# Utilizing Experimental Data in a Model-Free Data-Driven Method: The Case of Thermal Conduction

Jakob Platen, Atul Anantheswar, Felix Harazin, Johannes Storm, Michael Scherkenbach, Markus Oeser, Laurent Stainier, Michael Kaliske

## ► To cite this version:

Jakob Platen, Atul Anantheswar, Felix Harazin, Johannes Storm, Michael Scherkenbach, et al. Utilizing Experimental Data in a Model-Free Data-Driven Method: The Case of Thermal Conduction. International Journal for Numerical Methods in Engineering, 2025, 126 (22), pp.e70165. 10.1002/nme.70165. hal-05515223

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Submitted on 17 Feb 2026

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RESEARCH ARTICLE

OPEN ACCESS

# Utilizing Experimental Data in a Model-Free Data-Driven Method: The Case of Thermal Conduction

Jakob Platen¹ ID | Atul Anantheswar¹ ID | Felix Harazin¹ ID | Johannes Storm¹ ID | Michael Scherkenbach² |
Markus Oeser² | Laurent Stainier³ ID | Michael Kaliske¹ ID

¹Institute for Structural Analysis, Technische Universität Dresden, Dresden, Germany | ²BASt, Bundesanstalt für Strassen- und Verkehrswesen, Bergisch Gladbach, Germany | ³Institut de Recherche en Génie Civil et Mécanique (UMR 6183), Nantes Université—École Centrale de Nantes—CNRS, Nantes, France

**Correspondence:** Michael Kaliske (michael.kaliske@tu-dresden.de)

**Received:** 9 May 2025 | **Revised:** 8 October 2025 | **Accepted:** 10 October 2025

**Funding:** This work was supported by the Deutsche Forschungsgemeinschaft (Grant Nos. KA 1163/46-2, 453596084, KA 1163/37-2, GR 1504/11-2 and 414936990).

**Keywords:** data-driven method | experimental data | finite element method | thermal conduction

## ABSTRACT

The study at hand investigates the applicability of experimental data within a model-free data-driven solver. The heat conduction equation is facilitated and solved by the data-driven solver without the need for a constitutive law. At first, the results of an artificially created dataset are compared to a finite element solution to assess the quality of the approach. Subsequently, in-situ data is processed to create a dataset for the data-driven solver. This approach is validated by comparison to experimental data. The quality of the measured data is discussed, and challenges arising in utilizing experimental data are pointed out. Furthermore, an improved measuring concept for future experimental data is detailed and tested based on finite element simulations. Another finding is the ability of the data-driven solver to implicitly model changing material characteristics.

## 1 | Introduction

Many predictive methods rely on constitutive models to represent the material under investigation accurately. For instance, solid mechanics uses constitutive models to relate stress and strain measures. In heat conduction problems, Fourier's law is intensively used to model the relation between the temperature gradient and heat flux. These constitutive models are based upon material parameters, which are constants that need to be identified for each material separately. Therefore, experiments are simulated and the material parameters are adjusted until a good agreement of the simulation results with experimental results is achieved. An alternative approach is the model-free data-driven solver initially proposed by [1], which substitutes constitutive models with data. Nevertheless, partial differential equations are solved.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

© 2025 The Author(s). *International Journal for Numerical Methods in Engineering* published by John Wiley & Sons Ltd.
If at any point a constitutive relation would be required, a consistent pair of the thermodynamically conjugate state variables is selected from a dataset by a distance minimization procedure. This simulation technique is successfully applied to solid mechanics [1–8] and diffusion problems [9]. In the contribution at hand, the data-driven solver is applied to heat conduction problems.

Since the data-driven solver requires a certain dataset size to represent the material realistically, in many contributions, the datasets are artificially generated [1, 2, 4, 6–9]. Nevertheless, the main goal is the applicability of experimental data within the data-driven solver. Incorporating experimental data, pointing out challenges that arise due to the utilization of experimental data, and solution strategies for these challenges are another key aspect of the contribution at hand. Therefore, the data-driven solver is applied to thermal conduction problems in which the geometry can be represented by a 1-dimensional geometry. As a consequence, a framework is created in which the influence of the experimental data can be studied with relative ease.

Another challenge in constitutive modeling is the consideration of multi-physics. If multiple fields are considered, it is required to model the influence of one field variable on other state variables by means of constitutive equations. For instance, if temperature and solid mechanics are combined, thermal expansion and the change of elastic constants with respect to temperature are represented by constitutive models. In the contribution at hand, it is investigated that the data-driven solver can represent changes in material parameters by grouping data into sub-datasets if a distinguishing physical quantity exists. This grouping is considered to be a less complicated challenge.

To obtain datasets large enough that an analysis with the data-driven solver utilizing experimental data is possible, data from the duraBASt project is used. This project is meant to test asphalt in a one-to-one scale under predefined loads. A key advantage of utilizing this data is that measurements of the temperature along the asphalt are present over the course of more than two years. Hence, a large and reliable amount of data is generated, which is key for performing the study at hand. The importance of the correct prediction of asphalt arises naturally from the gigantic amounts of goods transported over roads around the world and the need for individual and public transportation arising from connecting communities and providing essential services throughout countries.

The contribution at hand is focused on two hypotheses. Firstly, can experimental data be incorporated into the data-driven solver and used for predictive simulations? Secondly, is the data-driven solver capable of modeling changing material characteristics implicitly by grouping data? To evaluate those hypotheses, the study is structured as follows. Firstly, the data acquisition of the experimental data and the processing of it are detailed in Section 2. Then, the data is analyzed using the finite element method in Section 3 to draw conclusions about the material under investigation. The data-driven solver for heat conduction problems is described in Section 4. Furthermore, the data-driven solver is validated for thermal problems. Subsequently, experimental data is applied in this method in Section 5. Furthermore, an optimized measuring concept is tested, and the implicit modeling of changing material characteristics is the subject of investigation.

## 2 | Materials and Methods

In this section, the acquisition of in-situ data, the outlier removal, and the analysis of the in-situ data with the finite element method are elaborated.

