# Cats, not CAT scans: a study of dataset similarity in transfer learning for
2D medical image classification

Irma van den Brandt1, Floris Fok1, Bas Mulders1, Joaquin Vanschoren1 Veronika
Cheplygina2,1

1Eindhoven University of Technology, The Netherlands  2IT University of
Copenhagen, Denmark

###### Abstract

Transfer learning is a commonly used strategy for medical image
classification, especially via pretraining on source data and fine-tuning on
target data. There is currently no consensus on how to choose appropriate
source data, and in the literature we can find both evidence of favoring large
natural image datasets such as ImageNet, and evidence of favoring more
specialized medical datasets. In this paper we perform a systematic study with
nine source datasets with natural or medical images, and three target medical
datasets, all with 2D images. We find that ImageNet is the source leading to
the highest performances, but also that larger datasets are not necessarily
better. We also study different definitions of data similarity. We show that
common intuitions about similarity may be inaccurate, and therefore not
sufficient to predict an appropriate source a priori. Finally, we discuss
several steps needed for further research in this field, especially with
regard to other types (for example 3D) medical images. Our experiments and
pretrained models are available via https://www.github.com/vcheplygina/cats-
scans

##  I Introduction

In medical image classification, labeled data is often scarce, inviting the
use of techniques such as transfer learning [26, 9, 52, 36, 28], where the
goal is to reuse information across datasets. When training a neural network,
transfer can be achieved by first training on a larger, source dataset (for
example, natural images such as cats) and then further fine-tuning on a
smaller, target dataset (such as a dataset of computed tomography (CT or CAT)
scans. This allows the network to reuse features it learned on the source
data, thus lowering the amount of target data needed.

A popular source dataset is ImageNet [15], although it has been argued whether
it is the best strategy for medical target data. For example, [32] argue that
the dataset properties (i.e. number of classes, level of granularity in
classes, size) of ImageNet and a medical target dataset may be too different
to allow effective feature reuse. More suitable features may be learned from
datasets more similar to the medical target dataset [43, 45]. Some studies
have indeed shown that the effectiveness of feature reuse decreased with
increasing differences between the source and target task [50, 42].

In a previous study we reviewed papers which compared medical to non-medical
source datasets for medical target data [8]. Out of 12 papers, three concluded
a medical source was best, three concluded that a non-medical source was best,
and two did not find differences. Others did not provide a definite
conclusion, but did provide other valuable insights. In general, the papers
seemed to agree that source datasets need to be “large enough” and “similar
enough”, but exact definitions of these properties were not given. Since each
paper used a different set of datasets, it was not possible to extract further
conclusions from the study.

Given a new target problem, we would like to identify what the best course of
action is without trying all possible source datasets. The goal of the study
is therefore two-fold:

  * •

Investigate the relationship between transfer learning performance and
properties such as dataset size or origin of images, and

  * •

Investigate whether a dataset similarity measure, based on such a meta-
representation, can be used to predict which source dataset is the most
appropriate for a particular target dataset.

We perform transfer learning experiments with nine datasets. We show that the
natural image dataset ImageNet, which is the largest, leads to the best
performances. However, even small datasets can be valuable for transfer
learning. We also examine two definitions of dataset similarity, and how it
relates to transfer learning performance. Both definitions show very weak to
weak correlations with the performance. However, the most similar datasets
appear to have the least effect on performance, which contrasts our results
from earlier findings.

##  II Related Work

###  II-A How can we do transfer learning?

Transfer learning [30] relies on the idea of transferring information from
related, but not the same, learning problems. In supervised classification
scenarios we normally assume that the training and test data are from the same
_domain_ $\mathcal{D}=(\mathcal{X},p(X)$), and _task_
$\mathcal{T}=(\mathcal{Y},f(\cdot))$, where $\mathcal{X}$ and $\mathcal{Y}$
are the feature and label spaces, $p(X)$ is the distribution of the feature
vectors, and $f$ the mapping of the features to labels. In transfer learning
scenarios, we assume that we are dealing either with different domains
$\mathcal{D}_{S}\neq\mathcal{D}_{T}$ and/or different tasks
$\mathcal{T}_{S}\neq\mathcal{T}_{T}$.

