Accepted Papers

• Fixing the Pitfalls of Probabilistic Time-Series Forecasting Evaluation by Kernel Quadrature.
  • Masaki Adachi, Masahiro Fujisawa, Michael A Osborne.
    Abstract

    Despite the significance of probabilistic time-series forecasting models, their evaluation metrics often involve intractable integrations. The most widely used metric, the continuous ranked probability score (CRPS), is a strictly proper scoring function; however, its computation requires approximation. We found that popular CRPS estimators—specifically, the quantile-based estimator implemented in the widely used GluonTS library and the probability-weighted moment approximation—both exhibit inherent estimation biases. These biases lead to crude approximations, potentially resulting in improper rankings of forecasting model performance. To address this, we introduced a kernel quadrature approach that leverages an unbiased CRPS estimator and employs cubature construction for scalable computation. Empirically, our approach consistently outperforms the two widely used CRPS estimators.

• GaussED: A Python Package for Sequential Experimental Design.
  • Matthew A Fisher, Onur Teymur, Chris J. Oates
    Abstract

    Sequential algorithms are popular for experimental design, enabling emulation, optimisation and inference to be efficiently performed. For most of these applications bespoke software has been developed, but the approach is general and many of the actual computations performed in such software are identical. Motivated by the diverse problems that can in principle be solved with common code, this paper presents GaussED, a high-level syntax coupled to a powerful experimental design engine in Python, which together automate sequential experimental design for approximating a (possibly nonlinear) quantity of interest in Gaussian processes models. Using a handful of commands, GaussED can be used to: solve linear partial differential equations, perform tomographic reconstruction from integral data, implement Bayesian optimisation with gradient data, and emulate a complex computer model.

• Adaptive Probabilistic ODE Solvers Without Adaptive Memory Requirements.
  • Nicholas Krämer
    Abstract

    Despite substantial progress in recent years, probabilistic solvers with adaptive step sizes can still not solve memory-demanding differential equations unless we care only about a single point in time (which is far too restrictive; we want the whole time series). Counterintuitively, the culprit is the adaptivity itself: Its unpredictable memory demands easily exceed our machine's capabilities, making our simulations fail unexpectedly and without warning. Still, dropping adaptivity would abandon years of progress, which can't be the answer. In this work, we solve this conundrum. We develop an adaptive probabilistic solver with fixed memory demands building on recent developments in robust state estimation. Switching to our method (i) eliminates memory issues for long time series, (ii) accelerates simulations by orders of magnitude through unlocking just-in-time compilation, and (iii) makes adaptive probabilistic solvers compatible with scientific computing in JAX.

• Propagating Model Uncertainty through Filtering-based Probabilistic Numerical ODE Solvers.
  • Dingling Yao, Filip Tronarp, Nathanael Bosch
    Abstract

    Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs), also known as ODE filters, have been established as efficient methods for quantifying numerical uncertainty in the solution of ODEs. In practical applications, however, the underlying dynamical system often contains uncertain parameters, requiring the propagation of this model uncertainty to the ODE solution. In this paper, we demonstrate that ODE filters, despite their probabilistic nature, do not automatically solve this uncertainty propagation problem. To address this limitation, we present a novel approach that combines ODE filters with numerical quadrature to properly marginalize over uncertain parameters, while accounting for both parameter uncertainty and numerical solver uncertainty. Experiments across multiple dynamical systems demonstrate that the resulting uncertainty estimates closely match reference solutions. Notably, we show how the numerical uncertainty from the ODE solver can help prevent overconfidence in the propagated uncertainty estimates, especially when using larger step sizes. Our results illustrate that probabilistic numerical methods can effectively quantify both numerical and parametric uncertainty in dynamical systems.

