# Bayesian Calibration of Imperfect Computer Models using Physics-Informed Priors

arXiv: https://arxiv.org/abs/2201.06463 · HTML: https://mlsanity.com/paper/2201.06463/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 51 references that are in the map.
- First posted on arXiv 17 Jan 2022.

## Builds on (most cited first, 11 of 11)

- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations](https://mlsanity.com/paper/1703.10230/) (2017)
- [Understanding Probabilistic Sparse Gaussian Process Approximations](https://mlsanity.com/paper/1606.04820/) (2016)
- [Local Gaussian process approximation for large computer experiments](https://mlsanity.com/paper/1303.0383/) (2013)
- [MCMC for Variationally Sparse Gaussian Processes](https://mlsanity.com/paper/1506.04000/) (2015)
- [Random Feature Expansions for Deep Gaussian Processes](https://mlsanity.com/paper/1610.04386/) (2016)
- [APIK: Active Physics-Informed Kriging Model with Partial Differential Equations](https://mlsanity.com/paper/2012.11798/) (2020)
- [Calibrating a large computer experiment simulating radiative shock hydrodynamics](https://mlsanity.com/paper/1410.3293/) (2014)
- [Uncertainty quantification for computer models with spatial output using calibration-optimal bases](https://mlsanity.com/paper/1801.08184/) (2018)
- [Computer model calibration with large non-stationary spatial outputs: application to the calibration of a climate model](https://mlsanity.com/paper/1604.04478/) (2016)
- [A composite likelihood approach to computer model calibration using high-dimensional spatial data](https://mlsanity.com/paper/1308.0049/) (2013)

## Built on by (most cited first, 9 of 9)

- [Gaussian processes for Bayesian inverse problems associated with linear partial differential equations](https://mlsanity.com/paper/2307.08343/) (2023)
- [Parameter Inference based on Gaussian Processes Informed by Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2212.11880/) (2022)
- [Learning Physics between Digital Twins with Low-Fidelity Models and Physics-Informed Gaussian Processes](https://mlsanity.com/paper/2206.08201/) (2022)
- [Geometry-Aware Post-Hoc Uncertainty Quantification in Operator Learning](https://mlsanity.com/paper/2606.17513/) (2026)
- [Quantifying Local Model Validity using Active Learning](https://mlsanity.com/paper/2406.07474/) (2024)
- [PIGPVAE: Physics-Informed Gaussian Process Variational Autoencoders](https://mlsanity.com/paper/2505.19320/) (2025)
- [Are Statistical Methods Obsolete in the Era of Deep Learning? A Study of ODE Inverse Problems](https://mlsanity.com/paper/2505.21723/) (2025)
- [Multifidelity-Augmented Gaussian Process Inputs for Surrogate Modeling from Scarce Data](https://mlsanity.com/paper/2603.22050/) (2026)
- [APIC: Amortized Physics-Informed Calibration using Neural Processes](https://mlsanity.com/paper/2606.03355/) (2026)

## Abstract (arXiv)

We introduce a computational efficient data-driven framework suitable for quantifying the uncertainty in physical parameters and model formulation of computer models, represented by differential equations. We construct physics-informed priors, which are multi-output GP priors that encode the model's structure in the covariance function. This is extended into a fully Bayesian framework that quantifies the uncertainty of physical parameters and model predictions. Since physical models often are imperfect descriptions of the real process, we allow the model to deviate from the observed data by considering a discrepancy function. For inference, Hamiltonian Monte Carlo is used. Further, approximations for big data are developed that reduce the computational complexity from O(N3) to O(N· m2), where m ll N. Our approach is demonstrated in simulation and real data case studies where the physics are described by time-dependent ODEs describe (cardiovascular models) and space-time dependent PDEs (heat equation). In the studies, it is shown that our modelling framework can recover the true parameters of the physical models in cases where 1) the reality is more complex than our modelling choice and 2) the data acquisition process is biased while also producing accurate predictions. Furthermore, it is demonstrated that our approach is computationally faster than traditional Bayesian calibration methods.