arXiv subjects

Mark Girolami

Publications and source records attributed to Mark Girolami.

At least 19 recordsLinked to original sources

arXiv2026

Beyond Laplace: Closed-form wrapped Gaussian posterior approximations on statistical manifolds

In Bayesian statistics, the Laplace approximation provides a computationally efficient approximation to posterior distributions. However, its Gaussian form restricts it to elliptical shapes, limiting its ability to capture important posterior features such as skewness, heavy tails, and narrow high-probability regions. Recent work has addressed this limitation by exploiting Riemannian geometry to push forward Gaussian distributions from the tangent space to a chosen manifold, referred to wrapped Gaussians. While offering greater flexibility, they introduce substantial computational challenges. Sampling requires solving geodesic equations through the exponential map and density evaluation additionally depends on the logarithmic map and Jacobi fields, involving costly differential equation solvers alongside full matrices inverse operations, Christoffel symbols and curvature tensors. To overcome these limitations, we employ the theory of contrast functions to derive tractable approximations of the logarithmic and exponential maps on statistical manifolds endowed with the Fisher--Rao metric and the prior distribution geometry. The resulting methodology bypass the need to compute these geometric quantities and numerical solvers thereby removing the principal computational bottlenecks of existing wrapped Gaussian approaches. Empirical results across a range of models demonstrate that the proposed approximation captures complex posterior geometries while remaining orders of magnitude faster than current geometric state-of-the-art approximation.

stat.ME↗

arXiv2026

On the Geometric Ergodicity of Hamiltonian Monte Carlo

We establish general conditions under which Markov chains produced by the Hamiltonian Monte Carlo method will and will not be geometrically ergodic. We consider implementations with both position-independent and position-dependent integration times. In the former case we find that the conditions for geometric ergodicity are essentially a gradient of the log-density which asymptotically points towards the centre of the space and grows no faster than linearly. In an idealised scenario in which the integration time is allowed to change in different regions of the space, we show that geometric ergodicity can be recovered for a much broader class of tail behaviours, leading to some guidelines for the choice of this free parameter in practice.

stat.CO↗

arXiv2026

Autoencoders in Function Space

Autoencoders have found widespread application in both their original deterministic form and in their variational formulation (VAEs). In scientific applications and in image processing it is often of interest to consider data that are viewed as functions; while discretisation (of differential equations arising in the sciences) or pixellation (of images) renders problems finite dimensional in practice, conceiving first of algorithms that operate on functions, and only then discretising or pixellating, leads to better algorithms that smoothly operate between resolutions. In this paper function-space versions of the autoencoder (FAE) and variational autoencoder (FVAE) are introduced, analysed, and deployed. Well-definedness of the objective governing VAEs is a subtle issue, particularly in function space, limiting applicability. For the FVAE objective to be well defined requires compatibility of the data distribution with the chosen generative model; this can be achieved, for example, when the data arise from a stochastic differential equation, but is generally restrictive. The FAE objective, on the other hand, is well defined in many situations where FVAE fails to be. Pairing the FVAE and FAE objectives with neural operator architectures that can be evaluated on any mesh enables new applications of autoencoders to inpainting, superresolution, and generative modelling of scientific data.

stat.ML↗

arXiv2026

Distributional Inverse Homogenization

For many materials, macroscopic mechanical behavior is determined by an intricate microstructure. Understanding the relation between these two scales helps scientists and engineers design better materials. The relation which maps microstructure to bulk material properties can be understood via the well-established theory of homogenization. However inverting the homogenization process, to recover microstructural information from measured macroscopic properties, is fraught with difficulties because of the averaging processes that underlie homogenization. Therefore, scientists and engineers usually need recourse to more invasive, often highly localized, investigations to estimate the microstructure. In this work, we develop a noninvasive methodology by which one can leverage large collections of measured bulk material properties to infer information about the statistics of microstructure at a global level. We call this, distributional inverse homogenization. We study this problem in one and two dimensions, considering both periodic and stochastic homogenization. We demonstrate the methodology in the context of 2D Voronoi constructions and underpin the observed empirical success with theory in 1D. We also show how the natural spatial variability of microstructure can be exploited to gather data that enables distributional inversion. And we concurrently learn a surrogate model, approximating the homogenization map, that accelerates the resulting computations in this setting. The work formulates a new class of inverse problems, bridging ideas from probability and homogenization to facilitate the inference of microstructural material variability from macroscopic measurements.

