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# Physics-Informed Gaussian Processes model for heat source identification

Émile Roux, Pascale Balland, Christian Elmo, Ludovic Charleux

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## To cite this version:

Émile Roux, Pascale Balland, Christian Elmo, Ludovic Charleux. Physics-Informed Gaussian Processes model for heat source identification. Materials Today Communications, 2025, 47, pp.112892. 10.1016/j.mtcomm.2025.112892. hal-05109647

HAL Id: hal-05109647

https://hal.science/hal-05109647v1

Submitted on 12 Jun 2025

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# Physics-Informed Gaussian Processes model for heat source identification

Emile Roux ID*, Pascale Balland ID, Christian Elmo ID, Ludovic Charleux ID

University of Savoie Mont Blanc, SYMME, FR-74000 Annecy, France

## ARTICLE INFO

**Keywords:**

Gaussian process

Machine learning

Heat source identification

## ABSTRACT

Thermal source identification is crucial in various engineering fields, particularly in material science, where understanding heat generation during mechanical loading is essential for comprehending material behavior. This study introduces a novel method for identifying self-heating sources in steel under mechanical loading by leveraging infrared thermography and Physics-Informed Gaussian Processes (PIGP). The method addresses the limitations of traditional filtering techniques, which struggle with noise sensitivity and lack integration with physical laws. PIGP uses Gaussian process regression to model noisy temperature data while embedding the heat equation directly into its framework, enabling robust and physically consistent estimation of heat sources. We present the theoretical formulation, training, and inference process of PIGP in detail. Synthetic data experiments demonstrate that PIGP delivers accurate heat source estimations even under significant infrared measurement noise ($\sigma = 0.084$ K), with estimation errors remaining below 5%. These results make PIGP a highly competitive alternative to traditional filtering-based approaches, primarily due to its reduced number of tuning parameters. These results highlight PIGP's potential as a robust and accurate tool for thermal source identification in complex, noisy environments.

## 1. Introduction

Thermal source identification is a critical problem across various engineering fields, particularly in material science, where accurately determining heat generation during mechanical loading is essential for understanding material behavior and the onset of material failure.

Infrared thermography has become a widely adopted technique in this field, offering high-resolution, real-time monitoring of surface temperature variations. This non-invasive approach enables the study of thermomechanical interactions in materials subjected to loading conditions. However, the challenge lies in interpreting these temperature fields to accurately identify the underlying heat sources, especially when the data is corrupted by noise. Estimating heat sources from noisy data becomes a non-trivial inverse problem, often requiring advanced modeling and filtering techniques to ensure reliable results. As reported by Woodbury et al. [1], the inversion of heat transfer problem is a well-known problem, with already identified difficulties (ill-conditioning, non-linearity, noise, etc.).

Several studies have specifically addressed the use of filtering techniques to enhance the identification of heat sources from infrared thermography data affected by noise. A chronological review of these contributions highlights both methodological advances and application-specific improvements in the processing of thermal data. Chrysochoos et al. [2] laid the groundwork by introducing an infrared image processing technique to analyze the calorific effects during strain localization

in materials, using filtering methods to reduce noise in thermographic data and to improve the accuracy of heat source estimation. A direct derivation method is proposed based on Fourier's decomposition of the temperature field. Authors highlight the instability and noise sensitivity of methods based on direct differentiation. It shows how filtering (based on Fourier), introduces intrinsic limitations such as attenuation and spreading of the source. Rainieri et Pagliarini [3] go in deep on the filtering approaches by testing different filters (Butterworth, Chebyshev, Elliptic, Median), they conclude that Weiner filter is a good solution to estimate the Laplacian of the temperature field. The study directly emphasizes the need to adjust the filter parameter based on the noise level, and report the side effects of the filtering process. Berthel et al. [4] use a local filtering methods based on least square approximation of the temperature field to estimate its Laplacian. This local approach shows its limit when the signal over noise ratio became defavorable. Moreover the authors highlight the need to adjust the filter parameters based on the noise level, which can be challenging in practice. Delpueyo et al. [5] developed a post-processing technique based on an optimized derivative Gaussian filter to reconstruct heat source fields from temperature measurements. The authors report the difficulties to find the optimal filter parameters with no prior knowledge of the source term (dimension and magnitude).

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: emile.roux@univ-smb.fr (E. Roux).</footnote>
The work of Ranc et al. [6] confirms the strong sensibility of the classical methods presented above to noise, and the need of fine adjustment of the filter parameters. From this state, the authors propose a new approach to filter the temperature field before applying the classical differentiation method to estimate the heat source. The proposed method is based on a Proper Orthogonal Decomposition (POD) technique, which effectively reduces noise in the temperature field. By applying POD to the temperature data, the authors demonstrate that the filtering process significantly enhances the accuracy of heat source estimation. This study provides valuable insights into the limitations of classical methods and emphasizes the importance of careful tuning in filtering techniques.

