Finite-element Gaussian processes for the machine learning of steady-state linear partial differential equations - Research article overview

Aug 5, 2026

Introduces finite-element Gaussian processes (FEGPs) as a physics-informed machine learning framework for inverse problems governed by steady-state, linear PDEs. The method combines a Gaussian process prior for the unknown solution with a likelihood enforcing the PDE through its weak form, approximated using finite-element basis functions. By avoiding explicit kernel differentiation required by strong-form physics-informed GPs (PIGPs), the approach improves computational scalability. Numerical experiments on synthetic benchmarks show better results than PIGPs and competitiveness with physics-informed neural networks, with improved uncertainty quantification.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages22

Reading Time55 min

AccessFree to download

Introduction

Physics-informed machine learning context

Physics-informed neural networks and uncertainty

Background

Governing equations

| Finite-element Gaussian processes for the machine learning of steady-state linear partial diﬀerential equations
David Dalton∗, Hao Gao , Dirk Husmeier
University of Glasgow, G12 8QQ, United Kingdom | | | | | --- | --- | --- | --- | | a r t i c l e i n f o | | a b s t r a c t | | | Keywords:
Physics-informed machine learning Gaussian processes
Finite-elements
Inverse problems | | We introduce ﬁnite-element Gaussian processes (FEGPs), a novel physics-informed machine learning approach for solving inverse problems involving steady-state, linear partial diﬀerential equations (PDEs). Our framework combines a Gaussian process prior for the unknown solution function with a likelihood that incorporates the PDE in its weak form, using a ﬁnite-element approximation. This approach oﬀers signiﬁcantly better scalability than physics-informed Gaussian processes (PIGPs), which rely on the strong form of the PDE. Through numerical experiments on a range of synthetic benchmark problems, we show that FEGPs oﬀer results which outperform PIGPs, and are competitive with physics-informed neural networks (PINNs) with improved uncertainty quantiﬁcation. | | 1. Introduction Over the past generation, scientiﬁc research has witnessed substantial advances in the modelling of multiphysics phenomena, with applications spanning chemistry, molecular biology, geophysics, and climate forecasting [1]. These models are typically expressed in the form of partial diﬀerential equations (PDEs), which are often suﬃciently complex to necessitate numerical solution techniques. Despite parallel progress in the development of numerical PDE solvers-such as the ﬁnite-element and ﬁnite-diﬀerence methods [2] -signiﬁcant challenges persist in eﬀectively integrating these solvers with observational data. These challenges are particularly acute in inverse problems, where the aim is to infer latent physical properties of a system, such as material stiﬀness or thermal conductivity. In such cases, conventional solvers can become computationally prohibitive [3], and can struggle to adapt to ill-posed settings, for instance where boundary condition information is incomplete. Physics-informed machine learning (PIML) has recently emerged as a promising alternative framework for modelling physical systems. PIML encompasses a variety of techniques that integrate data-driven machine learning models with physics-based mathematical descriptions, facilitating a coherent fusion of empirical observations with established physical laws [4]. These methods address several limitations of traditional numerical solvers, particularly when dealing with ill-posed problems [3]. Among these approaches, physics-informed neural networks (PINNs) [5] have gained prominence due to their ﬂexibility and computational eﬃciency [6,7]. PINNs approximate the unknown state of a physical system using a neural network, which is trained to minimise a composite loss function that penalises both deviations from observational data and violations of the governing physical laws, typically encoded as PDEs. In practical applications, accounting for predictive uncertainty-particularly that arising from incomplete or noisy data-is often essential. This need has led to the development of probabilistic PIML methods, among which Gaussian process (GP) regression has ∗ Corresponding author. E-mail addresses: [david.dalton@glasgow.ac.uk](david.dalton@glasgow.ac.uk) (D. Dalton), [hao.gao@glasgow.ac.uk](hao.gao@glasgow.ac.uk) (H. Gao), [dirk.husmeier@glasgow.ac.uk](dirk.husmeier@glasgow.ac.uk) (D. Husmeier). [https://doi.org/10.1016/j.cma.2025.118580](https://doi.org/10.1016/j.cma.2025.118580) Received 21 July 2025; Received in revised form 15 October 2025; Accepted 11 November 2025 Available online 20 December 2025 0045-7825/© 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license ([http://creativecommons.org/licenses/by/4.0/](http://creativ

FAQs

What are finite-element Gaussian processes (FEGPs) in this work?

FEGPs are a probabilistic physics-informed machine learning approach that uses a Gaussian process prior and a PDE-informed likelihood built from the weak form using finite-element basis functions.

How do FEGPs differ from physics-informed Gaussian processes (PIGPs)?

FEGPs integrate the PDE via its weak form, avoiding the need to explicitly differentiate the GP kernel, whereas PIGPs rely on the strong form and require kernel differentiation.

How is model performance and uncertainty assessed?

Performance and efficiency are evaluated theoretically and through synthetic numerical experiments, showing FEGPs outperform PIGPs and are competitive with physics-informed neural networks while providing improved uncertainty quantification.

Aug 5, 2026

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