Transfer learning can be achieved via different strategies, of which a popular
one is to pretrain on the source data, and then use the pretrained network
either for extracting off-the-shelf features from the target data, or for
further fine-tuning on the target data. In the pretraining/fine-tuning
scenario, both the domain and the task can be different. Still, transfer
learning can be beneficial, and various studies have looked at how to do this
successfully [42, 19, 21, 37]. Some general findings are for example, that
fine-tuning tends to be more beneficial than extracting off-the-shelf
features, and that fine-tuning only the last layers is more effective [42].

###  II-B How are source datasets selected in medical imaging?

Since the introduction of transfer learning in medical imaging, transferring
weights from ImageNet has become popular - [28] review 102 papers which do
this. However, there are arguments that the size of images and number of
classes might not be suitable for medical imaging, where large (possibly
higher-dimensional) images of few classes are typically used [49, 32].

Although ImageNet is often used, it is rare to find papers comparing different
types of sources. We presented the details of 12 such comparisons in our
previous study [8], with a brief summary of the reviewed papers here for
completeness:

  * •

[35, 27, 33] conclude that a natural source dataset is better

  * •

[38, 23, 49] conclude that a medical source dataset is better

  * •

[7, 16] do not find differences

  * •

[39, 11, 29] show that larger source datasets are not necessarily better

  * •

[42, 51] suggest similarity to the target is important (but do not compare
natural to medical sources)

Such studies often mention the concept of dataset similarity, but this
definition is often not given or quantified. Similarity can be referred to in
terms of the visual image properties [27], feature representations [23],
number and variety of classes [5], and other characteristics.

###  II-C How can we predict an appropriate source dataset?

A way to predict an effective source dataset could be based on meta-learning
[47], where meta-data about existing datasets or methods and their performance
is available. Such a meta-dataset would allow training a model to predict how
well a particular dataset or method would perform in a previously unseen
situation. One specific example, and the example that we address, is
predicting the suitability of a source dataset. This could be done by only
considering the source and target datasets (but not the transfer learning
method), which we call “model-independent”.

Dataset source selection has been studied for various applications. In an
example from natural language processing [34], the authors learn a model-
independent measure of dataset similarity, weighting meta-features using
examples from the target dataset. They combine similarity measures based on
comparing distributions of words or topics with diversity features based on
the source data, and show that diversity combines complementary information.
Another example of dataset source selection is [3] where the authors propose a
distance measure based on the feature-label distributions (i.e., considering
both the domain and the task) of the datasets. They show it is predictive of
the transferability of datasets, defined by the relative decrease in error
when a source dataset is used. They show this for several applications,
although it is worth noting that for natural images, only versions of two
related datasets (Tiny ImageNet and CIFAR-10) are examined.

An example from computer vision is Task2Vec [1], a popular approach to encode
tasks (feature-label distributions). The idea behind Task2Vec is to obtain
embeddings of classification tasks so that the relationship between the tasks
can be analyzed, even if the datasets have different characteristics such as
number of classes or image sizes. They investigate both a symmetric and an
asymmetric version of the Task2Vec distance and show that the asymmetric
version correlates with transferability between tasks.

Compared to other applied fields, in medical imaging, the application of meta-
learning is relatively recent. To date it has been used for predicting
segmentation scores in photos of wounds [6], predicting if two datasets
originate from the same source [10], initializing weights for fine-tuning an
algorithm [18] and predicting segmentation scores of segmentation algorithms
across 13 datasets [40]. None of these studies have looked at dataset source
selection.