More recently Jaya Seelan et al. [7] highlights the applicability of advanced thermographic image processing techniques (building on prior work [2,4]) to characterize microstructural heterogeneities in complex materials such as welds, particularly by analyzing the dissipative heat source measured under elastic loading. The effectiveness of this approaches heavily relies on a fine-tuning process, which often requires detailed thermal signal analysis or prior knowledge of the heat source to optimize filtering parameters. This dependency on accurate adjustments and sensitivity to noise introduces limitations to the broader applicability of these techniques.

Zálešák et al. [8] report in their state-of-the-art review that the use of machine learning in this context is rather rare. The recent work of Guirguis et al. [9] presents a machine learning approach to estimate the heat source in real-time in the context of additive manufacturing. Similarly, Dong et al. [10] introduce a data-driven distortion compensation framework for the laser powder bed fusion (L-PBF) process. This approach aligns with our objective of improving thermal analyses in metal through advanced computational techniques. Durgam et al. [11] explore the prediction of temperatures of rectangular silicon heat sources cooled by forced convection using various machine learning algorithms. The study reveals that ensemble learning combined with an artificial neural network (ANN) provides the most accurate temperature predictions compared to individual algorithms such as support vector regression (SVR), but they not tackle directly the heat sources identification.

Looking at the fields of machine learning the Gaussian Process (GP) models, due to nature, are very good candidates to handle noisy data. Compared to classical filtering methods the GP theory provides auto-calibration method (the training stage based on the minimization of the negative log marginal likelihood) which allows learning the tuning parameters directly from the data set. The GP could be therefore seen as filtering methods with a law number of tuning parameters, which is able to produce a smooth temperature field. The smooth temperature field can be then derived respect to time and space to estimate the heat source. Pfortner et al. [12] highlight that purely data-driven models fail to extrapolate beyond the training domain. Incorporating physics improves generalization.

In a recent work by Raissi et al. [13] it has been demonstrated that GP regression can handle linear differential equations. In this work Raissi et al. demonstrated the interest of the method to various fields of science. One of the examples treats about inferring thermal parameters from a few temperatures and heat source measurements. This method embeds the physics of the problem into the GP framework, ensuring that the estimated parameters are consistent with the governing equations of heat conduction. The concept of Physics-Informed Gaussian Processes (PIGP) is introduced. It offers a promising solution by embedding these laws into the machine learning framework to identify heat sources based on noisy observations.

While Raissi et al.'s work presents, as an example of the method, the identification of thermal coefficients from a limited number of measurements, our approach extends this methodology to tackle more complex real-world scenarios, such as the full identification of heat source fields.

The proposed work addresses the challenges highlighted in previous methods, particularly the sensitivity to noise and the dependency on precise parameter tuning, as discussed previously. By bridging the gap between the simplified example provided by Raissi et al. [13] and more complex scenarios, this approach aims to identify the full heat source field while mitigating the limitations of noise sensitivity and reducing the reliance on extensive parameter adjustments.

This paper begins by presenting the theoretical background of the PIGP model, detailing its formulation, training, and prediction processes. We then validate the method through synthetic data experiments, accompanied with a sensitivity analysis. Finally, we apply the PIGP model to real-world tensile test data, demonstrating its effectiveness in estimating heat sources under noisy conditions.

## 2. Inverse problem statement

For this work, the heat conduction problem is simplified to a one-dimensional problem. This assumption is commonly made in the analysis of tensile tests in solid mechanics. It assumes that the thermomechanical state (stresses, strains, and temperature) is homogeneous throughout the cross-sectional area of the sample.

With this assumption, the three-dimensional heat conduction problem reduces to a one-dimensional case, as described in [2]. The heat conduction equation along the x-axis is given by:

$$ \frac{\partial \theta(x, t)}{\partial t} - \frac{k}{\rho c_p} \frac{\partial^2 \theta(x, t)}{\partial x^2} + \frac{h}{A} \frac{l}{A} \theta(x, t) = q(x, t) \quad (1) $$

Where $\rho$ is the density, $c_p$ is the specific heat capacity, $k$ is the thermal conductivity of the tested material, $x$ is the space coordinate, $\theta(x, t)$ is the variation of temperature field according to the initial temperature $T_0$ of the sample: $\theta(x, t) = T(x, t) - T_0$.

The source term $s(x, t)$ is introduced by the means of:

$$ q(x, t) = \frac{s(x, t)}{\rho c_p} \quad (2) $$

As reports in multiple studies [2,5,14], authors prefer to work with the expression of $q(x, t)$ rather than $s(x, t)$. The $q(x, t)$ is express in $K \cdot s^{-1}$ while the true source term $s(x, t)$ is express in $W \cdot m^3$. In the following, $q(x, t)$ will be named "heat source" for the sake of simplicity.