physics.comp-ph↗

arXiv2026

CoBit: Language Modeling with Bitstream Diffusion

Diffusion language models (DLMs) promise parallel, order-agnostic generation, but on standard benchmarks they have historically lagged behind autoregressive models in sample quality and diversity. Recent continuous flow and diffusion approaches have narrowed this gap. In this work, we further close the autoregressive gap by modeling text as a continuous diffusion process over fixed-width binary bitstreams. We refer to the resulting model as CoBit (Continuous Bitstream Diffusion). Our approach represents semantic tokens as analog bit sequences and uses a matched-filter residual parameterization to isolate contextual learning from analytic independent-bit posteriors. Crucially, we adopt a stochastic sampler that applies Langevin-type corrections gated by the entropy-rate profile, concentrating stochasticity in high-information regions while remaining nearly deterministic elsewhere. On LM1B, our 130M-parameter model reaches a generative perplexity (GenPPL) of 59.76 at matched real-data entropy (4.31) using 256 neural function evaluations (NFEs), outperforming prior DLM baselines and reaching the autoregressive reference. On OpenWebText (OWT), our sampler establishes a new continuous-DLM Pareto frontier, achieving GenPPL 27.06 at entropy 5.26 using 4x fewer steps than previous 1024-NFE baselines. Scaling the same recipe to a 462M-parameter model (CoBit-M) further improves the OWT GenPPL-entropy frontier over the 130M model (CoBit-S) and over medium-scale continuous and discrete DLM baselines, reaching GenPPL 19.5 at entropy 5.40, near real-data entropy (5.44), and approaching pretrained GPT-2 Medium over the high-quality region. As an additional benefit, bitstream diffusion removes the O(V) vocabulary scaling bottleneck of standard DLMs: by predicting O(log V) bitwise logits via semantic bit-patching, it lowers memory and raises throughput, a scalable paradigm as vocabulary sizes grow.

cs.CL↗

arXiv2026

Geometric Autoencoder Priors for Bayesian Inversion: Learn First Observe Later

Uncertainty Quantification (UQ) is paramount for inference in engineering. A common inference task is to recover full-field information of physical systems from a small number of noisy observations, a usually highly ill-posed problem. Sharing information from multiple distinct yet related physical systems can alleviate this ill-posedness. Critically, engineering systems often have complicated variable geometries prohibiting the use of standard multi-system Bayesian UQ. In this work, we introduce Geometric Autoencoders for Bayesian Inversion (GABI), a framework for learning geometry-aware generative models of physical responses that serve as highly informative geometry-conditioned priors for Bayesian inversion. Following a ''learn first, observe later'' paradigm, GABI distills information from large datasets of systems with varying geometries, without requiring knowledge of governing PDEs, boundary conditions, or observation processes, into a rich latent prior. At inference time, this prior is seamlessly combined with the likelihood of a specific observation process, yielding a geometry-adapted posterior distribution. Our proposed framework is architecture-agnostic. A creative use of Approximate Bayesian Computation (ABC) sampling yields an efficient implementation that utilizes modern GPU hardware. We test our method on: steady-state heat over rectangular domains; Reynolds-Averaged Navier-Stokes (RANS) flow around airfoils; Helmholtz resonance and source localization on 3D car bodies; RANS airflow over terrain. We find: the predictive accuracy to be comparable to deterministic supervised learning approaches in the restricted setting where supervised learning is applicable; UQ to be well calibrated and robust on challenging problems with complex geometries.

stat.ML↗

arXiv2026

Koopman Autoencoders with Continuous-Time Latent Dynamics for Fluid Dynamics Forecasting