##  III Datasets

In total, we use nine datasets derived from six data sources in the
experiments. Some derived datasets are available online, others are created by
sampling from one of the data sources as shown Figure 2. Table I provides an
overview of the datasets and their properties. In Figure 1, three random
examples from every dataset are given. The non-medical datasets and medical
datasets PCam-small and KimiaPath 960 are used as source datasets only, the
other medical datasets ISIC2018, Chest X-rays and PCam-middle are used as
source and target datasets in the transfer learning process. Also,
KimiaPath960 is used as source dataset for PCam-middle only.

Figure 1: Three random images of every dataset used in the experiments. From
left to right: ImageNet (A), STL-10 (B), STI-10 (C), DTD (D), ISIC2018 (E),
Chest X-rays (F), PCam-middle and PCam-small (G), and KimiaPath960 (H). The
images shown for ImageNet subsets STL-10 (B) and STI-10 (C) are also included
in ImageNet (A).

####  III-1 ImageNet

The ImageNet database contains $1,431,167$ natural, colored images of size
$256$x$256$ from $1,000$ different classes [15]. On average every class
contains $1,000$ images.

####  III-2 STL-10

STL-10 is subset of ImageNet consisting of $10$ classes: airplane, bird, car,
cat, deer, dog, horse, monkey, ship and truck [13]. Each class contains
$1,300$ $96$x$96$ colored images.

####  III-3 Subset Textures ImageNet (STI-10)

The Subset Textures ImageNet (STI-10) dataset is a subset of ImageNet created
for this paper. The idea is to subsample ImageNet to have the same number of
images, but not focus on classes depicting textures. The subset consists of
$10$ classes representing a textural pattern, which are chosen pseudo-randomly
from the ImageNet hierarchy [41]. The $10$ classes included are the following:
brick, stone wall, flower, honeycomb, egg, rock, fabric, cloud, chain and
ocean. On average every class contains $900$ colored images of size
$112$x$112$.

####  III-4 Describable Textures Dataset (DTD)

The Describable Textures Dataset (DTD) contains images of textural patterns
labeled according to $47$ different human-centric attributes inspired by the
perceptual characteristics of the patterns [12]. Every class consists of $120$
colored images with size ranging from $300$x$300$ to $640$x$640$. The textural
pattern is displayed on at least $90\%$ of the surface of the image.

####  III-5 ISIC 2018 - Task 3 (Training set)

The ISIC 2018 database consists of $10,015$ dermoscopic images of skin lesions
collected by the International Skin Imaging Collaboration (ISIC) [14, 46]. The
images are categorized according to seven categories such as melanoma and
benign nevus which can be detected from the image. On average every class
contains $1,430$ colored images of size $600$x$450$. The maximum class
contains $6,705$ images, the minimum class only $115$. This dataset was
released by ISIC as a training dataset for the skin lesion classification task
(task 3) in the ISIC Challenge 2018. The test set is not included in this
paper since ISIC has never published the associated ground truth labels.

####  III-6 Chest X-rays

The Chest X-rays dataset contains $5,863$ binary-labeled (Pneumonia/Normal)
chest x-ray images from pediatric patients aged one to five years old
collected by the Guangzhou Women and Children’s Medical Center [20]. From the
$5,863$ images $1,583$ are labeled normal and $4,273$ are labeled with
pneumonia. Image sizes range from $72$x$72$ to $2,916$x$2,583$.

####  III-7 PatchCamelyon-100,000 (PCam-middle)

PatchCamelyon (PCam) is a classification dataset consisting of $237,680$
colored images of size $96$x$96$ displaying histopathologic scans of lymph
node sections [48]. The images are labeled positive depending on the presence
of metastatic tissue, with $163,818$ positive and $163,862$ negative images. A
random, stratified subset of $100,000$ images is created to speed up training.

####  III-8 PatchCamelyon-10,000 (PCam-small)

Like PCam-middle, PCam-small is a stratified subset of the original
PatchCamelyon dataset. The total number of images in this subset is 10% of
PCam-middle, namely $10,000$.