To account for heat loss by convection through the sides of the sample, the term $h \frac{l}{A} \theta(x, t)$ is added to this equation as reported in [15]. In this context, $h$ represents the convective heat transfer coefficient, $l$ represents the perimeter of the sample, and $A$ represents the cross-sectional area.

If the temperature field $\theta(x, t)$ is known over time and space, one can directly compute the Laplacian term and the derivative of the temperature with respect to time from the heat conduction equation. This computation would provide an estimation of the source term $q(x, t)$. However, the measurement of the temperature using an IR camera introduces noise, making the computation of the derivatives with respect to time and space very challenging.

The inverse problem focuses on identifying the heat source term $q(x, t)$ using temperature measurements subject to noise.

## 3. The Physics-Informed Gaussian processes

This section provides an overview of Gaussian processes, followed by a detailed explanation of the physical informed Gaussian process introduced by Raissi et al. [13].

### 3.1. Outline of Gaussian process

A Gaussian process GP is a collection of random variables which have a joint Gaussian distribution [16]. If we assume that the function $f(x)$ is a Gaussian process, this Gaussian process is fully specified by its
means function $\mu(x)$ (usually taken as $\mu(x) = 0$ for simplicity) and its covariance function $k(x, x', p)$:

$$f(x) \sim GP(\mu(x), k(x, x', p)) \qquad (3)$$

$k(x, x', p)$ is the covariance function, or so-called the kernel function, which expresses the covariance between the function values $f(x)$ and $f(x')$ at two different inputs $x$ and $x'$. $p$ is the set of hyperparameters of the covariance function.

The kernel function $k(x, x', p)$ is a measure of the similarity between $x$ and $x'$, and it plays a crucial role in the behavior and predictive capabilities of the Gaussian process. The choice of the kernel function is problem-dependent and should be selected based on prior knowledge of the data. This stage is often named as the prior tailoring.

Model calibration is an essential step in the use of Gaussian processes. It involves adjusting the hyperparameters set $p$ of the covariance function $k(x, x', p)$ to best fit the observed data. The observed data are defined by a set of observation values $\mathbf{y} = \{y_1, y_2, \dots, y_n\}$ at input locations $\mathbf{X} = \{x_1, x_2, \dots, x_n\}$, where $n$ is the number of observations. This is typically done by minimizing the log negative marginal likelihood of the observed data under the Gaussian process model. The negative log marginal likelihood ($\mathcal{N}LM\mathcal{L}$) is given by:

$$\mathcal{N}LM\mathcal{L}(\mathbf{y}, \mathbf{K}, p) = \frac{1}{2} \mathbf{y}^T \mathbf{K}^{-1} \mathbf{y} + \frac{1}{2} \log |\mathbf{K}| + \frac{n}{2} \log(2\pi) \qquad (4)$$

Where $\mathbf{K}$ is the covariance matrix, and $|\mathbf{K}|$ denote the determinant of the $\mathbf{K}$ matrix.

The covariance matrix $\mathbf{K}$ is constructed by evaluating the covariance function $k(x, x', p)$ at all pairs of observed inputs. Specifically, the element at the $i$th row and $j$th column of $\mathbf{K}$ is given by:

$$K_{ij} = k(x_i, x_j, p) \qquad (5)$$

Where $x_i$ and $x_j$ are the $i$th and $j$th observed inputs, respectively.

Once the model is calibrated, we can make predictions at new input locations. The Gaussian process provides a distribution over function values at these new locations. This distribution is Gaussian, with a mean and standard deviation given by:

$$\bar{\mu}(x_*) = k(x_*, \mathbf{X})[k(\mathbf{X}, \mathbf{X})]^{-1}\mathbf{y} \qquad (6)$$

$$\bar{\sigma}^2(x_*) = k(x_*, x_*) - k(x_*, \mathbf{X})[k(\mathbf{X}, \mathbf{X})]^{-1}k(\mathbf{X}, x_*) \qquad (7)$$

Here, $x_*$ are the new input locations.

## 3.2. Physics-Informed Gaussian processes

The described methodology aims to extend the work of Raissi et al. [13] to the case of heat conduction problems.

### Problem statement

We assume that we have two kinds of observed data:

* **Temperature observations:** a set of observations $y_\theta = \{\theta_1, \theta_2, \dots, \theta_n\}$ at input locations $X_\theta = \{(x_1, t_1), (x_2, t_2), \dots, (x_n, t_n)\}$. $n$ is the number of temperature observations.

* **Source observations:** a set of observations $y_q = \{q_1, q_2, \dots, q_m\}$ at input locations $X_q = \{(x_1, t_1), (x_2, t_2), \dots, (x_m, t_m)\}$. $m$ is the number of source observations.

Knowing that this two observation sets are linked by the means of the heat equation (Eq. (1)), we aim at building a Gaussian process model both for the temperature field and the source term.

### Prior tailoring

In this section we will define the prior of the data-driven algorithm that learns from the linear operator defined by the heat equation (Eq. (1)).