Forecasting physical systems over long horizons from irregularly sampled observations demands models that are stable, computationally efficient, and free of fixed-timestep assumptions. We address this with a continuous-time Koopman autoencoder whose latent dynamics obey $dz/dt = \mathbf{K}_{\mathrm{cont}} z$, yielding closed-form inference via $z(τ) = \exp(\mathbf{K}_{\mathrm{cont}} τ) z(0)$ at any horizon $τ$ in a single step. This decouples forecast cost from forecast length at inference time and supports data assimilation as gradient-based optimization with cost independent of the assimilation window. However, scaling continuous-time Koopman dynamics to high-dimensional chaotic systems causes severe latent instability, including spectral collapse and trajectory divergence over long horizons. In contrast, discrete Koopman methods train an operator $\mathbf{A}$ such that $z_{t+Δt} = \mathbf{A} z_t$; recovering the continuous generator could be theoretically done through matrix logarithm but requires conditions not guaranteed by training, and approximation errors grow with the $Δt$ imposed by the training data. These methods also require fixed, regular timesteps. We identify an empirically effective set of structural constraints -- rollout training, forward-backward consistency, latent regularization, and physics-conditioned LoRA -- sufficient for stable long-horizon latent dynamics. On challenging fluid benchmarks, our method outperforms strong diffusion and operator-learning baselines on long-horizon forecasting while achieving a 110$\times$ inference speedup.

cs.LG↗

arXiv2026

Efficient Deconvolution in Populational Inverse Problems

This work is focussed on the inversion task of inferring the distribution over parameters of interest leading to multiple sets of observations. The potential to solve such distributional inversion problems is driven by increasing availability of data, but a major roadblock is blind deconvolution, arising when the observational noise distribution is unknown. However, when data originates from collections of physical systems, a population, it is possible to leverage this information to perform deconvolution. To this end, we propose a methodology leveraging large data sets of observations, collected from different instantiations of the same physical processes, to simultaneously deconvolve the data corrupting noise distribution, and to identify the distribution over model parameters defining the physical processes. A parameter-dependent mathematical model of the physical process is employed. A loss function characterizing the match between the observed data and the output of the mathematical model is defined; it is minimized as a function of the both the parameter inputs to the model of the physics and the parameterized observational noise. This coupled problem is addressed with a modified gradient descent algorithm that leverages specific structure in the noise model. Furthermore, a new active learning scheme is proposed, based on adaptive empirical measures, to train a surrogate model to be accurate in parameter regions of interest; this approach accelerates computation and enables automatic differentiation of black-box, potentially nondifferentiable, code computing parameter-to-solution maps. The proposed methodology is demonstrated on porous medium flow, damped elastodynamics, and simplified models of atmospheric dynamics.

stat.ML↗

arXiv2026

Towards Efficient and Stable Ocean State Forecasting: A Continuous-Time Koopman Approach

We investigate the Continuous-Time Koopman Autoencoder (CT-KAE) as a lightweight surrogate model for long-horizon ocean state forecasting in a two-layer quasi-geostrophic (QG) system. By projecting nonlinear dynamics into a latent space governed by a linear ordinary differential equation, the model enforces structured and interpretable temporal evolution while enabling temporally resolution-invariant forecasting via a matrix exponential formulation. Across 2083-day rollouts, CT-KAE exhibits bounded error growth and stable large-scale statistics, in contrast to autoregressive Transformer baselines which exhibit gradual error amplification and energy drift over long rollouts. While fine-scale turbulent structures are partially dissipated, bulk energy spectra, enstrophy evolution, and autocorrelation structure remain consistent over long horizons. The model achieves orders-of-magnitude faster inference compared to the numerical solver, suggesting that continuous-time Koopman surrogates offer a promising backbone for efficient and stable physical-machine learning climate models.

cs.LG↗

arXiv2026

Riemannian Laplace Approximation with the Fisher Metric

Laplace's method approximates a target density with a Gaussian distribution at its mode. It is computationally efficient and asymptotically exact for Bayesian inference due to the Bernstein-von Mises theorem, but for complex targets and finite-data posteriors it is often too crude an approximation. A recent generalization of the Laplace Approximation transforms the Gaussian approximation according to a chosen Riemannian geometry providing a richer approximation family, while still retaining computational efficiency. However, as shown here, its properties depend heavily on the chosen metric, indeed the metric adopted in previous work results in approximations that are overly narrow as well as being biased even at the limit of infinite data. We correct this shortcoming by developing the approximation family further, deriving two alternative variants that are exact at the limit of infinite data, extending the theoretical analysis of the method, and demonstrating practical improvements in a range of experiments.

cs.LG↗

arXiv2026

Hierarchical Inference and Closure Learning via Adaptive Surrogates for ODEs and PDEs