• Fast Gaussian process regression for high dimensional functions with derivative information.
  • Aleksei Sorokin, Pieterjan Robbe, Fred J Hickernell
    Abstract

    Gaussian process regression (GPR) is the backbone of many methods in probabilistic numerics. Exact GPR on \(n\) sampling locations in \(d\) dimensions generally costs \(\mathcal{O}(n^3)\) to fit and requires \(\mathcal{O}(n^2)\) storage. Using certain pairings of sampling locations and kernels induces nice structure into the Gram matrix and enables accelerated exact GPR. One popular pairing uses Cartesian grids with product kernels to induce Kronecker structure in the Gram matrix. This reduces exact GP fitting costs to \(\mathcal{O}(d n^{3/d})\) and storage requirements to \(\mathcal{O}(d n^{2/d})\), but quickly becomes intractable when the dimension exceeds a few dozen. Recent work has shown that pairings of certain low-discrepancy sequences with special kernels enable GPR to scale like \(\mathcal{O}(n \log n)\) for fitting costs and \(\mathcal{O}(n)\) for storage requirements. We describe an extension of these methods to problems which observe \(m\) derivative multi-indices at each of the \(n\) low-discrepancy sampling locations. By exploiting similar structure across blocks of the Gram matrix, we are able to reduce the GP fitting cost from \(\mathcal{O}(n^3 m^3)\) to \(\mathcal{O}(n^2 \log n + m^3 n)\) and reduce the storage requirements from \(\mathcal{O}(n^2 m^2)\) to \(\mathcal{O}(n m^2)\). We explore a number of synthetic benchmarks to illustrate the potential of the proposed approach.

• Natural Evolutionary Search meets Probabilistic Numerics.
  • Pierre Osselin, Masaki Adachi, Xiaowen Dong, Michael A Osborne
    Abstract

    Zeroth-order local optimisation algorithms are essential for solving real-valued black-box optimisation problems. Among these, Natural Evolution Strategies (NES) represent a prominent class, particularly well-suited for scenarios where prior distributions are available. By optimising the objective function in the space of search distributions, NES algorithms naturally integrate prior knowledge during initialisation, making them effective in settings such as semi-supervised learning and user-prior belief frameworks. However, due to their reliance on random sampling and Monte Carlo estimates, NES algorithms can suffer from limited sample efficiency. In this paper, we introduce a novel class of algorithms, termed Probabilistic Natural Evolutionary Strategy Algorithms (ProbNES), which enhance the NES framework with Bayesian quadrature. We show that ProbNES algorithms consistently outperforms their non-probabilistic counterparts as well as global sample efficient methods such as Bayesian Optimisation (BO) or $\pi$BO across a wide range of tasks, including benchmark test functions, data-driven optimisation tasks, user-informed hyperparameter tuning tasks and locomotion tasks.

• Randomised Postiterations for Calibrated BayesCG.
  • Niall Vyas, Disha Hegde, Jon Cockayne
    Abstract

    The Bayesian conjugate gradient method offers probabilistic solutions to linear systems but suffers from poor calibration, limiting its utility in uncertainty quantification tasks. Recent approaches leveraging postiterations to construct priors have improved computational properties but failed to correct calibration issues. In this work, we propose a novel randomised postiteration strategy that enhances the calibration of the BayesCG posterior while preserving its favourable convergence characteristics. We present theoretical guarantees for the improved calibration, supported by results on the distribution of posterior errors. Numerical experiments demonstrate the efficacy of the method in both synthetic and inverse problem settings, showing enhanced uncertainty quantification and better propagation of uncertainties through computational pipelines.

• A Dictionary of Closed-Form Kernel Mean Embeddings.
  • Francois-Xavier Briol, Toni Karvonen, Alexandra Gessner, Maren Mahsereci
    Abstract

    Kernel mean embeddings -- integrals of a kernel with respect to a probability distribution -- are essential in Bayesian quadrature, but also widely used in other computational tools for numerical integration or methods for statistical inference. These methods often require, or are enhanced by, the availability of a closed-form expression for the kernel mean embedding. However, deriving such expressions can be challenging, limiting the applicability of kernel-based techniques when practitioners do not have access to a closed-form embedding. This paper addresses this limitation by providing a comprehensive dictionary of known kernel mean embeddings, along with practical tools for deriving new embeddings from known ones. We also provide a Python library that includes minimal implementations of the embeddings.