Firstly, the linear operator $\mathcal{L}_{(x,t)}^{p_{\text{phi}}}$ is defined by:

$$\mathcal{L}_{(x,t)}^{p_{\text{phi}}} \theta(x,t) := \frac{\partial \theta(x,t)}{\partial t} - \frac{k}{\rho c_p} \frac{\partial^2 \theta(x,t)}{\partial x^2} + \frac{h}{\rho c_p} \frac{l}{A} \theta(x,t) = q(x,t) \qquad (8)$$

Where $p_{\text{phi}}$ is the set of physical parameters involved in the physical problem ($p, c_p, k, h, l, A$).

An assumption to build the prior is that the relative temperature $\theta(x, t)$ is a Gaussian process defined by a zero mean and a covariance function $k_{\theta\theta}(x, t, x', t'; p)$:

$$\theta(x, t) \sim GP(0, k_{\theta\theta}(x, t, x', t'; p)) \qquad (9)$$

$p$ is the set of hyperparameters of the covariance function $k_{\theta\theta}(x, t, x', t', p)$, $(x, t)$ and $(x', t')$ are two input locations where the covariance function is evaluated.

Without loss of generality, Gaussian process priors used in this work are assumed to have a squared exponential covariance function involving two hyperparameters $l_x$ and $l_t$ for the spatial and temporal dimensions, respectively. The covariance function is defined as:

$$k_{\theta\theta}(x, t, x', t'; p) = \exp \left( - \frac{(x - x')^2}{l_x^2} - \frac{(t - t')^2}{l_t^2} \right) \qquad (10)$$

A key observation is that any linear transformation of a Gaussian process, such as differentiation or integration, remains a Gaussian process [13]. Therefore,

$$\mathcal{L}_{x,t}^{p_{\text{phi}}} \theta(x, t) = q(x, t) \sim GP(0, k_{qq}(x, t, x', t'; p, p_{\text{phi}})) \qquad (11)$$

with the following fundamental relationship between the kernels $k_{\theta\theta}$ and $k_{qq}$:

$$k_{qq}(x, t, x', t'; p, p_{\text{phi}}) = \mathcal{L}_{x,t}^{p_{\text{phi}}} \mathcal{L}_{x',t}^{p_{\text{phi}}} k_{\theta\theta}(x, t, x', t'; p) \qquad (12)$$

And the cross relationship between $k_{\theta\theta}$, $k_{qq}$ and $k_{\theta q}$:

$$k_{\theta q}(x, t, x', t'; p, p_{\text{phi}}) = \mathcal{L}_{x',t}^{p_{\text{phi}}} k_{\theta\theta}(x, t, x', t'; p) \qquad (13)$$ $$k_{q\theta}(x, t, x', t'; p, p_{\text{phi}}) = \mathcal{L}_{x,t}^{p_{\text{phi}}} k_{\theta\theta}(x, t, x', t'; p)$$

### Training

The training of the model is done by minimizing the negative log marginal likelihood as defined in Eq. (4). To perform this minimization we have to define the vector of observations $\mathbf{y}$:

$$\mathbf{y} = \begin{bmatrix} \mathbf{y}_\theta \\ \mathbf{y}_q \end{bmatrix} \qquad (14)$$

And the covariance matrix $\mathbf{K}$:

$$\mathbf{K} = \begin{bmatrix} k_{\theta\theta}(\mathbf{X}_{\theta}, \mathbf{X}_{\theta}) + \sigma_{\theta}^2 \mathbf{I}_n & k_{\theta q}(\mathbf{X}_{\theta}, \mathbf{X}_q) \\ k_{q\theta}(\mathbf{X}_q, \mathbf{X}_{\theta}) & k_{qq}(\mathbf{X}_q, \mathbf{X}_q) + \sigma_q^2 \mathbf{I}_m \end{bmatrix} \qquad (15)$$

In order to note overload, the notation of the dependencies to the hyperparameters $p$ and $p_{\text{phi}}$ are not explicitly written in the covariance matrix $\mathbf{K}$. Here, $\sigma_{\theta}^2$ and $\sigma_q^2$ are the noise variances of the temperature and source observations, respectively, and $\mathbf{I}_n$ and $\mathbf{I}_m$ are the identity matrices of size $n$ and $m$, respectively.

The minimization of the negative log marginal likelihood may allow us to infer from the data the following parameters:

* The hyperparameters $p$ of the covariance functions $k_{\theta\theta}$ and $k_{qq}$,

* the physical parameters $p_{\text{phi}}$ of the linear operator $\mathcal{L}_{x,t}^{p_{\text{phi}}}$,

* the noise variances $\sigma_{\theta}^2$ and $\sigma_q^2$.