####  III-9 KimiaPath960

The KimiaPath960 classification dataset contains 960 histopathology images of
size $308$x$168$ labeled according to 20 different classes [22]. The twenty
classes are based on the image selection process: out of 400 whole slide
images of muscle, epithelial and connective tissue 20 images visually
representing different pattern types were chosen. After this, 48 regions of
interest of the same size were selected and downsampled to $308$x$168$ from
all 20 WSI scans.

Dataset | Origin | Size | Classes | Image Size | Channels
---|---|---|---|---|---
ImageNet [15] | Non-medical | 1,431,167 | 1,000 |  $256$x$256$ | 3
STL-10 [13] | Non-medical | 13,000 | 10 |  $96$x$96$ | 3
Subset Textures ImageNet (STI-10) | Non-medical | 8,060 | 10 |  $112$x$112$ | 3
Describable Textures Dataset (DTD) [12] | Non-medical | 5,640 | 47 | [$300$x$300$, $640$x$640$] | 3
ISIC 2018 - Task 3 (Training set) [14, 46] | Medical | 10,015 | 7 |  $600$x$450$ | 3
Chest X-rays [20] | Medical | 5,863 | 2 | [$72$x$72$, $2916$x$2583$] | 1
PatchCamelyon-100,000 (PCam-middle) | Medical | 100,000 | 2 |  $96$x$96$ | 3
PatchCamelyon-10,000 (PCam-small) | Medical | 10,000 | 2 |  $96$x$96$ | 3
KimiaPath960 [22] | Medical | 960 | 20 |  $308$x$168$ | 3
Table I: Datasets used in the experiments with properties name, medical or
non-medical origin, size, number of classes, image sizes, and number of
channels (RGB/Grey-scale). The non-medical datasets ImageNet, STL-10, STI-10,
DTD, and the medical datasets PCam-small and KimiaPath960 are used as source
dataset only, the other medical datasets ISIC2018, Chest X-rays and PCam-
middle are used as source and target dataset.

##  IV Methods

###  IV-A Transfer learning

The first part of our experiments consists of transfer learning experiments.
We investigate how different networks, pretrained on the source data with
different properties (such as dataset size), perform on the target data.

An overview summarizing the transfer learning experiments different
subsections is given in Figure 2.

Figure 2: Overview of the experiment setup. Data-related entities are denoted
by cylinders, code-entities by rectangles. Red entities are stored on a
private server, blue entities on Github, green entities on Open Science
Framework (OSF) and orange entities on a Neptune Observer using Sacred.

We perform all experiments with the ResNet50-like architecture, as our focus
is on the influence of dataset properties on transfer learning, rather than on
optimizing performance for a specific dataset. We chose ResNet50 due to their
accessibility and ease of optimization [17]. We initially also included an
EfficientNet-architecture [44], with similar results but slower training, thus
we did not continue with these experiments.

The first step is to pretrain the models. For ImageNet, we used publicly
available pretrained models. For the other datasets which are smaller in size,
we slightly modify the ResNet50 architecture by replacing the fully-connected
layers by a GlobalAveragePooling2D, Dropout and Dense layer for predictions,
in order to reduce overfitting [25]. We aimed to keep other training settings
(dropout rate, data augmentation, image size, batch size, learning rate,
training epochs) constant across experiments, but allowed modifications in
cases of severe over- or underfitting. The full list of settings can be found
in Appendix Appendix A - Pretraining settings. These settings were chosen
using only the training and validation splits of each dataset.

The second step is to fine-tune the pretrained models with medical target
images. For this we adapt the prediction layer of the pretrained model to the
target dataset. All other layers are initialized with the resulting weights
from pretraining. The whole model is then fine-tuned, as deeper fine-tuning is
recommended when the source and target data are from different domains [21,
42]. Again, we kept the other settings as constant as possible.

We evaluate the models using 5-fold cross-validation with $80\%-10\%-10\%$
splits into a training, validation and test sets. We first examined the
performance difference of the pretrained models on these sets to diagnose
over- or underfitting. We used accuracy for natural images, and AUC (area
under the curve) for the medical images. For multi-class problems, we used a
one-against-all AUC, weighted by class priors [31]. For evaluating the fine-
tuned models on the target datasets, we used 5-fold cross-validation and
reported the average AUC-score on the target dataset.

