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Bayesian grey-box identification of nonlinear convection effects in heat transfer dynamics

Wouter M. Kouw, Caspar Gruijthuijsen, Lennart Blanken, Enzo Evers, Timothy Rogers

TL;DR

The paper tackles identifying nonlinear convection effects in lumped-element heat transfer by introducing a Gaussian process latent force model (GPLFM) that augments known conduction and linear convection with a GP for nonlinear convection. Through Bayesian smoothing, it obtains state estimates for temperatures and GP states, while a Laplace approximation yields approximate posterior hyperparameters; a Bayesian polynomial regression then recovers the nonlinear convective function $\hat{r}(\cdot,\cdot)$. The method is validated on both simulated data and measurements from a heated-rod assembly, showing accurate nonlinear-convection recovery and improved forward predictions over baseline models. This uncertainty-aware grey-box approach supports more reliable control and cooling strategies in thermally constrained systems.

Abstract

We propose a computational procedure for identifying convection in heat transfer dynamics. The procedure is based on a Gaussian process latent force model, consisting of a white-box component (i.e., known physics) for the conduction and linear convection effects and a Gaussian process that acts as a black-box component for the nonlinear convection effects. States are inferred through Bayesian smoothing and we obtain approximate posterior distributions for the kernel covariance function's hyperparameters using Laplace's method. The nonlinear convection function is recovered from the Gaussian process states using a Bayesian regression model. We validate the procedure by simulation error using the identified nonlinear convection function, on both data from a simulated system and measurements from a physical assembly.

Bayesian grey-box identification of nonlinear convection effects in heat transfer dynamics

TL;DR

The paper tackles identifying nonlinear convection effects in lumped-element heat transfer by introducing a Gaussian process latent force model (GPLFM) that augments known conduction and linear convection with a GP for nonlinear convection. Through Bayesian smoothing, it obtains state estimates for temperatures and GP states, while a Laplace approximation yields approximate posterior hyperparameters; a Bayesian polynomial regression then recovers the nonlinear convective function . The method is validated on both simulated data and measurements from a heated-rod assembly, showing accurate nonlinear-convection recovery and improved forward predictions over baseline models. This uncertainty-aware grey-box approach supports more reliable control and cooling strategies in thermally constrained systems.

Abstract

We propose a computational procedure for identifying convection in heat transfer dynamics. The procedure is based on a Gaussian process latent force model, consisting of a white-box component (i.e., known physics) for the conduction and linear convection effects and a Gaussian process that acts as a black-box component for the nonlinear convection effects. States are inferred through Bayesian smoothing and we obtain approximate posterior distributions for the kernel covariance function's hyperparameters using Laplace's method. The nonlinear convection function is recovered from the Gaussian process states using a Bayesian regression model. We validate the procedure by simulation error using the identified nonlinear convection function, on both data from a simulated system and measurements from a physical assembly.
Paper Structure (15 sections, 37 equations, 4 figures)

This paper contains 15 sections, 37 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model specification
  3. Heat transfer dynamics
  4. Temporal Gaussian Processes
  5. Augmented discrete-time model
  6. Probabilistic state-space model
  7. Inference
  8. State estimation
  9. Hyperparameter estimation
  10. Static nonlinearity estimation
  11. Experiments
  12. Simulated data
  13. Measured data
  14. Discussion
  15. Conclusions

Figures (4)

  • Figure 1: (Top) Photo of heated rod demonstrator, with 3 blocks, 2 insulation discs, 13 temperature sensors and 1 heater. (Bottom left) Data simulated according to Eq. \ref{['eq:heatdynamics']} with parameters described in Sec. \ref{['sec:simdata']}. (Bottom right) Data measured from demonstrator with parameters described in Sec. \ref{['sec:measureddata']}.
  • Figure 2: Experiment on data from simulated system. Top row: the first three subplots (left-to-right) show the value of the nonlinear convection function over visited temperature states, the GP estimates of this function and the posterior predictive distribution. The fourth subplot shows a forward simulation with the mean posterior predictive and the fifth subplot shows the absolute error (lower is better) between simulation with the true and identified convection function, i.e., $|$ states true - states identified $|$. Bottom row: first three subplots show standard polynomials fitted to residuals left by a state-space model without nonlinear convection terms. The fourth subplot shows a forward simulation using the regression function fit to the residuals, which deviates far from the forward simulation with the true convection function. The absolute error between the two is plotted in the fifth subfigure.
  • Figure 3: Prior and approximate posterior distributions under Laplace's method for the kernel hyperparameters $l$ and $\gamma$.
  • Figure 4: (Top row) GP states as a function of temperature states for sensors $4$ (red), $9$ (blue), and $12$ (orange), with 3d-order polynomial fits (mean posterior predictive distributions; black dashed). (Bottom row) Absolute simulation error, $|$ measurement - mean prediction $|$, for GPLFM versus SSM method.