Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers

Pförtner, Marvin; Steinwart, Ingo; Hennig, Philipp; Wenger, Jonathan

doi:10.48550/arxiv.2212.12474

Cited by 1 publication

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“…Another area that has combined physics and GPs is in the modeling of ordinary and partial differential equations (ODE/PDEs). In the work of Pförtner et al (2022), GPs conditioned on observations resulting from linear operators are shown to provide a general framework for solving linear PDEs, accommodating scenarios where physics are partially known (i.e., the full governing equations are unavailable), as well as propagating error resulting from both parameter and measurement uncertainty. In a similar vein, Raissi et al (2017a and demonstrate that by constructing a joint distribution between successive time or spatial points, a GP prior can be constructed that can explicitly learn solutions of partial differential equations, coining the term "numerical Gaussian processes.…”

Section: Related Workmentioning

confidence: 99%

A spectrum of physics-informed Gaussian processes for regression in engineering

Cross,

Rogers,

Pitchforth

et al. 2024

DCE

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Despite the growing availability of sensing and data in general, we remain unable to fully characterize many in-service engineering systems and structures from a purely data-driven approach. The vast data and resources available to capture human activity are unmatched in our engineered world, and, even in cases where data could be referred to as “big,” they will rarely hold information across operational windows or life spans. This paper pursues the combination of machine learning technology and physics-based reasoning to enhance our ability to make predictive models with limited data. By explicitly linking the physics-based view of stochastic processes with a data-based regression approach, a derivation path for a spectrum of possible Gaussian process models is introduced and used to highlight how and where different levels of expert knowledge of a system is likely best exploited. Each of the models highlighted in the spectrum have been explored in different ways across communities; novel examples in a structural assessment context here demonstrate how these approaches can significantly reduce reliance on expensive data collection. The increased interpretability of the models shown is another important consideration and benefit in this context.

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Section: Related Workmentioning

confidence: 99%