The minimization of the negative log marginal likelihood is known to be ill-conditioned and may have multiple local minima. Therefore, an ad-hoc optimization algorithm needs to be used. In this work, the Nelder-Mead simplex algorithm [17] is used, which has been shown to
efficiently handle this type of optimization problem [18,19]. To address the issue of ill-conditioning, bounds were imposed on the hyperparameters and physical parameters during the optimization process. These bounds were carefully chosen based on the physical interpretation of the parameters and the expected values derived from the data. Special attention was given to the lower bounds of the noise parameter, which can be estimated in a preliminary stage. The lower bound was set to a value ten times smaller than the estimated noise level, providing a safeguard against overfitting in the PIGP model.

### Prediction

Having trained the model, we can predict the values $\theta(x, t)$ and $q(x, t)$ at any new input locations $(x, t)$ by writing the posterior distributions.

The mean and the variance of the temperature field $\theta(x, t)$ at the new input locations $(x, t)$ are given by:

$$ \begin{aligned} \bar{\theta}(x, t) &= c_u^T \mathbf{K}^{-1} \mathbf{y} \\ \bar{\sigma}_{\theta}^2(x, t) &= k_{\theta\theta}(x, t, x, t) - c_u^T \mathbf{K}^{-1} c_u \end{aligned} \quad (16) $$

The mean and the variance of the source term $q(x, t)$ at the new input locations $(x, t)$ are given by:

$$ \begin{aligned} \bar{q}(x, t) &= (c_v^T \mathbf{K}^{-1} \mathbf{y}) \\ \bar{\sigma}_q^2(x, t) &= (k_{qq}(x, t, x, t) - c_v^T \mathbf{K}^{-1} c_v) \end{aligned} \quad (17) $$

Where $c_u$ and $c_v$ are the vectors of covariances between the new input locations $(x, t)$ and the observed input locations $X_\theta$ and $X_q$, respectively:

$$ \begin{aligned} c_{\theta}^{T} &= [k_{\theta\theta}(x, t, \mathbf{X}_{\theta}) \quad k_{\theta q}(x, t, \mathbf{X}_{\theta})] \\ c_{q}^{T} &= [k_{q\theta}(x, t, \mathbf{X}_{\theta}) \quad k_{qq}(x, t, \mathbf{X}_{q})] \end{aligned} \quad (18) $$

The presented framework is implemented using the python language and more precisely the JAX module [20]. The JAX module is a high-performance for machine learning library that allows for automatic differentiation and just-in-time compilation. Here we take advantage of automatic differentiation capability, which is very efficient to compute the kernel derivatives that are mandatory in Eq. (12).

The physical informed Gaussian process model presented in this section is used to infer the source term of the heat equation (Eq. (1)) from temperature measurements.

This strategy is firstly validated on synthetic data and secondly applied to real data obtained during a tensile test in the following section.

## 4. Validation tests on synthetic data

### Synthetic data generation

To build the synthetic data, the heat equation (Eq. (1)) is solved with a known source term $q(x, t)$. A one-dimensional finite difference scheme is used to solve the heat equation.

A null flux boundary condition is applied at the two ends of the domain. The values of the involved parameters are presented in Table 1.

The synthetic source function is defined by the following equation:

$$ q(x, t) = q_0 \exp \left( - \frac{(x-x_c)^2}{\xi_w^2} + \frac{(t-t_c)^2}{\xi_w^2} \right) \quad (19) $$

Where $q_0$ is the amplitude of the source, $x_c$ and $t_c$ are the center of the source in space and time, and $x_w$ and $t_w$ are the width of the source in space and time. The initial temperature field of the sample is set to $\theta(x, t = 0) = 0$. Parameters values use for the source definition are given in Table 1. These values have been chosen to be representative of a typical tensile test on a steel material under free convection.

Fig. 1 shows the source term $q(x, t)$ (Eq. (19)) on the left and the computed temperature field $\theta(x, t)$ on the right. Each field is represented

<table><caption>**Table 1**
Parameter values used for synthetic data generation.</caption><thead><tr><th>Param</th><th>Value</th><th>Unit</th><th></th></tr></thead><tbody><tr><td>L</td><td>120</td><td>mm</td><td>Length of the domain</td></tr><tr><td>e</td><td>2.5</td><td>mm</td><td>Thickness of the sample</td></tr><tr><td>w</td><td>20</td><td>mm</td><td>Width of the sample</td></tr><tr><td>t<sub>end</sub></td><td>30</td><td>s</td><td>Duration of the test</td></tr><tr><td>ρ</td><td>7840</td><td>kg/m<sup>3</sup></td><td>Density</td></tr><tr><td>c<sub>p</sub></td><td>449</td><td>J/kg/K</td><td>Specific heat capacity</td></tr><tr><td>k</td><td>69.5</td><td>W/m/K</td><td>Thermal conductivity</td></tr><tr><td>h</td><td>10</td><td>W/m<sup>2</sup>/K</td><td>Heat transfer coefficient</td></tr><tr><td>q<sub>0</sub></td><td>3.</td><td>K<sup>-1</sup></td><td>Amplitude of the source</td></tr><tr><td>x<sub>c</sub></td><td>6</td><td>mm</td><td>Center of the source in space</td></tr><tr><td>t<sub>c</sub></td><td>10</td><td>s</td><td>Center of the source in time</td></tr><tr><td>x<sub>w</sub></td><td>0.02</td><td>mm</td><td>Width of the source in space</td></tr><tr><td>t<sub>w</sub></td><td>4</td><td>s</td><td>Width of the source in time</td></tr></tbody></table>

by an isovalue plot, with the space coordinate $x$ on the horizontal axis and the time coordinate $t$ on the vertical axis.