### 2.1 | In In-Situ Data Acquisition

The underlying data acquisition was realized on duraBASt. This is the BASt demonstration, investigation, and reference area, located at the Köln-Ost motorway junction (BAB A3/A4). The investigation and demonstration section offers 6 areas with a maximum length of 100 meters and a width of 4 and 5 meters. The test object was constructed on area DU-Z5 in September 2019 as part of the project Testing the cold recycling construction method with foamed bitumen on the duraBASt in conjunction with accelerated pavement testing using the MLS30 in cooperation with the Wirtgen Group. Two-thirds of the surface was constructed using the cold recycling method, while the last third was constructed in accordance with the standard for comparison purposes (40 mm asphalt surface course SMA 8 N 50/70, 140 mm asphalt base course AC22 T N 70/100, 380 mm frost protection layer FSS). Accompanying the planned load tests, instrumented measuring fields (3 pieces, each 1 m × 3.5 m, symmetrical to the center line) were laid out.
<figure><img src="image_3.png" /><figcaption>**FIGURE 1** | Calibration curve of T-sensors.</figcaption></figure>

The measuring field in the standard-compliant area contains the temperature sensors analyzed in this project. These sensors were integrated during the field setup and lay on top of each other at depths of 400 mm, 180 mm, 40 mm, and 0 mm (surface). The geo-reference data for the center of the measuring field is latitude 50.95208353552195 and longitude 7.0373307615439185.

The sensors were of the DS18B20 type. These are digital sensors that transmit the measured values via a (digital) 1-Wire interface. The measuring range is -55°C to 125°C, whereby the accuracy in the range -10°C to 85°C is specified at +/-0.5°C. The sensors are also characterized by an individual serial number (ID), which is used for communication. The factory calibration of the sensors was checked before installation with the aid of a water bath and the use of a reference sensor in the relevant range of 5°C to 40°C. The good agreement of different sensor IDs with the reference can be seen in Figure 1.

A single-board computer was extended with a 1-Wire interface to record the measured values and used for continuous measurement every 5 min using a Python program, starting in March 2020. The measured values were written to files on a daily basis, which were then regularly backed up via USB stick and transferred to an SQL Postgres database using another Python script.

At the same time as the data was recorded, the MLS30 loading machine began to load the test field. The relevant load times for the measurement field under consideration here are shown in Figure 2.

In addition to the temperature values and load times, the database contains other relevant project data, such as information on the structure and data from other integrated sensors, as well as the results of additional, cyclical measurements of pavement-related responses.

## 2.2 | Data Preprocessing

It is a well-established fact that real-world data is often not directly applicable as input in simulations and requires preprocessing to render it suitable. In the context of this work, the real-world data refers to temperature readings from sensors at specific locations within the pavement, collected by BAST over a period of about 120 weeks, and stored in an SQL database. The structure of the pavement and the location of the temperature sensors are detailed in the schematic shown in Figure 3. The data in the SQL database requires processing before it is possible to use it in further downstream simulation applications. In this case, the processing involved outlier detection and interpolation for missing data. To keep the time consistency of the dataset, the eliminated data points are then replaced by interpolated values.
<figure><img src="image_4.png" /><figcaption>**FIGURE 2** | MLS loading with 50 kN in hours.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**FIGURE 3** | Schematic pavement structure and temperature sensor locations.</figcaption></figure>

### 2.2.1 | Outlier Detection

The collected data from sensor ID 8 shows a single possible outlying value of 42°C, while the immediate neighbors are around 20°C. This data point is assumed to be erroneous, since no change in the experimental environments is known or detected, and is manually deleted.

### 2.2.2 | Interpolation

The collected data is time-discrete for each sensor, and the sensors do not measure at exactly the same time. Thus, the data of the sensors cannot be compared directly. There are also periods of time where the sensor data is absent due to sensor failures. Therefore, linear interpolation is conducted to ensure continuously available data to compare different sensors. For that, linear interpolation is applied between neighboring data points, so the gaps are filled and arbitrary points in time are selectable. This ensures that the data is available at different times and is consistent within regular intervals. The interpolated, continuous dataset is applicable to downstream simulations.

After the preprocessing, the data is available at regular intervals and is synchronized. Thus, it can be used as input into simulations. The obtained data after processing is plotted in Figure 4.
<figure><img src="image_6.png" /><figcaption>FIGURE 4 | Temperature data at various depths after processing.</figcaption></figure>

## 3 | Analysis of Experimental Data with the Finite Element Method

In this section, an analysis of the in-situ data with the finite element method is carried out. Furthermore, it is shown that the material parameters change over the course of the measured time.

### 3.1 | Finite Element Solution

To simulate the experiments by the finite element method, the heat conduction equation

$$ \rho c \dot{\theta} = - \operatorname{div}(q) \qquad (1) $$
is utilized [10–12]. In Equation (1), $\rho$ is the density, $c$ is the mass specific heat capacity, $\theta$ is the temperature, and $q$ is the heat flux. For the abbreviation of notation, the volume specific heat capacity is introduced as $C = \rho c$. Equation (1) is used to obtain the unknown field of temperature. Hence, a constitutive law is required, within the finite element method, to relate the heat flux to the temperature. Therefore, Fourier's law

$$q = -k \mathrm{grad}(\theta) \qquad (2)$$

is used, which introduces the thermal conductivity $k$ as another material parameter. Equation (1) cannot be solved analytically for arbitrary geometries. Hence, the heat conduction equation is processed by the finite element method. Therefore, Equation (1) is integrated over the volume and multiplied by a time-independent test function $\delta\theta$ reading

$$\int_B \delta\theta C \dot{\theta} dv + \int_B \delta\theta \mathrm{div}(q) dv = 0, \qquad (3)$$

where $B$ denotes the domain of the body. Incorporating the divergence theorem and Gauss's theorem leads to

$$\int_B \delta\theta C \dot{\theta} dv + \int_{\partial B} \delta\theta(q \cdot n) da - \int_B \mathrm{grad}(\delta\theta) \cdot q dv = 0. \qquad (4)$$