###  IV-B Meta-learning

In our meta-learning experiments, rather than directly searching for a meta-
learning model to predict the most suitable source, we focus on the
intermediate step of defining an meta-representation for each dataset, which
allows us to measure their similarity. We use two types of meta-features,
based on experts, and based on Task2Vec [1].

The expert features are inspired by high-level properties that are sometimes
mentioned when image datasets are described. We asked five biomedical
engineering students, familiar with transfer learning, to look at nine random
images from the dataset, and quantify six properties: natural or medical
images, grayscale or color images, objects or textures, the type of objects or
textures shown, the diversity of images, and the number of classes. The
observers’ answers are combined via majority vote, and dataset similarity is
calculated via the Hamming distance.

For Task2Vec we use a ResNet50 model pretrained on ImageNet as the probe
network. This architecture is chosen for consistency with the transfer
learning experiments. For each dataset we then create 100 different subsets
using sample weights from the original dataset and the subset index (i.e.,
1-100) as random state. The size of each subset is equal to 1% of the size of
the original dataset. The settings such as number of epochs and optimizer used
by Task2Vec during training are equal to those in [1]. Running Task2Vec thus
creates 100 embeddings per dataset. We calculate the dataset distance as the
average cosine distance between five random pairs of embeddings.

To evaluate the quality of these representations, we examine the Spearman
correlation between the dataset distance in meta-feature space, and
transferability of the datasets. The transferability measures the relative
improvement on the target dataset, when a particular source dataset is used.
We adapt the measure from [3], which is based on the classification error, but
due to the use of medical target datasets we use the AUC:

$\mathcal{T}(\mathcal{D}_{S}\rightarrow\mathcal{D}_{T})=\frac{AUC(\mathcal{D}_{S}\rightarrow\mathcal{D}_{T})-AUC(\mathcal{D}_{T})}{AUC(\mathcal{D}_{T})}\times
100$ (1)

where $AUC(\mathcal{D}_{S}\rightarrow\mathcal{D}_{T})$ is the performance when
using transfer learning from source to target, and $AUC(\mathcal{D}_{T})$ is
the performance of the baseline without transfer learning, i.e. only using the
target data.

##  V Results

###  V-A Transfer learning

####  V-A1 Pretraining

Table II shows the results for the source datasets alone, as a baseline to
establish that acceptable scores are obtained before moving on to fine-tuning.
Note that here single splits are used.

For the non-medical datasets we evaluate the accuracy. For STL-10, based on
natural images, we obtain reasonable accuracy. The accuracy for the texture
dataset DTD is rather low - classifying 47 different textural patterns with
only 120 images per class is a difficult task. The same holds for STI-10, even
though there are now 10 classes with 900 images per class.

For the medical datasets, AUC is the most relevant, although we report the
accuracy for completeness. The AUCs show fairly good discrimination between
classes. The difference in PCam-middle and PCam-small shows the effect of
reducing the number of images by a factor of 10. Interestingly, the accuracy
of both models shows overfitting that is more pronounced than in the other
sources.

Pretraining Results
---
Dataset | Train Acc | Valid Acc | Test Acc | Test AUC
STL-10 | 0.749 | 0.783 | 0.712 |
STI-10 | 0.719 | 0.607 | 0.614 |
DTD | 0.633 | 0.672 | 0.638 |
ISIC2018 | 0.568 | 0.510 | 0.499 | 0.848
Chest X-rays | 0.810 | 0.804 | 0.790 | 0.912
PCam-middle | 0.801 | 0.709 | 0.710 | 0.834
PCam-small | 0.741 | 0.683 | 0.689 | 0.790
KimiaPath960 | 0.842 | 0.833 | 0.854 | 0.997
Table II: Training, validation and test accuracy scores from pretraining on
source datasets. Additionally, for the medical source datasets the AUC-score
is given. These are used as a baseline to establish that good enough models
are trained on the sources.