Before using this temperature fields later on, the field is then corrupted by a Gaussian noise with a standard deviation equal to 0.08 K. This value of the noise standard deviation is chosen to be representative of the noise level of the IR camera.

### Physics-Informed Gaussian processes training

Using this corrupted temperature field the PIGP is trained. For this first test, all available temperature data are used to train the model. We also include the knowledge of the source at the initial time and the final time in the training dataset.

To train the model, we need to determine the hyperparameters $l_x$ and $l_t$ of the $k_{\theta\theta}$ kernel function (Eq. (10)) and the noise variance $\sigma_{\theta}^2$ by the means of the negative log marginal likelihood minimization (Eq. (4)).

The hyperparameters obtain after the training phase are found to be equal to $l_x = 0.021$ m, $l_t = 4.62$ s, and the noise standard deviation $\sigma_{\theta} = 0.080^{\circ}$. The noise standard deviation obtained after training is very close to the value used to corrupt the temperature field. It shows the ability of the PIGP model to estimate the noise level of the data set.

### Physics-Informed Gaussian processes: source estimation

Once the PIGP is trained, we can predict the source term $\bar{q}(x, t)$ (Eq. (17)), and the temperature field $\bar{\theta}(x, t)$ (Eq. (16)) at any new input locations $(x, t)$.

Fig. 2 shows the isovalue plot of the predicted temperature field using the PIGP model, while Fig. 3 presents the corresponding isovalue plot for the predicted source field.

Fig. 2b is the main isovalue plot that shows how the temperature field $\bar{\theta}(x, t)$ evolves over both space and time. The x-axis represents space (position along the domain), and the y-axis represents time. The top graph (Fig. 2a) is a snapshot of the center graph at a particular time. It provides a cross-sectional view of the temperature along the domain at a specific moment ($t = 10, 20, \text{and } 30$ s). The left graph (Fig. 2c) shows the temporal evolution of the temperature at a particular spatial point (the center of the domain).

Fig. 3 shows the isovalue plot of the predicted source field using the PIGP model, the same organization as Fig. 2 is used.

Both figures demonstrate the model's capability to accurately reconstruct the temperature and source fields from noisy temperature measurements. The smoothness and continuity of the fields in both figures indicate the PIGP's ability to handle noise effectively and produce consistent estimates of the underlying heat source distribution.
<figure><img src="image_5.png" /><figcaption>Fig. 1. Synthetic data generated by direct solving the heat equation - isovalue of the source term (left) and temperature field (right).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 2. PIGP on synthetic data test case - (a) temperature profile at different times over space, (b) main isovalue of the temperature field over space and time, (c) temperature evolution at the center of the sample over time. Subplots (a) and (c) are cross-sectional views of the main isovalue plot (b).</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 3. PIGP on synthetic data test case - (a) source profile at different times, (b) isovalue of source field, (c) source evolution at the center of the sample. Subplots (a) and (c) are cross-sectional views of the main isovalue plot (b).</figcaption></figure>

## Sensitivity analysis

Since both the noise and amount of training points have an impact on the accuracy of the model, a systematic study is carried out to evaluate the impact of those parameters on the quality of the model. The data are corrupted with different noise standard deviations from 0.01 K to 1.0 K. Which makes the signal-to-noise ratio (SNR) varying from 3.8 to 50 for this temperature profile. The signal-to-noise ratio is defined as:

$$SNR = \frac{\sqrt{\frac{1}{n} \sum_{i=1}^{n} (\theta_{(x_i, t_i)} - \bar{\theta})^2}}{\sigma} \qquad (20)$$

Where σ is the added noise standard deviation, $\bar{\theta}$ the mean value of θ and n is the number of observations. It is important to note that for a large number of training points, a lower bound is imposed on the noise level to prevent overfitting. This lower bound is set to a value approximately ten times smaller than the roughly estimated noise level. The relative root mean squared (RMS) error is computed for each test:

$$RMS = \frac{\sqrt{\frac{1}{n} \sum_{i=1}^{n} (q_{(x_i, t_i)} - \bar{q}_{(x_i, t_i)})^2}}{\sqrt{\frac{1}{n} \sum_{i=1}^{n} q_{(x_i, t_i)}^2}} \qquad (21)$$

The results are shown in Figs. 4 and 5.

