Markov chain Monte Carlo. We introduce perturbations of the data distribution by simulating a diffusion process on the discrete state space endowed with a graph structure. This allows us to inform the choice of perturbation from the structure of the modelled discrete variable, while the continuous time parameter enables fine-grained control of the perturbation. Empirically, we demonstrate the efficacy of the proposed approaches in a wide range of applications, including the estimation of discrete densities with non-binary vocabulary and binary image modelling. Finally, we train EBMs on tabular data sets with applications in synthetic data generation and calibrated classification.

stat.ML↗

arXiv2023

Energy Discrepancies: A Score-Independent Loss for Energy-Based Models

Energy-based models are a simple yet powerful class of probabilistic models, but their widespread adoption has been limited by the computational burden of training them. We propose a novel loss function called Energy Discrepancy (ED) which does not rely on the computation of scores or expensive Markov chain Monte Carlo. We show that ED approaches the explicit score matching and negative log-likelihood loss under different limits, effectively interpolating between both. Consequently, minimum ED estimation overcomes the problem of nearsightedness encountered in score-based estimation methods, while also enjoying theoretical guarantees. Through numerical experiments, we demonstrate that ED learns low-dimensional data distributions faster and more accurately than explicit score matching or contrastive divergence. For high-dimensional image data, we describe how the manifold hypothesis puts limitations on our approach and demonstrate the effectiveness of energy discrepancy by training the energy-based model as a prior of a variational decoder model.

stat.ML↗

arXiv2023

System Effects in Identifying Risk-Optimal Data Requirements for Digital Twins of Structures

Structural Health Monitoring (SHM) technologies offer much promise to the risk management of the built environment, and they are therefore an active area of research. However, information regarding material properties, such as toughness and strength is instead measured in destructive lab tests. Similarly, the presence of geometrical anomalies is more commonly detected and sized by inspection. Therefore, a risk-optimal combination should be sought, acknowledging that different scenarios will be associated with different data requirements. Value of Information (VoI) analysis is an established statistical framework for quantifying the expected benefit of a prospective data collection activity. In this paper the expected value of various combinations of inspection, SHM and testing are quantified, in the context of supporting risk management of a location of stress concentration in a railway bridge. The Julia code for this analysis (probabilistic models and influence diagrams) is made available. The system-level results differ from a simple linear sum of marginal VoI estimates, i.e. the expected value of collecting data from SHM and inspection together is not equal to the expected value of SHM data plus the expected value of inspection data. In summary, system-level decision making, requires system-level models.

stat.AP↗

arXiv2023

A Fourier representation of kernel Stein discrepancy with application to Goodness-of-Fit tests for measures on infinite dimensional Hilbert spaces

Kernel Stein discrepancy (KSD) is a widely used kernel-based measure of discrepancy between probability measures. It is often employed in the scenario where a user has a collection of samples from a candidate probability measure and wishes to compare them against a specified target probability measure. KSD has been employed in a range of settings including goodness-of-fit testing, parametric inference, MCMC output assessment and generative modelling. However, so far the method has been restricted to finite-dimensional data. We provide the first analysis of KSD in the generality of data lying in a separable Hilbert space, for example functional data. The main result is a novel Fourier representation of KSD obtained by combining the theory of measure equations with kernel methods. This allows us to prove that KSD can separate measures and thus is valid to use in practice. Additionally, our results improve the interpretability of KSD by decoupling the effect of the kernel and Stein operator. We demonstrate the efficacy of the proposed methodology by performing goodness-of-fit tests for various Gaussian and non-Gaussian functional models in a number of synthetic data experiments.

math.ST↗

arXiv2023

Training Discrete Energy-Based Models with Energy Discrepancy

Training energy-based models (EBMs) on discrete spaces is challenging because sampling over such spaces can be difficult. We propose to train discrete EBMs with energy discrepancy (ED), a novel type of contrastive loss functional which only requires the evaluation of the energy function at data points and their perturbed counter parts, thus not relying on sampling strategies like Markov chain Monte Carlo (MCMC). Energy discrepancy offers theoretical guarantees for a broad class of perturbation processes of which we investigate three types: perturbations based on Bernoulli noise, based on deterministic transforms, and based on neighbourhood structures. We demonstrate their relative performance on lattice Ising models, binary synthetic data, and discrete image data sets.