Inverse problems are the task of calibrating models to match data. They play a pivotal role in diverse engineering applications by allowing practitioners to align models with reality. In many applications, engineers and scientists do not have a complete picture of i) the detailed properties of a system (such as material properties, geometry, initial conditions, etc.); ii) the complete laws describing all dynamics at play (such as friction laws, complicated damping phenomena, and general nonlinear interactions). In this paper, we develop a principled methodology for leveraging data from collections of distinct yet related physical systems to jointly estimate the individual model parameters of each system, and learn the shared unknown dynamics in the form of an ML-based closure model. To robustly infer the unknown parameters for each system, we employ a hierarchical Bayesian framework, which allows for the joint inference of multiple systems and their population-level statistics. To learn the closures, we use a maximum marginal likelihood estimate of a neural network embeded within the ODE/PDE formulation of the problem. To realize this framework we utilize the ensemble Metropolis-Adjusted Langevin Algorithm (MALA) for stable and efficient sampling. To mitigate the computational bottleneck of repetitive forward evaluations in solving inverse problems, we introduce a bilevel optimization strategy to simultaneously train a surrogate forward model alongside the inference. Within this framework, we evaluate and compare distinct surrogate architectures, specifically Fourier Neural Operators (FNO) and parametric Physics-Informed Neural Network (PINNs).

cs.LG↗

arXiv2026

Bayesian modelling and quantification of Raman spectroscopy

Raman spectroscopy can be used to identify molecules such as DNA by the characteristic scattering of light from a laser. It is sensitive at very low concentrations and can accurately quantify the amount of a given molecule in a sample. The presence of a large, nonuniform background presents a major challenge to analysis of these spectra. To overcome this challenge, we introduce a sequential Monte Carlo (SMC) algorithm to separate the observed spectrum into a series of peaks plus a smoothly-varying baseline, corrupted by additive white noise. The peaks are modelled using Lorentzian or Gaussian broadening functions, while the baseline is estimated using a penalised cubic spline. This latent continuous representation accounts for differences in resolution between measurements. By incorporating this representation in a Bayesian model, we can quantify the relationship between molecular concentration and peak intensity, thereby providing an improved estimate of the limit of detection (LOD), which is of major importance in analytical chemistry.

stat.AP↗

arXiv2025

Retrieval-augmented reasoning with lean language models

This technical report details a novel approach to combining reasoning and retrieval augmented generation (RAG) within a single, lean language model architecture. While existing RAG systems typically rely on large-scale models and external APIs, our work addresses the increasing demand for performant and privacy-preserving solutions deployable in resource-constrained or secure environments. Building on recent developments in test-time scaling and small-scale reasoning models, we develop a retrieval augmented conversational agent capable of interpreting complex, domain-specific queries using a lightweight backbone model. Our system integrates a dense retriever with fine-tuned Qwen2.5-Instruct models, using synthetic query generation and reasoning traces derived from frontier models (e.g., DeepSeek-R1) over a curated corpus, in this case, the NHS A-to-Z condition pages. We explore the impact of summarisation-based document compression, synthetic data design, and reasoning-aware fine-tuning on model performance. Evaluation against both non-reasoning and general-purpose lean models demonstrates that our domain-specific fine-tuning approach yields substantial gains in answer accuracy and consistency, approaching frontier-level performance while remaining feasible for local deployment. All implementation details and code are publicly released to support reproducibility and adaptation across domains.

cs.CL↗

arXiv2025

Error analysis for a statistical finite element method

The recently proposed statistical finite element (statFEM) approach synthesises measurement data with finite element models and allows for making predictions about the unknown true system response. We provide a probabilistic error analysis for a prototypical statFEM setup based on a Gaussian process prior under the assumption that the noisy measurement data are generated by a deterministic true system response function that satisfies a second-order elliptic partial differential equation for an unknown true source term. In certain cases, properties such as the smoothness of the source term may be misspecified by the Gaussian process model. The error estimates we derive are for the expectation with respect to the measurement noise of the $L^2$-norm of the difference between the true system response and the mean of the statFEM posterior. The estimates imply polynomial rates of convergence in the numbers of measurement points and finite element basis functions and depend on the Sobolev smoothness of the true source term and the Gaussian process model. A numerical example for Poisson's equation is used to illustrate these theoretical results.