Fig. 4 displays the relative Root Mean Squared error (Eq. (21)) of the source term estimation as a function of the added noise σ and the amount of training points. This figure provides a sensitivity analysis of the PIGP model, showing how the accuracy of the heat source estimation decreases with increasing noise levels and improves with a higher number of training points. Some key configurations of the source thermal estimation (highlighted as red dots in Fig. 4) are further illustrated in Fig. 5. As expected, the source thermal profile is accurately identified when a large number of training points is used and when the noise magnitude is low. Conversely, under poor conditions, the estimation is inaccurate: oscillations around zero are observed for times between 20 and 30 s. This figure highlights the importance of increasing the number of training points and minimizing measurement noise as much as possible.

The plot highlights the model's robustness at lower noise levels, but it also demonstrates the expected degradation in performance as noise increases. In the work of Delpueyo et al. [5], the noise level is lower ($\sigma$ = 0.02 K), and the tested source exhibits higher stiffness. Similarly, the work of Ranc et al. [6] reports comparable accuracy. The PIGP approach offers an advantage in that it eliminates the need for a fine-tuning stage, simplifying the process.
<figure><img src="image_8.png" /><figcaption>Fig. 4. Relative root mean squared error on the source therm estimation as a function of the added noise σ over the amount of training points. The line representing σ = 0.084 K is plotted, this noise is representative of real IR measurements. The red dots represent key configuration that are detailed in Fig. 5.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 5. PIGP outputs for different key values of noise and number of training points (nbTP). The key points are highlighted in Fig. 4. The temperature and source are plotted over time on the middle line of the sample.</figcaption></figure>

Furthermore, it underscores the importance of selecting an adequate amount of training points to achieve reliable predictions, particularly in noisy environments.

5. Application to tensile test data

As a demonstration of the effectiveness of the method on real data,
we have applied the method to the data of a tensile test. The data are

obtained by measuring the temperature field on the surface of a DP1000
steel sample during an uniaxial tensile test. The needed parameters to
perform the analysis are summarized in Table 2.

It is worth noting that the physical parameters *p*<sub>phi</sub> play a crucial
role in accurately identifying the heat source term *q*(x, t). In the present
work, the values of these parameters are taken from the literature [21].
To go further and identify these parameters during the training stage,
one would need measurement data where both θ and *q* are known. A
Physics-Informed Gaussian Process (PIGP) could then be trained to infer
<figure><img src="image_10.png" /><figcaption>Fig. 6. Temperature field measured during a tensile test. (a) view of the IR images at *t* = 36 s (b) temperature field evolution respect to time.</figcaption></figure>

<table><caption>**Table 2**

Parameter values used for real test case with DP1000 steel. DP1000 material constants are taken from [21].</caption><thead><tr><th>Param</th><th>Value</th><th>Unit</th><th></th></tr></thead><tbody><tr><td>L</td><td>44</td><td>mm</td><td>Length of the domain</td></tr><tr><td>e</td><td>2.5</td><td>mm</td><td>Thickness of the sample</td></tr><tr><td>l</td><td>15</td><td>mm</td><td>Width of the sample</td></tr><tr><td>t<sub>end</sub></td><td>42</td><td>s</td><td>Duration of the test</td></tr><tr><td colspan="4" style="text-align: center;">DP1000 thermal parameter</td></tr><tr><td>ρ</td><td>7840</td><td>kg/m<sup>3</sup></td><td>Density</td></tr><tr><td>c<sub>p</sub></td><td>449</td><td>J/kg/K</td><td>Specific heat capacity</td></tr><tr><td>k</td><td>69.5</td><td>W/m/K</td><td>Thermal conductivity</td></tr><tr><td>h</td><td>10</td><td>W/m<sup>2</sup>/K</td><td>Heat transfer coefficient</td></tr></tbody></table>

the parameters $p_{\text{phi}}$, but this aspect lies beyond the scope of the current study.

We use a FLIR A655sc camera to measure IR field. This camera has a resolution of 640 × 480 pixels². It is mounted with a 25° field of view objectives, completed with a close-up IR lens 5.8×. This leads to a typical region of interest of 64 × 48 mm² at a working distance of 172 mm. The conversion of the IR field to temperature field is done by the camera software. The face of the sample is covered by a black paint with a known and quasi-constant emissivity in the specter of IR camera, this emissivity of the paint is documented to be equal to 0.97 [22].

The temperature measurements are given in Fig. 6. The gray dotted line in Fig. 6a represents the line along which the temperature field is extracted from the IR images. Fig. 6b represents the evolution of the temperature field along this line. The temperature over the line is stacked over time.