$\partial B$ is the surface of the domain and $n$ is the outward normal vector. The term containing $q \cdot n$ is identified as the boundary term. The temperature field, as well as the test function, is discretized by shape functions $N$ and their gradients $B$ according to

$$\theta = N^I \theta^I, \qquad (5)$$

$$\mathrm{grad}(\theta) = B^I \theta^I, \qquad (6)$$

$$\delta\theta = N^I \delta\theta^I \qquad (7)$$

and

$$\mathrm{grad}(\delta\theta) = B^I \delta\theta^I, \qquad (8)$$

where $\theta^I$ and $\delta\theta^I$ are the nodal values of the temperature and test function, respectively. The superscript $I$ denotes the node. For abbreviation, $g = \mathrm{grad}(\theta)$ is introduced. The matrix containing the gradient of the shape functions is specified for a linear one-dimensional element containing two nodes as $B_{1d} = \begin{bmatrix} \frac{-1}{L^e} & 0 \\ 0 & \frac{1}{L^e} \end{bmatrix}$, where $L^e$ denotes the length of the element. For the general, three-dimensional case, the matrix is stated as

$$B^I = \begin{bmatrix} N_1^I & 0 & 0 \\ 0 & N_2^I & 0 \\ 0 & 0 & N_3^I \\ N_2^I & N_1^I & 0 \\ 0 & N_3^I & N_2^I \\ N_3^I & 0 & N_1^I \end{bmatrix}, \quad \text{where } N_i^I = \frac{\partial N^I}{\partial X_i}. \qquad (9)$$

To discretize an arbitrary domain by finite elements, the current domain is subdivided into finite domains by the assembly operator, reading

$$B = \sum_{E=1}^{N} B_{E}. \qquad (10)$$

The residual for each finite element can be stated as

$$0 = \int_{B_E} N^I C N^J \dot{\theta}^J dv - \int_{B_E} B^I \cdot q dv. \qquad (11)$$

Since the resulting system of equations is linear, the solution is not further detailed in the contribution at hand. For all subsequent sections, it is assumed that a constant cross-section of 1 m² is considered. Consequently, the discretization is carried out with linear one-dimensional elements. In the contribution at hand, the thermal advection is solved for
<figure><img src="image_7.png" /><figcaption>FIGURE 5 | Flowchart describing the algorithm to identify material parameters.</figcaption></figure>

the one-dimensional case. Hence, the integration over the volume is simplified to $\int_{B_E} dv = \int_{B_E} Adl$ an integration over the length $l$ for a constant cross-section $A$. For roads, a cross-section of $1 \times 1 \text{ m}^2$ is considered as representative. The integration is conducted numerically by multiplication with integration weights $\omega$.

## 3.2 | Simulation of In-Situ Data to Identify Material Parameters

With the processed data available, it is possible to run simulations with the finite element framework FEAP [13], such that the temperature at the surface is used as boundary conditions. Furthermore, for all simulations in the contribution at hand, the temperature in 12 m depth is fixed as 10°C, according to [14]. Then, material parameters can be identified by optimization, such that the resulting temperatures at interior points of the pavement match the sensor data [15]. The flowchart that details the strategy to identify material parameters is provided in Figure 5. The optimization procedure is coded in Python, based on the Optuna framework [16]. The metric to check if optimization is necessary can be expressed as

$$ ||T_{\text{sens}} - T_{\text{sim}}|| \le T_{\text{tol}}, \qquad (12) $$

where $T_{\text{sens}}$ refers to the temperature data recorded by sensors, $T_{\text{sim}}$ denotes the temperature resulting from simulation, and $T_{\text{tol}}$ is the tolerance which should not be exceeded (otherwise an optimization would need to be performed). The tolerance can be set individually for each layer of the pavement, and the values for sensor ID 4 are 3°C, for sensor ID 8 are 1°C, and for sensor ID 10 are 1.5°C.

The objective function to be minimized during the optimization with the Optuna framework is expressed as

$$ \sum_{k=1}^{n} (T_{\text{sens}}^{k} - T_{\text{sim}}^{k})^{2} \rightarrow \min, \qquad (13) $$

which is the square of differences between the sensor values and simulated values summed over each step of the simulation—in this case, each week of the sensor data. Further, the default Optuna Tree-structured Parzen Estimator is run for 2000 iterations to find suitable material parameters. The parameters are only updated if they fit better than the previously identified parameters, subject to the following metric

$$ Update \Rightarrow \left( current \frac{\sum_{k=1}^{n} RMS_k}{n} \right) \le \left( previous \frac{\sum_{k=1}^{n} RMS_k}{n} + \Delta_{tol} \right), \qquad (14) $$

where $RMS_k$ denotes the square root of the mean squared distances between temperatures averaged for the $n$ internal sensor locations, subjected to a tolerance of $\Delta_{tol}$. In the study reported in this manuscript, the value of $\Delta_{tol} = 0.25$. The optimization procedure then stores the best-fitting parameters for each week. The results of the optimization are plotted in Figure 6.
<figure><img src="image_8.png" /><figcaption>**FIGURE 6** | Specific heat capacity and thermal conductivity identified for the different layers of the pavement.</figcaption></figure>

It is apparent that the identified material parameters in each material layer can change significantly. This could be because the combination of identified material parameters that result in a suitable simulation solution is not unique. Thus, more than one set of material parameters could lead to matching results. The changes in material parameters could also be induced by climate effects such as the ingress of moisture, which can change the physical properties of the soil and subsoil layers. The changes in material parameters may also indicate deterioration of the pavement materials. In any case, the identification of a unique combination of material parameters for each pavement layer is not a trivial task and is beyond the scope of the current work.

Therefore, in the following studies, the weeks 90 to 100, where the parameters are constant, are further utilized to validate the data-driven framework. This period of interest is also highlighted in Figure 6 by vertical lines. Week 90 is used for validation, while weeks 91 to 100 are used for identification and data generation in the subsequent sections. The results for the identification period simulated by the finite element method are shown in Figure 7. As can be seen, a good agreement with reality has been achieved by the material parameter identification. Furthermore, the validation period is shown in Figure 8. For this week, a good agreement with reality is also observed. Hence, the finite element solution is seen as a reference for the data-driven method in the subsequent sections.

# 4 | Data-Driven Method for Heat Conduction

In this section, the application of the data-driven method for thermal conduction is discussed, derived, and compared to experiments.