• Effects of Interpolation Error and Bias on the Random Mesh Finite Element Method for Inverse Problems.
  • Anne Poot, Iuri Rocha, Pierre Kerfriden, Frans van der Meer
    Abstract

    Bayesian inverse problems are an important application for probabilistic solvers of partial differential equations: when fully resolving numerical error is computationally infeasible, probabilistic solvers can be used to consistently model the error and propagate it to the posterior. In this work, the performance of the random mesh finite element method (RM-FEM) is investigated in a Bayesian inverse setting. We show how interpolation error negatively affects the RM-FEM posterior, and how these negative effects can be diminished. In scenarios where FEM is biased for a quantity of interest, we find that RM-FEM struggles to accurately model this bias.

• Learning to Solve Related Linear Systems.
  • Disha Hegde, Jon Cockayne
    Abstract

    Solving multiple parametrised related systems is an essential component of many numerical tasks. Borrowing strength from the solved systems and learning will make this process faster. In this work, we propose a novel probabilistic linear solver over the parameter space. This leverages information from the solved linear systems in a regression setting to provide an efficient posterior mean and covariance. We advocate using this as companion regression model for the preconditioned conjugate gradient method, and discuss the favourable properties of the posterior mean and covariance as the initial guess and preconditioner. We also provide several design choices for this companion solver. Numerical experiments showcase the benefits of using our novel solver in a hyperparameter optimisation problem.

• Bayesian autoregression to optimize temporal Matérn-kernel Gaussian process hyperparameters.
  • Wouter M. Kouw
    Abstract

    We present a probabilistic numerical procedure for optimizing Matérn-class temporal Gaussian processes with respect to the kernel covariance function's hyperparameters. It is based on casting the optimization problem as a recursive Bayesian estimation procedure for the parameters of an autoregressive model. The recursive nature means that there is a initial value that should improve after every update, much like iterative local optimization techniques. We demonstrate that the proposed procedure outperforms the standard maximum marginal likelihood-based approach in both runtime and ultimate root mean square error in Gaussian process regression.

• Solving Einstein’s equations as Bayesian regression.
  • Frederik De Ceuster, Tom Colemont, Tjonnie G.F. Li
    Abstract

    Gravitational waves (GWs) are revolutionising our fundamental understanding of physics and cosmology. However, the numerical modelling required to turn their measurement into a scientific detection poses a formidable computational challenge. In this paper, we explore whether a Bayesian view can help enhance the computational efficiency of GW source models. As a proof-of-principle, we pose the solution of the Einstein equations, which relate the dynamics of spacetime to its matter content, as a Bayesian regression problem. By choosing natural priors, based on Green's functions of the relevant operators, we open up ways to better target computing power in our numerical models. Therefore, we conclude that probabilistic numerics is a promising approach to overcome the computational challenges in GW science.

• Online Conformal Probabilistic Numerics via Adaptive Edge-Cloud Offloading.
  • Qiushuo Hou, Sangwoo Park, Matteo Zecchin, Yunlong Cai, Guanding Yu, Osvaldo Simeone
    Abstract

    Consider an edge computing setting in which a user submits queries for the solution of a linear system to an edge processor, which is subject to time-varying computing availability. The edge processor applies a probabilistic linear solver (PLS) so as to be able to respond to the user’s query within the allotted time and computing budget. Feedback to the user is in the form of an uncertainty set. Due to model misspecification, the uncertainty set obtained via a direct application of PLS does not come with coverage guarantees with respect to the true solution of the linear system. This work introduces a new method to calibrate the uncertainty sets produced by PLS with the aim of guaranteeing long-term coverage requirements. The proposed method, referred to as online conformal prediction-PLS (OCP-PLS), assumes sporadic feedback from cloud to edge. This enables the online calibration of uncertainty thresholds via online conformal prediction (OCP), an online optimization method previously studied in the context of prediction models. The validity of OCP-PLS is verified via experiments that bring insights into trade-offs between coverage, prediction set size, and cloud usage. </p </details>