####  V-A2 Fine-tuning

The AUC scores for fine-tuning on the target datasets experiments are given in
Figure 3. For all three targets, ImageNet weights lead to the highest scores.
STL-10 and STI-10 both lead to lower performances, although the difference
from ImageNet depends on the target dataset. This suggests reducing the number
of classes in ImageNet decreases performance, which is consistent with results
in [19].

The texture dataset DTD has the second highest scores for ChestX-Ray and PCam-
middle, but is not particularly good for ISIC2018. This suggests that the
visual importance of textures may be more important for ChestX-Ray and PCam-
middle, than for ISIC2018. The high scores on ChestX-Ray and PCam-middle are
interesting given the difference in size between DTD and ImageNet. Acquiring
more images for all classes in DTD and improving the pretrained model could be
worthwhile here. STI-10 performs similarly or worse than DTD in all cases,
suggesting that the granularity of DTD might be important. Another potential
difference is the resolution of the images, which is lower for STI-10.

Figure 3: Mean AUC-scores and error bars of five-fold cross validation
transfer learning experiments. Source datasets are encoded by a different
color bars, target datasets ISIC2018, Chest X-rays and PCam-middle are shown
on the x-axis. Using ImageNet weights results in highest scores across all
target datasets.

Overall, the AUC scores suggest it is not necessarily better to train on
medical sources when the target dataset is medical, as we obtained the highest
scores with ImageNet. These results do not necessarily align with those of
earlier studies. For example, for PCam-middle, [45] had better results when
training on the full KimiaPath dataset (available only on request and hence
not used here) plus the full PCam dataset, suggesting that increasing the size
could improve performance.

On the other hand, our results also show that a large dataset is not a
prerequisite for good performance, as shown by training on DTD. This could
mean there are opportunities for curating source datasets such that they are a
good fit for specific types of target problems, but also more require less
training resources due to their smaller size.

###  V-B Meta-learning

We show the relationship between the dataset distances and the transferability
(defined in Eq. 1) in Fig. 4. The expert assessments show a weak correlation
($\rho=0.24$), and the task2vec representation shows a very weak negative
correlation ($\rho=-0.11$). Neither correlation is significant, if we use 0.05
as the significance threshold.

There are, however, some trends we can observe in the plots. A general trend
is that smaller dataset distances do not affect the transferability a lot.
Although research often refers to “similar” datasets being effective,
intuitively the most similar dataset is the target dataset itself, which would
not add new information to the learning problem.

On the other hand, larger dataset distances can impact the transferability
both positively and negatively. While ImageNet has a higher distance to most
datasets, presumably its size and diversity ensure positive transferability
scores for most datasets. On the other hand, Chest-XRay might be too different
from the other datasets we used, possibly because of grayscale rather than RGB
images, and sometimes even leads to negative transferability scores.

Figure 4: Transferability versus dataset distance, defined by (top) experts,
and (bottom) task2vec.

##  VI Discussion

To summarize, our results show that:

  * •

ImageNet weights lead to the best AUC scores for all three medical targets

  * •

The dataset size and number of classes are important, but there are
exceptions, such as with the texture dataset DTD

  * •

Medical sources do not necessarily outperform non-medical sources

  * •

“Similar” datasets often do not improve training, “dissimilar” datasets can
affect training both positively and negatively

There are a number of important considerations when interpreting these
results, which we discuss below.

###  VI-A Transfer learning

We estimated transfer learning performance using a modified ResNet50 model,
and allowed for minor changes to the parameters to prevent severe over- or
under-fitting. Despite allowing for this, a few models did not converge during
training. Since we tried to avoid exhaustively optimizing for each dataset,
this is a feature rather than a bug. We also had to make some concessions
(such as resizing the images) due to computational resource constraints. On
the other hand, for ImageNet we used publicly available weights, where more
resources have been allocated to optimization. Therefore, our results might
not necessarily reflect the performances typically found in the literature,
when less datasets are examined at a time.