## 4.1 | Data-Driven Solver

The data-driven solver is based upon the idea that the constitutive law Equation (2) can be replaced by a dataset *D*, containing a collection of (*g*, *q*) tuples of temperature gradients and corresponding heat fluxes. The model-free
<figure><img src="image_9.png" /><figcaption>**FIGURE 7** | Results of weeks 91 to 100 by the finite element method.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**FIGURE 8** | Results of week 90 by the finite element method.</figcaption></figure>
data-driven paradigm is then based on the idea that the solution corresponds to admissible states, i.e., verifying conservation equation Equation (1), as close as possible to the material dataset $D$. To define the closest point in the dataset, the following distance function between two states $z^*$ and $z^L$ is defined

$$d(z^*, z^L) = \frac{1}{2} \left[ K_{DD}(g^* - g^L) \cdot (g^* - g^L) + \frac{1}{K_{DD}} (q^* - q^L) \cdot (q^* - q^L) \right], \quad (15)$$

where $z^* = \{g^*, q^*\}$ denotes a state from the dataset, and $z^L = \{g^L, q^L\}$ denotes an admissible state. The scalar $K_{DD}$ is a weighting factor, ensuring dimensional homogeneity of the distance function. It could be taken as a second-order tensor, but is chosen here as a scalar value. Considering the boundary-value problem, two solution spaces are defined. First, $D_g$ denotes the discrete space corresponding to the assignment of one element of $D$ to each of the $M$ integration points (in an FE setting), reading

$$D_g = \underbrace{D \times D \times \cdots \times D}_{M \text{ times}}. \quad (16)$$

Second, the space $\mathcal{E}$ of temperatures and fluxes verifying the discretized balance equations, for a given timestep, is defined by

$$C_{IJ} \frac{\theta^J - \theta_n^J}{\Delta t} - \sum_{e=1}^{M} \omega_e \mathbf{B}_e^I \cdot \mathbf{q}_e = 0 \quad \forall I = 1, \dots, N_{\text{nodes}} \quad (17)$$

where $\omega_e$ denote integration weights, and boundary conditions. The subscript $\square_e$ denotes the integration point $e$, and $C_{IJ}$ denotes the components of the capacity matrix $\mathbf{C}$. The problem can then be rewritten as

$$\min_{z^* \in D_g} \min_{(\theta, q) \in \mathcal{E}} \sum_{e=1}^{M} \omega_e d(z_e^*, (\mathbf{B}_e \theta, q_e)), \quad (18)$$

where $z_e^*$ is the state from the dataset assigned to integration point $e$ and $\theta$ are the nodal temperatures. The constraints Equation (17) can be treated through the use of Lagrange multipliers $\eta$ (the treatment of Neumann boundary conditions will not be detailed here, but follows a similar logic). For a given assignment $z^*$, stationarity conditions of the Lagrangian with respect to $\theta$, $q$ and $\eta$ then yield

$$\sum_{e=1}^{M} \omega_e K_{DD} \mathbf{B}_e^T (\mathbf{B}_e \theta - \mathbf{g}_e^*) + \frac{1}{\Delta t} \mathbf{C} \eta = 0, \quad (19)$$

$$\frac{\omega_e}{K_{DD}} (\mathbf{q}_e - \mathbf{q}_e^*) - \omega_e \mathbf{B}_e \eta = 0, \quad (20)$$

$$\frac{1}{\Delta t} \mathbf{C}(\theta - \theta_n) - \sum_{e=1}^{M} \omega_e \mathbf{B}_e^T \mathbf{q}_e = 0, \quad (21)$$

which can be rearranged as a system for $\theta$ and $\eta$ as

$$\Delta t \mathbf{K} \theta + \mathbf{C} \eta = \sum_{e=1}^{M} \omega_e K_{DD} \mathbf{B}_e^T \mathbf{g}_e^*, \quad (22)$$

$$\mathbf{C} \theta - \Delta t \mathbf{K} \eta = \mathbf{C} \theta_n + \Delta t \sum_{e=1}^{M} \omega_e \mathbf{B}_e^T \mathbf{q}_e^*, \quad (23)$$

and an update formula for the fluxes

$$\mathbf{q}_e = \mathbf{q}_e^* + K_{DD} \mathbf{B}_e \eta, \quad (24)$$

where $\mathbf{K} = \sum_{e=1}^{M} \omega_e K_{DD} \mathbf{B}_e^T \mathbf{B}_e$ is a pseudo-conductivity matrix and $\square^T$ denotes the transpose. As discussed in Reference [1], the minimization with respect to $z^*$ amounts to a nearest neighbor search for each integration point. The model-free data-driven solver then consists of fixed-point iterations alternating the resolution of the above equations and nearest-neighbor searches. Convergence is attained when nearest-neighbor searches yield the same result as in the previous iteration.
<table><caption>TABLE 1 | Parameters for data-driven simulations with artificial data.</caption><thead><tr><th>Parameter</th><th>n<sub>1</sub></th><th>n<sub>2</sub></th><th>n<sub>3</sub></th><th>n<sub>4</sub></th><th>n<sub>5</sub></th><th>K<sub>DD<sub>1</sub></sub></th><th>K<sub>DD<sub>2</sub></sub></th><th>K<sub>DD<sub>3</sub></sub></th><th>K<sub>DD<sub>4</sub></sub></th><th>K<sub>DD<sub>5</sub></sub></th><th>Δt</th></tr></thead><tbody><tr><td>Unit</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>s</td></tr><tr><td>Value</td><td>2</td><td>4</td><td>5</td><td>3</td><td>16</td><td>0.32</td><td>0.88</td><td>1</td><td>1</td><td>0.96</td><td>60</td></tr></tbody></table>