At the beginning of the test no load is applied, the temperature field is therefore homogeneous and constant, these data are used to measure the initial temperature $T_0$ and to qualify the noise level of measurement. The Fig. 7 shows a histogram of the temperature field during this period. A normal distribution is fitted to the data, the standard deviation of the noise is found to be 0.084 K. The signal-to-noise ratio can be therefore evaluated, it is close to $SNR = 46$. The analysis of the

<figure><img src="image_11.png" /><figcaption>Fig. 7. Evaluation of the noise level of the IR camera.</figcaption></figure>

noise distribution also confirms the Gaussian nature of the noise, which is a key assumption of the PIGP model. A specific attention should be paid if the noise differs from a Gaussian distribution, in this case more advanced GP framework may be used [23]. According to the sensitivity analysis (Fig. 4), the model should be able to provide an estimation of the source term with accuracy better than 5%.

The PIGP is then trained using all the temperature observations and the source observations at the initial time (the source is null at the beginning of the test). The hyperparameters of the model are found to be $l_x = 0.0121$ m, $l_t = 5.38$ s, and the noise standard deviation on the wall data set $\sigma_\theta = 0.0719$ K.

Figs. 8 and 9 show respectively the isovalue plot of the predicted temperature and source field using the PIGP model, the same organization as Fig. 2 is used.

Since the heat source is unknown, to assess the quality of the PIGP model, a rough estimation of the source is made using a finite difference scheme. Before applying the finite difference scheme, the temperature field is smoothed with a Gaussian filter having a standard
<figure><img src="image_12.png" /><figcaption>Fig. 8. PIGP on DP1000 test case - temperature field - (a) temperature profile at different times, (b) isovalue of temperature field, (c) temperature evolution at the center of the sample.</figcaption></figure>

deviation of 1 mm in space and 0.5 s in time. A comparison between the PIGP and the finite difference scheme is shown in Fig. 10. The finite difference scheme provides a non-continuous field, as expected, and requires a fine-tuning stage of the Gaussian filter to smooth the field. The important point is that both approaches provide similar results, offering additional validation of the PIGP model, which complements the validation performed on synthetic data in the previous section.

On Figs. 8 one can see the ability of the PIGP to smooth the temperature field and to provide a continuous field. On Fig. 9 the estimated source field is presented. Tree stages can be observed. The first stage ($t < 15$ s) where the source is homogeneous and close to zero or slightly negative. It corresponds to the elastic stage, where a cooling is observed under tensile loading as reports in [24]. The second stage ($15 s < t < 35$ s) where the source increased homogeneously, it may correspond to the homogeneous plastic deformation along the sample. And the third stage ($t > 35$ s) where the source field becomes non-homogeneous, a peak of sources appears it is located at $x = 10$ mm, it may correspond to the beginning of strain localization phenomena.

This example shows the ability of the method to deal with real data set and its potential to provide insight on the heat generation during mechanical loading.

## 6. Conclusion

The proposed Physics-Informed Gaussian Processes model successfully integrates the physics of heat conduction with machine learning, offering a robust method for heat source identification in the presence of noisy data. The model exploits the Gaussian process framework, which allows for both flexible noise handling and the incorporation of physical constraints through the heat equation. The results from synthetic data experiments demonstrate that PIGP accurately reconstructs temperature and heat source fields, even under significant noise conditions.

The proposed methodology is currently implemented for one-dimensional heat conduction problems. However, extending it to two-dimensional problems is expected to be straightforward. The main challenge would involve designing kernels that appropriately account for boundary conditions, such as Neumann conditions. Additionally, careful consideration should be given to the computational cost associated with Gaussian Processes when dealing with a large number of training points.

In addition, the model's application to real-world tensile test data illustrates its relevance in a practical context, suggesting its potential applicability to other problems involving thermomechanical coupling.
<figure><img src="image_13.png" /><figcaption>Fig. 9. PIGP on DP1000 test case - source field - (a) source profile at different times, (b) isovalue of source field, (c) source evolution at the center of the sample.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 10. Heat source profile over time identified by the PIGP and roughly estimation by a finite difference scheme.</figcaption></figure>

While further validation across different materials, geometries, and 3D configurations is needed, this preliminary work lays the foundation for broader use within specific domains of materials science and mechanical engineering.

By embedding physical principles into the machine learning framework, PIGP provides reliable and physically consistent estimation techniques. One of the strengths of the method is to avoid the tuning stage of the hyperparameters, the model is able to learn the noise level and the other needed hyperparameters directly from the data set. This methodology offers a promising tool for analyzing heat generation during mechanical loading.

### CRediT authorship contribution statement

**Emile Roux:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Pascale Balland:** Writing – review & editing, Funding acquisition. **Christian Elmo:** Writing – review & editing, Software. **Ludovic Charleux:** Writing – review & editing, Software, Conceptualization.

### Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Emile Roux reports financial support was provided by University Savoie Mont Blanc. Emile Roux reports a relationship with University Savoie
Mont Blanc that includes: employment. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Data availability

Data will be made available on request.

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