math.ST↗

arXiv2025

Statistical Finite Elements via Interacting Particle Langevin Dynamics

In this paper, we develop a class of interacting particle Langevin algorithms to solve inverse problems for partial differential equations (PDEs). In particular, we leverage the statistical finite elements (statFEM) formulation to obtain a finite-dimensional latent variable statistical model where the parameter is that of the (discretised) forward map and the latent variable is the statFEM solution of the PDE which is assumed to be partially observed. We then adapt a recently proposed expectation-maximisation like scheme, interacting particle Langevin algorithm (IPLA), for this problem and obtain a joint estimation procedure for the parameters and the latent variables. We consider three main examples: (i) estimating the forcing for linear Poisson PDE, (ii) estimating diffusivity for linear Poisson PDE, and (iii) estimating the forcing for nonlinear Poisson PDE. We provide computational complexity estimates for forcing estimation in the linear case. We also provide comprehensive numerical experiments and preconditioning strategies that significantly improve the performance, showing that the proposed class of methods can be the choice for parameter inference in PDE models.

stat.CO↗

arXiv2025

Efficient Prior Calibration From Indirect Data

Bayesian inversion is central to the quantification of uncertainty within problems arising from numerous applications in science and engineering. To formulate the approach, four ingredients are required: a forward model mapping the unknown parameter to an element of a solution space, often the solution space for a differential equation; an observation operator mapping an element of the solution space to the data space; a noise model describing how noise pollutes the observations; and a prior model describing knowledge about the unknown parameter before the data is acquired. This paper is concerned with learning the prior model from data; in particular, learning the prior from multiple realizations of indirect data obtained through the noisy observation process. The prior is represented, using a generative model, as the pushforward of a Gaussian in a latent space; the pushforward map is learned by minimizing an appropriate loss function. A metric that is well-defined under empirical approximation is used to define the loss function for the pushforward map to make an implementable methodology. Furthermore, an efficient residual-based neural operator approximation of the forward model is proposed and it is shown that this may be learned concurrently with the pushforward map, using a bilevel optimization formulation of the problem; this use of neural operator approximation has the potential to make prior learning from indirect data more computationally efficient, especially when the observation process is expensive, non-smooth or not known. The ideas are illustrated with the Darcy flow inverse problem of finding permeability from piezometric head measurements.

stat.ML↗

arXiv2025

A Primer on Variational Inference for Physics-Informed Deep Generative Modelling

Variational inference (VI) is a computationally efficient and scalable methodology for approximate Bayesian inference. It strikes a balance between accuracy of uncertainty quantification and practical tractability. It excels at generative modelling and inversion tasks due to its built-in Bayesian regularisation and flexibility, essential qualities for physics related problems. For such problems, the underlying physical model determines the dependence between variables of interest, which in turn will require a tailored derivation for the central VI learning objective. Furthermore, in many physical inference applications this structure has rich meaning and is essential for accurately capturing the dynamics of interest. In this paper, we provide an accessible and thorough technical introduction to VI for forward and inverse problems, guiding the reader through standard derivations of the VI framework and how it can best be realized through deep learning. We then review and unify recent literature exemplifying the flexibility allowed by VI. This paper is designed for a general scientific audience looking to solve physics-based problems with an emphasis on uncertainty quantification

stat.ML↗

arXiv2025

Targeted Separation and Convergence with Kernel Discrepancies

Maximum mean discrepancies (MMDs) like the kernel Stein discrepancy (KSD) have grown central to a wide range of applications, including hypothesis testing, sampler selection, distribution approximation, and variational inference. In each setting, these kernel-based discrepancy measures are required to (i) separate a target P from other probability measures or even (ii) control weak convergence to P. In this article we derive new sufficient and necessary conditions to ensure (i) and (ii). For MMDs on separable metric spaces, we characterize those kernels that separate Bochner embeddable measures and introduce simple conditions for separating all measures with unbounded kernels and for controlling convergence with bounded kernels. We use these results on $\mathbb{R}^d$ to substantially broaden the known conditions for KSD separation and convergence control and to develop the first KSDs known to exactly metrize weak convergence to P. Along the way, we highlight the implications of our results for hypothesis testing, measuring and improving sample quality, and sampling with Stein variational gradient descent.

stat.ML↗