## 4.2 | Validation

To validate the data-driven approach, the weeks 91 to 100 are simulated with artificially generated data. Since the temperature is assumed not to change over the cross-section, a 1-D discretization is used. The first domain spans 0.04 m and represents SMA, while the second domain is 0.14 m and represents AC. The third and fourth layers resemble FSS and span 0.22 m and 0.16 m, respectively. The last layer is subsoil and spans 11.44 m. Based upon [14], in this depth, the average temperature is 10°C. This temperature is prescribed at the lower end of the line. The top temperature is another boundary condition of the simulation. Here, the temperature measured by sensor 3 is prescribed. The values between two measured values are linearly interpolated. The simulation parameters (number of elements per section, denoted as $n_i$, the layer-specific metric $K_{DD}$, and the time step $\Delta t$) are adjusted until a good agreement with reality is achieved. The parameters used are listed in Table 1. The values for $K_{DD}$ are obtained by multiplying the thermal conductivity of each layer by a factor of 0.8. The dataset for each layer is generated by using Fourier's law. Since the finite element method can simulate reality for these time periods with high accuracy, the data-driven approach should be capable of the same. For all materials, the temperature gradient is sampled in the interval [-100, 100] with 1E + 5 samples. It is assumed that this amount of data points for each material is sufficient. To obtain the heat fluxes, the temperature gradient is multiplied by the conductivity $k$ for each material. The results for weeks 91 to 100 are shown in Figure 9. As can be seen, the agreement with reality is similar if compared to the finite element method. The relative errors are reported to better assess the quality of the fit. This error measure is calculated as $\epsilon_{rel} = \frac{|T_{sim} - T_{sens}|}{T_{sim}}$. Since the simulation utilizes a time increment smaller than the measurements, the values of the simulation are interpolated to the points in time where experimental data is available. For the calculation, the temperatures are converted to Kelvin to avoid division by zero and the disproportionately high influence of values around 0. Furthermore, the average of the relative error is calculated for each sensor. For sensor 4, the average is 0.38%, for sensor 8, the average relative error is 0.09%, and for sensor 10, the average is 0.23%. Furthermore, week 90 is simulated as validation, and the results are depicted in Figure 10. Also, for this week, the relative errors are computed. The averages are given by 0.37% for sensor 4, 0.15% for sensor 8, and 0.38% for sensor 10. From this data, it can be concluded that the agreement with reality is satisfactory. Hence, it can be concluded that the data-driven method is capable of simulating heat conduction. Furthermore, the amount of 1E + 5 samples is sufficient. Therefore, the incorporation of real-world data remains.

# 5 | Data-Driven Method With Real-World Data

In this section, the validated formulation of the data-driven solver is used with in-situ data to demonstrate the capabilities of this approach to rely on experimental data.

## 5.1 | Utilization of In-Situ Data

To use a dataset in the data-driven solver, two quantities are required. Firstly, the temperature gradient, which can be determined directly from data measured, and secondly, the corresponding flux. The temperature gradient can be calculated as an average value for each layer as

$$ \text{grad}(\theta)^i = \frac{\theta^{i+1} - \theta^i}{L^i}, \qquad (25) $$

where $\theta^{i+1}$ is the temperature value of the sensor, which is placed after the material layer, $\theta^i$ is the temperature value of the sensor placed at the start of the material layer $i$, and $L^i$ is the thickness of the layer. Since sensor 10 is placed not at the interface of FSS and the subsoil, it is assumed that the 16 cm of FSS are negligible compared to the 11.44 m of subsoil. Hence, the length of the last layer in the calculation of the dataset is 11.6 m. In the subsequent simulations, the 16 cm of FSS are assigned the dataset generated by sensors 8 and 10 in conjunction with a length of 22 cm. The temperature in 12 m is again assumed to be constant 10°C, and takes the place of $\theta^{i+1}$ for the last layer. To obtain a larger dataset, which is generally better for the data-driven solver, the values for the opposite coordinate direction grad($\theta$)$^i = \frac{\theta^i - \theta^{i+1}}{L^i}$ are
<figure><img src="image_11.png" /><figcaption>FIGURE 9 | Results of data-driven solver by artificial data for weeks 91 to 100.</figcaption></figure>

included as well. No heat fluxes are directly measured. Therefore, another assumption has to be invoked. It is assumed that the thermal conductivity of the first layer (SMA) is known. Hence, the heat flux in the first layer $q^1$ is calculated by Fourier's law. For the subsequent layers, further assumptions are utilized. Firstly, each section between sensors (and ground temperature) is discretized by one line element, and secondly, a lumped heat capacity matrix is assumed. As a consequence, the heat flux in the next section can be approximated by

$$q^{i+1} = q^i - \frac{1}{\Delta t} \left[ \frac{C_i L^i + C_{i+1} L^{i+1}}{2} \right] (\theta_{n+1}^{i+1} - \theta_n^{i+1}). \quad (26)$$
<figure><img src="image_12.png" /><figcaption>**FIGURE 10** | Results of validation by the data-driven solver with artificial data.</figcaption></figure>

In Equation (26), $\theta_{n+1}^{i+1}$ is the temperature of the next known time step of the next node, and $\theta_{n+1}^{i+1}$ is the temperature of the current time step of the next node. For the temperature values, the data measured by the sensors is used. The time step is the measuring interval, and the geometry as well as the heat capacity is assumed to be known.

Based upon these assumptions and Equations (25) and (26), the dataset for each layer is generated. Since a one-dimensional discretization is utilized, the datasets can be plotted as three-dimensional histograms. The resulting datasets are depicted in Figure 11. For layer 1 (Figure 11a), the resulting dataset is allocated along a line, which is centered at the coordinate origin. This result is to be expected, since Fourier's law is applied in this layer. Furthermore, the dataset exhibits a point symmetry in the origin, which originates from invoking both coordinate directions in the dataset.
<figure><img src="image_13.png" /><figcaption>b)</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>c)</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>d)</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>e)</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**FIGURE 11** | Three-dimensional histograms of the datasets generated from in-situ data and Equations (25) and (26), which are used for (a) layer 1, (b) layer 2, (c) layer 3, (d) layer 4, and (e) layer 5.</figcaption></figure>

The other datasets are more diffuse and do not show such a clear trend. This is caused by the assumptions invoked. This finding becomes more apparent in the next section. Also, for close values of the temperature gradient, the corresponding heat fluxes tend to fluctuate, which is indicated since no clear line can be identified. This can lead to erratic behavior in the data-driven simulations. No check for duplicates is conducted to determine how good the data-driven approach can handle such diffuse datasets. The values for temperature gradient in the last layer (Figure 11d) are limited to small values. This finding originates from the large length (11.6 m) and the fixed temperature at one side. A limited dataset causes another challenge in simulations utilizing the data-driven solver. Increasing the number of elements used can lead to
<figure><img src="image_18.png" /><figcaption>**FIGURE 12** | Graphical representation of the influence of element length.</figcaption></figure>

<table><caption>**TABLE 2** | Parameters for data-driven simulations with in-situ data.</caption><thead><tr><th>Parameter</th><th>n<sub>1</sub></th><th>n<sub>2</sub></th><th>n<sub>3</sub></th><th>n<sub>4</sub></th><th>n<sub>5</sub></th><th>K<sub>DD</sub></th><th>Δt</th></tr></thead><tbody><tr><td>Units</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>s</td></tr><tr><td>Value</td><td>3</td><td>7</td><td>14</td><td>13</td><td>11</td><td>7.30558112</td><td>514.54</td></tr></tbody></table>

worse agreements with experimental data for certain boundary conditions. For instance, if a node undergoes rapid heating or cooling (intense sunlight or sudden rainfall), the temperature of this node changes significantly more than the neighboring nodes. If small elements are used in this area, the gradients are high and may lie outside the utilized dataset. Hence, in such cases, a coarser discretization distributes the gradient more, resulting in smaller values, which can be represented by a limited dataset. This relation is graphically depicted in Figure 12.