We only report results for a ResNet50-like architecture, however in early
experiments, we obtained similar results with an EfficientNet, which was less
computationally efficient, which is why we did not continue with the
experiments. Despite ResNet’s suitability for transfer learning, for smaller
datasets with fewer classes like the medical datasets, over-parameterization
is a serious issue with this model [32], which is consistent with our results.
Based on these findings, it would be worthwhile to repeat the experiment using
smaller, more lightweight models [24].

###  VI-B Meta-Learning

We used two different ways to quantify the similarity of datasets - based on a
data-driven Task2Vec representation, and based on human assessments of high-
level features of the datasets, such as the type of images. The Task2Vec
representation may not be the best choice, because it is based on an ImageNet-
trained network, possibly biasing the results. The expert representation is
based on selected properties, and would be different if other factors were
included.

Finally, our experiments encompass a limited selection of (types) of datasets.
In particular, we chose to focus on 2D datasets due to the ease of adaptation
of training the models, and all datasets except Chest-XRay consisted of RGB
images. As such, the transferability plots likely only show a part of the
whole picture. Including other modalities, particularly 3D images such as
computed tomography (CT) or magnetic resonance (MR) images, would increase the
range of dataset distances, and possibly also transferability scores.

Even with an increased range of dataset distances and transferability scores,
we cannot expect generalization to all datasets and tasks. Locally, however,
we expect there to be useful patterns, that can reduce the computational
resources needed when selecting a source dataset for a previously unseen
target medical imaging problem.

###  VI-C Future directions

Next to studying other transfer learning and meta-learning methods, we expect
that the most informative experiments will involve using other datasets. These
include other medical imaging modalities such as CT or MR, but also including
modified versions of the source datasets. It would be valuable to have more
fine-grained variations in various dataset properties, such as the size.

There is also a connection between data augmentation and transfer learning
that is worth investigating. Introducing more diversity into the training
data, as data augmentation does, can be beneficial, but can also be
detrimental if the changes are too large. As such it would be interesting to
interpolate between data augmentation and transfer from a different source,
for example by creating “mix-and-match” source datasets.

These experiments are resource-intensive, especially if larger datasets are
used. We would therefore emphasize the importance of researchers sharing their
pretrained models. This could improve performance across datasets, but also
helps with reducing the overall carbon footprint of our field [4] and prevents
its de-democratization [2]. Ideally we envision a platform like
OpenML111https://www.openml.org, where sharing of datasets, experiments and
results would be possible. Until then we encourage the use of persistent
repositories, such as Github and OSF. Our own code and models can be accessed
via https://www.github.com/vcheplygina/cats-scans.

##  VII Conclusions

We studied transfer learning from natural and medical 2D image datasets, and
investigated dataset similarity in relation to the performance after transfer.
We show that ImageNet leads to the best performances after transfer, but our
results also suggest that even small datasets can be valuable for transfer. We
also show that our intuition about dataset similarity is not necessarily
predictive of transfer learning performance. We expect that further insights
can be gained by studying more diverse datasets and connections with for
example data augmentation.

## Acknowledgments

We thank Colin Nieuwlaat, Felix Schijve and Thijs Kooi for contributing to the
discussions about this project. Veronika Cheplygina is supported by the
NovoNordisk Foundation (starting package grant) from May 2021, however, the
work in this paper was done previously to this award.

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## Appendix A - Pretraining settings