stat.ML↗

arXiv2023

A High-dimensional Convergence Theorem for U-statistics with Applications to Kernel-based Testing

We prove a convergence theorem for U-statistics of degree two, where the data dimension $d$ is allowed to scale with sample size $n$. We find that the limiting distribution of a U-statistic undergoes a phase transition from the non-degenerate Gaussian limit to the degenerate limit, regardless of its degeneracy and depending only on a moment ratio. A surprising consequence is that a non-degenerate U-statistic in high dimensions can have a non-Gaussian limit with a larger variance and asymmetric distribution. Our bounds are valid for any finite $n$ and $d$, independent of individual eigenvalues of the underlying function, and dimension-independent under a mild assumption. As an application, we apply our theory to two popular kernel-based distribution tests, MMD and KSD, whose high-dimensional performance has been challenging to study. In a simple empirical setting, our results correctly predict how the test power at a fixed threshold scales with $d$ and the bandwidth.

math.ST↗

arXiv2023

Using Perturbation to Improve Goodness-of-Fit Tests based on Kernelized Stein Discrepancy

Kernelized Stein discrepancy (KSD) is a score-based discrepancy widely used in goodness-of-fit tests. It can be applied even when the target distribution has an unknown normalising factor, such as in Bayesian analysis. We show theoretically and empirically that the KSD test can suffer from low power when the target and the alternative distributions have the same well-separated modes but differ in mixing proportions. We propose to perturb the observed sample via Markov transition kernels, with respect to which the target distribution is invariant. This allows us to then employ the KSD test on the perturbed sample. We provide numerical evidence that with suitably chosen transition kernels the proposed approach can lead to substantially higher power than the KSD test.

stat.ML↗

arXiv2023

Batch Bayesian Optimization via Particle Gradient Flows

Bayesian Optimisation (BO) methods seek to find global optima of objective functions which are only available as a black-box or are expensive to evaluate. Such methods construct a surrogate model for the objective function, quantifying the uncertainty in that surrogate through Bayesian inference. Objective evaluations are sequentially determined by maximising an acquisition function at each step. However, this ancilliary optimisation problem can be highly non-trivial to solve, due to the non-convexity of the acquisition function, particularly in the case of batch Bayesian optimisation, where multiple points are selected in every step. In this work we reformulate batch BO as an optimisation problem over the space of probability measures. We construct a new acquisition function based on multipoint expected improvement which is convex over the space of probability measures. Practical schemes for solving this `inner' optimisation problem arise naturally as gradient flows of this objective function. We demonstrate the efficacy of this new method on different benchmark functions and compare with state-of-the-art batch BO methods.

stat.ML↗

arXiv2022

Minimum Stein Discrepancy Estimators

When maximum likelihood estimation is infeasible, one often turns to score matching, contrastive divergence, or minimum probability flow to obtain tractable parameter estimates. We provide a unifying perspective of these techniques as minimum Stein discrepancy estimators, and use this lens to design new diffusion kernel Stein discrepancy (DKSD) and diffusion score matching (DSM) estimators with complementary strengths. We establish the consistency, asymptotic normality, and robustness of DKSD and DSM estimators, then derive stochastic Riemannian gradient descent algorithms for their efficient optimisation. The main strength of our methodology is its flexibility, which allows us to design estimators with desirable properties for specific models at hand by carefully selecting a Stein discrepancy. We illustrate this advantage for several challenging problems for score matching, such as non-smooth, heavy-tailed or light-tailed densities.

math.ST↗

arXiv2022

Spatial anomaly detection with optimal transport

This manuscript outlines an automated anomaly detection framework for jet engines. It is tailored for identifying spatial anomalies in steady-state temperature measurements at various axial stations in an engine. The framework rests upon ideas from optimal transport theory for Gaussian measures which yields analytical solutions for both Wasserstein distances and barycenters. The anomaly detection framework proposed builds upon our prior efforts that view the spatial distribution of temperature as a Gaussian random field. We demonstrate the utility of our approach by training on a dataset from one engine family, and applying them across a fleet of engines -- successfully detecting anomalies while avoiding both false positives and false negatives. Although the primary application considered in this paper are the temperature measurements in engines, applications to other internal flows and related thermodynamic quantities are made lucid.

stat.AP↗