To obtain an optimal fit of the data-driven simulation with experimental data, an optimization is carried out utilizing the NSGA-II optimization strategy from Reference [17]. The number of elements for each section, the metric, and the time step size are adjusted until the optimal fit is achieved. This agreement is quantified by the square root of the sum of the point-wise difference squared of the temperature measured in experiments and simulated by the data-driven solver for each sensor. Hence, three objective functions are defined by sensors 4, 8, and 10. The optimization is run for 150 generations. No other criteria are set for the determination of the optimization. The identified simulation parameters are listed in Table 2. For the optimization, only weeks 91 to 100 are used. The resulting agreement with reality is shown in Figure 13. As can be seen, the overall agreement is not so close compared to the previous simulations. This phenomenon can be explained by multiple reasons. Firstly, the datasets, generated by Equations (25) and (26), are diffuse, since no fluxes are measured, and, consequently, assumptions are made to overcome this issue. Secondly, the datasets are also sparse in certain regions. The dataset describing the first layer (Figure 11a) is sparse in the origin, while the dataset for the fifth layer is sparse between 0 and 0.5 for grad(θ). Another challenge is that the datasets contain four points, which are close in the temperature gradient, but show a significant difference in the heat flux. These challenges lead to overestimations of the temperature change in Figure 13 for the data-driven simulation if compared to sensor data.

To assess the quality of the simulation with the data-driven solver utilizing in-situ data, week 90 is also simulated with the same simulation parameters and datasets. The results are shown in Figure 14. As can be seen, the temperature fluctuations are even more pronounced this week. Also, a significant drop in temperature can be found for the data-driven simulation compared to sensor 10 after 400,000 s. Both can be explained by the datasets shown in Figure 11. Both phenomena can be caused by points in the dataset that are close in the temperature gradient but exhibit significant differences in the heat flux. For the fluctuations, the temperature gradient changes in such a way that the next time step, a larger flux with the opposing sign is chosen from the dataset. Hence, a rapid change with increasing magnitude is found in the temperature-time curves. For the steep drop in the data-driven solution, which is compared to sensor 10, data points are selected, which lead to heat fluxes that transport heat into the subsoil. As a consequence, the temperature throughout all observed points decreases. Lower layers are more affected by this effect, since the high heat fluxes occur in the lowest layer. This finding is further supported by Figure 11e, since here the largest values for q are found. One main conclusion to draw from these findings is that the utilization of the data-driven solver with in-situ data requires a sophisticated measuring concept. Limiting the measurements to just one physical quantity of the dataset is not enough, even if assumptions are utilized to construct the other quantity (see Equations (25) and (26)). Another important conclusion is that the measuring interval of one measurement every 10 min for 10 weeks per layer is sufficient for in-situ data. For a good choice of simulation parameters, all simulated weeks (90 to 100) agree sufficiently with experimental data. Hence, this number can serve as a lower bound for further experimental designs. The uncertainty of the data is not assessed in the contribution at hand. The assessment of uncertainty and the application of appropriate uncertainty-reducing measurements could reduce the fluctuations of
<figure><img src="image_19.png" /><figcaption>**FIGURE 13** | Results of data-driven solver by in-situ data processed by Equations (25) and (26) for weeks 91 to 100.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>**FIGURE 14** | Results of validation by the data-driven solver with in-situ data processed by Equations (25) and (26).</figcaption></figure>
the heat flux within the dataset. Though it is more likely that the assumptions of considering the temperature at the interfaces of the layer, therefore, homogenizing the layer, and the lumped heat capacity matrix are more influential.

One important aspect that occurred during the optimization is that the objective function is quite sensitive to changes in the simulation parameters for the in-situ dataset. In the contribution at hand, this finding is demonstrated by simulating weeks 91 to 100 with slightly modified simulation parameters, which are given in Table 3. Only the number of elements in the second layer is increased by 2. The results of the simulation with the data-driven solver are depicted in Figure 15. As can be seen, the change in the number of elements is significant. The data-driven simulation is close to reality for sensor 4. This point of measurement is mostly influenced by layers 1 and 2. The datasets for these layers (Figure 11a,b) show only limited fluctuations in the heat flux. Consequently, no data points are selected, which transport significant amounts of heat into subsequent layers. Since more elements in layer 2 are used, the temperature gradients are different, which leads to different data points being selected in the solution procedure. These cause different temperatures and temperature gradients throughout the discretized geometry. For layers 3 to 5, large values of $q$ are contained in the datasets. As a consequence, heat is taken away from the material points at locations of sensors 8 and 10. This finding is most pronounced for the temperatures at the location of sensor 10. Since the boundary conditions are the temperature measured by sensor 3 at the top and a constant average soil temperature of 10°C at the bottom, the obtained values of around -60°C are considered unphysically. This finding further highlights the importance of high-quality datasets for the data-driven solver. Hence, an improved measuring concept is proposed and tested in the next section.

## 5.2 | Enrichment of In-Situ Data

To achieve better results, a different measuring concept is proposed. The concept outlined in Section 2.1 generates data, which describes the individual layer in a smeared manner. As pointed out in the previous section, this smeared representation leads to certain problems within the data-driven solver. To obtain a more realistic representation of the material itself, two temperatures are measured at two closely adjacent points. The gradient is calculated as the difference in temperature divided by the distance, see Equation (25). Due to the small distance, larger gradients can be more easily included in the dataset. Furthermore, the heat flux needs to be included in the dataset to reduce the amount of assumptions required to generate the dataset significantly. An overview of heat flux measurement techniques is presented in Reference [18]. The heat flux needs to be measured between the two closely adjacent points of temperature measurement. In this case, the heat fluxes can be seen as directly linked to the measured temperature gradient.

The proposed measurement concept is tested by means of finite element simulations. The simulation of Figure 7 is used to generate the data. Temperatures and fluxes are measured and saved for each layer individually. The flux is measured at the middle point of each layer, and the temperature is recorded from the two closest nodes. FSS is subdivided into two layers, one spanning 22 cm above sensor 10, and one spanning 16 cm below sensor 10. The number of elements for the 5 layers used is stated in Table 4. A high number of elements is used to obtain closely adjacent nodes.

The datasets obtained from the finite element simulation are depicted in Figure 16. As can be seen, the data now closely aligns with straight lines, which is to be expected, since Fourier's law is utilized in the simulation. The dataset size is half compared to the dataset used in Section 5.1, since only one coordinate direction is considered. Hence, no point symmetry can be found in any dataset. Furthermore, the datasets for layers 3 and 4 are different. Both layers resemble FSS in the experiments. Another benefit is that the fluxes do not fluctuate so much for small variations in the gradient. Therefore, a less erratic behavior is expected. In the dataset representing subsoil, there is a large range of temperature gradients. Still, the range is limited compared to the other layers. This could be overcome by measuring at a different point. The fluctuations of the temperature are so rapid that the induced heat flux heats the subsequent layers, but induces not much change of temperature in the lower subsoil regions. Hence, measuring close to the interface of FSS and subsoil may lead to a broader dataset. In the contribution at hand, it is studied whether this limited dataset still improves the overall quality of the data-driven solver. No investigation regarding the optimal points of measurement is conducted.

Firstly, the weeks 91 to 100, which are used to generate the data, are simulated. The simulation parameters are optimized by [17] to achieve the best fit with sensor data. The resulting simulation parameters are listed in Table 5. Since no assumptions regarding the heat conductivity are made within the data-driven simulation, the same metric $K_{DD}$ is applied for all layers. The results for the best fitting parameters are depicted in Figure 17. As can be seen, the agreement with reality is high.
<table><caption>**TABLE 3** | Parameters for data-driven simulations with in-situ data to study the sensitivity of the results.</caption><thead><tr><th>Parameter</th><th>n<sub>1</sub></th><th>n<sub>2</sub></th><th>n<sub>3</sub></th><th>n<sub>4</sub></th><th>n<sub>5</sub></th><th>K<sub>DD</sub></th><th>Δt</th></tr></thead><tbody><tr><td>Units</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>s</td></tr><tr><td>Value</td><td>3</td><td>9</td><td>14</td><td>13</td><td>11</td><td>7.30558112</td><td>514.54</td></tr></tbody></table>

<figure><img src="image_21.png" /><figcaption>**FIGURE 15** | Results of changed simulation parameters for the data-driven solver with in-situ data.</figcaption></figure>
<table><caption>**TABLE 4** | Number of elements for the finite element simulation testing the improved measuring concept.</caption><thead><tr><th>Parameter</th><th>n<sub>1</sub></th><th>n<sub>2</sub></th><th>n<sub>3</sub></th><th>n<sub>4</sub></th><th>n<sub>5</sub></th></tr></thead><tbody><tr><td>Unit</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Value</td><td>10</td><td>20</td><td>30</td><td>20</td><td>150</td></tr></tbody></table>

<figure><img src="image_22.png" /><figcaption>**FIGURE 16** | Three-dimensional histograms of the datasets generated from finite element simulation and used for (a) layer 1, (b) layer 2, (c) layer 3, (d) layer 4, and (e) layer 5.</figcaption></figure>
<table><caption>**TABLE 5** | Parameters for data-driven simulations with enhanced data.</caption><thead><tr><th>Parameter</th><th>n<sub>1</sub></th><th>n<sub>2</sub></th><th>n<sub>3</sub></th><th>n<sub>4</sub></th><th>n<sub>5</sub></th><th>K<sub>DD</sub></th><th>Δt</th></tr></thead><tbody><tr><td>Unit</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>s</td></tr><tr><td>Value</td><td>3</td><td>12</td><td>14</td><td>15</td><td>16</td><td>3.401391981</td><td>353.15</td></tr></tbody></table>

<figure><img src="image_23.png" /><figcaption>**FIGURE 17** | Results of data-driven solver by enhanced data for weeks 91 to 100.</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>**FIGURE 18** | Results of validation by the data-driven solver with enhanced data.</figcaption></figure>

Furthermore, a validation is carried out by simulating week 90. The same simulation parameters and datasets are used. A validation can be carried out, since the data points are not created by data from week 90, and the simulation parameters are not adjusted in week 90. The results are shown in Figure 18. As can be seen, the agreement with reality is good. Hence, it is assumed that the new measuring concept leads to better datasets, which are likely to work well with in-situ data as well. To address this point, more research and more experiments need to be conducted. On the other hand, it can be seen that the agreement of the data-driven simulation with enhanced data is very similar to the agreement with the finite
<figure><img src="image_25.png" /><figcaption>**FIGURE 19** | Distance of the data-driven solver over the simulated body and time.</figcaption></figure>

element solution and the experimental data. This behavior is to be expected. The data is generated by the finite element method. Consequently, the agreement of the finite element simulation with reality can be seen as an upper bound for the agreement of the data-driven solver with enhanced data. Another conclusion is that the amount of data points (10 weeks of constant measurement with one measurement every 10 min, resulting in 10,080 data points) can be seen as an estimator of the dataset size, which is required by experimental data for a one-dimensional heat conduction problem.

Another analysis is carried out by investigating the distance of each element over the simulated specimen and time. This relation is depicted in Figure 19. Here, the decimal logarithm of the distance is shown for weeks 91 to 100. The datasets are created by utilizing the values of a finite element simulation in the middle of each layer. As can be seen, the distance is significantly smaller in these regions. The highest distance is measured at the interfaces of layers. There, more than one material influences the temperature gradients and heat fluxes. Consequently, the datasets can represent these situations not optimally. Hence, the measuring concept could be further improved by obtaining values in these regions. Another finding is that the distances in the subsoil are the smallest. This behavior can be explained by the dataset. From Figure 16, it becomes apparent that layer 5 has the most dense dataset, since the interval for grad(θ) and *q* are the smallest intervals of any layer, and layer 5 obtains the same amount of data points as the other layers. As a consequence, the distances are smallest. Furthermore, the distances are highest around the interface of FSS and subsoil. This finding correlates with the agreement of the data-driven simulation with experimental data in Figure 17. For sensor 10, the largest difference is observed. This sensor is influenced most by the high distances in the simulation.

## 5.3 | Implicit Modeling of Changing Material Characteristics

As discussed in Section 3.2, material parameters change over time. In the contribution at hand, it is studied whether the data-driven approach is capable of handling such changes in the material behavior without explicit modeling. The specific heat capacity and the thermal conductivity of asphalt and soils can be influenced by other physical properties, for instance, moisture (see, e.g., [19]). In classical finite element simulations, these dependencies require modeling by constitutive functions. On the contrary, the data-driven approach could handle such relations without explicit modeling if the datasets represent such changes. The only requirement is a variable, which needs to be included in the dataset,
<table><caption>TABLE 6 | Parameters for data-driven simulations with changing datasets.</caption><thead><tr><th>Parameter</th><th>n<sub>1</sub></th><th>n<sub>2</sub></th><th>n<sub>3</sub></th><th>n<sub>4</sub></th><th>n<sub>5</sub></th><th>K<sub>DD</sub></th><th>Δt</th></tr></thead><tbody><tr><td>Unit</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>s</td></tr><tr><td>Value</td><td>3</td><td>12</td><td>14</td><td>15</td><td>16</td><td>3.401391981</td><td>353.15</td></tr></tbody></table>

<figure><img src="image_26.png" /><figcaption>FIGURE 20 | Results of three consecutive weeks with changing material characteristics simulated by the data-driven solver.</figcaption></figure>
that can differentiate between different states. For asphalt and soil, this variable could be the moisture content. Since the moisture content is not available, other quantities are used. The analysis by the finite element method reveals that the material parameters change over time, see Figure 6. Hence, time is chosen as the representative quantity. Alternatively, average temperatures could be used, since different moisture contents occur during different seasons.

To test the hypothesis that the data-driven solver is capable of handling changing material characteristics, the weeks 2 to 4 are analyzed in Figure 6. Firstly, the three weeks are simulated by the finite element method to produce datasets according to the method discussed in Section 5.2. The time step is set to 120 s. Therefore, 5040 data points are generated per layer and per week by the finite element method. The data generated by each week and each layer is grouped in one dataset. Hence, per layer, three datasets are generated, which originated from finite element simulations utilizing different material parameters. Subsequently, the three weeks are simulated in one data-driven solver simulation. The dataset is adjusted during the simulation when the time of the simulation passes into the next week. The simulation parameters used are given in Table 6. The results are depicted in Figure 20. As can be seen, the agreement of the data-driven solver with reality is sufficient. This finding leads to the conclusion that the data-driven solver is capable of handling changing material characteristics, if they can be represented in separate datasets without the need for explicit modeling and modeling assumptions, which yields a significant benefit of this simulation approach.

# 6 | Conclusion and Outlook

In the contribution at hand, a model-free data-driven solver is first applied to experimental data in a thermal conduction problem. Therefore, the heat conduction equation is successfully solved by the data-driven solver. Furthermore, experimental data is used within the data-driven solver, demonstrating that this simulation technique is capable of handling experimental data. Another key finding of the contribution at hand is that the measuring concept used to generate experimental data needs to be sophisticated and planned in detail before measurements can be undertaken. This aspect becomes apparent by comparing the simulation results of in-situ data with an optimized measuring concept proposed in the contribution at hand and tested by generating data with finite element simulations. Furthermore, the main downsides of experimental data are identified to be limited and diffuse datasets, leading to unphysical simulation results. If the data-driven solver is to be used with in-situ data, the measuring concept needs to represent the material behavior locally, needs to measure all quantities required for the dataset, and should rely on minimal assumptions in processing the data. Another finding of the study at hand is that the data-driven solver is capable of simulating implicitly changing material characteristics, if these characteristics can be described by a measurable physical quantity, eliminating the need for explicit modeling and modeling assumptions.

In future contributions, the proposed measuring concept can be tested with in-situ data. The amount of data points, which is required by the experimental design needs to be investigated further. 10,080 data points can be too many points to be feasible to be obtained by experiments. Though it is not tested whether the dataset is large enough to be generally applicable for asphalt throughout the entire year. Furthermore, other experimental designs can be tested. For different physical problems, e.g., mechanical problems, diffusion problems, etc., other measuring strategies need to be designed. The consideration of uncertainty within the data-driven solver [20] also needs to be extended to experimental data as well. Furthermore, other studies can be conducted on the phenomena that the data-driven solver can capture implicitly by datasets, which physical quantities are best at describing the implicit change of material characteristics, which needs to be addressed problem-specific, and how to apply this efficiently within simulations conducted by the data-driven solver.

The proposed method is applied in one-dimensional simulations. These simulations show that the model-free data-driven solver can be used with pure experimental data. Nevertheless, the framework can be extended to multiple dimensions in further research. The currently available experimental data does not contain a second or third direction. Therefore, assumptions about the material itself (for instance, isotropy) would need to be invoked, which are contrary to the main idea of the model-free data-driven solver. Hence, the simulations in the contribution at hand are restricted to one dimension to support the main finding, which is the applicability of experimental data within the model-free data-driven solver.

## Author Contributions

**Jakob Platen:** Conceptualization, Methodology, Software, Validation, Visualization, Writing – original draft. **Atul Anantheswar:** Methodology, Software, Writing – original draft. **Felix Harazin:** Methodology, Software, Validation, Visualization, Writing – review
andediting. **Johannes Storm:** Conceptualization, Supervision, Validation, Writing – review and editing. **Michael Scherkenbach:** Data curation, Investigation, Resources, Writing – original draft. **Markus Oeser:** Funding acquisition, Project administration, Resources, Supervision. **Laurent Stainier:** Conceptualization, Methodology, Software, Supervision, Writing – original draft, Writing – review and editing. **Michael Kaliske:** Funding acquisition, Project administration, Supervision, Writing – review and editing.

**Acknowledgments**

This work is supported by the German Research Foundation (Deutsche Forschungsgemeinschaft) under project KA 1163/46-2 (Projekt-ID: 414936990), Project No. 453596084 "SFB/TRR 339" (Project A01), and Grant Nos. KA 1163/37-2 and GR 1504/11-2 within Priority Program 1886. Open Access funding enabled and organized by Projekt DEAL.

**Conflicts of Interest**

The authors declare no conflicts of interest.

**Data Availability Statement**

The data that support the findings of this study are openly available in Data-driven-thermal-conduction at https://github.com/JP-TUD-ISD/Data-driven-thermal-conduction.

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