###  -A Pretraining

Source Dataset | Image size | Batch Size | Learning Rate | Learning Rate Scheduler | Epochs
---|---|---|---|---|---
STL-10 |  $96$x$96$ | 128 | 1.0$e-$3 | Yes, after 30 epochs | 50
STI-10 |  $112$x$112$ | 128 | 1.0$e-$4 | Yes, after 50 epochs | 70
DTD |  $300$x$300$ | 12 | 1.0$e-$3 | No | 60
ISIC2018 |  $112$x$112$ | 128 | 1.0$e-$5 | No | 70
Chest X-rays |  $112$x$112$ | 128 | 1.0$e-$6 | No | 70
PCam-middle |  $96$x$96$ | 128 | 1.0$e-$6 | No | 50
PCam-small |  $96$x$96$ | 128 | 1.0$e-$6 | No | 70
KimiaPath960 |  $308$x$168$ | 12 | 1.0$e-$5 | No | 100
Table III: Image sizes and compilation settings for all source datasets used
during pretraining. Image sizes are either based on the minimal image size in
the dataset, or to avoid long training times or unnecessary resizing. In case
of larger image sizes the batch size is minimal to avoid memory issues.
Compilation settings are chosen such that the model converges and overfitting
is minimized.

Generators feed the images in mini-batches to the model, the size of the mini-
batch given a specific source dataset is given in Table III. For DTD and
KimiaPath960 a small batch size of 12 is used due to the large image sizes,
for all other datasets a large batch size of 128 is used.

The different image sizes and model compilation settings for all source
datasets are given in Table III. In most cases the image size is based on the
minimal image size of the dataset (Table I). For ISIC2018 a smaller size is
chosen to speed up training, for Chest X-rays a larger size to avoid
unnecessary resizing.

The learning rate, used as argument for the Adam-optimizer, and number of
epochs are set to minimize overfitting and (almost) enable model convergence.
In some cases a learning rate scheduler, that multiplies the learning rate by
$0.5$ after $x$ amount of epochs, is used to slow down learning and avoid
overfitting. Potential class imbalance is minimized by passing class weights
to the model.

###  -B Fine-tuning

The images are fed in mini-batches of size 128 to the model after being
augmented with the same small augmentations as used during pretraining to
potentially improve transfer performance results [28].

All images are resized to $112$x$112$ for experiments where ISIC2018 or Chest
X-rays is used as target dataset. For experiments involving PCam-middle as
target dataset the images are not resized, their original size of $96$x$96$
remains unchanged. The Adam-optimizer is used with learning rates varying
between 1.0$e-$4 and 1.0$e-$6 for different target datasets. In experiments
involving target datasets ISIC2018 or Chest X-rays the number of epochs is set
to 50, for PCam-middle the number of epochs is 20 to avoid long training
times. The learning rates and number of epochs are chosen such that (severe)
overfitting is avoided and the model (almost) converges. Class weights are
provided to the model to minimize potential class imbalance.


# Transient non-Fourier behavior of large surface bodies

R. Kovács  1Department of Energy Engineering, Faculty of Mechanical
Engineering, Budapest University of Technology and Economics, Műegyetem rkp.
3., H-1111 Budapest, Hungary
2Department of Theoretical Physics, Wigner Research Centre for Physics,
Institute for Particle and Nuclear Physics, Budapest, Hungary
3Montavid Thermodynamic Research Group, Budapest, Hungary

###### Abstract.

The variety and complexity of heterogeneous materials in the engineering
practice are continuously increasing, open-cell metal foams filled with phase
change materials are typical examples. These are also having an impact on the
recent developments in the energy industry. Earlier room temperature heat
pulse experiments on macroscale foam samples showed non-Fourier over-diffusive
behavior on a particular time scale. Since there is a need to investigate such
complex structures on larger spatial scales and extend the one-dimensional
analysis on two-, and three-dimensional settings, here we develop a two-
dimensional analytical solution for the Guyer-Krumhansl and Jeffreys-type heat
equations in cylindrical coordinates to investigate the transient thermal
behavior of large bodies. We provide the steady-state and transient
temperature and heat flux distributions for a space-dependent heat source. The
solutions presented here will be helpful for the thermal characterization of
complex materials and for the validation of numerical methods.

##  1\. Introduction

Numerous experimental and theoretical studies emerged on room-temperature heat
conduction beyond Fourier in recent years. On the one hand, the nanoscale
effects result in the deviation from Fourier’s law, usually with the
appearance of ballistic heat conduction [1, 2, 3]. Furthermore, the size
dependence of thermal conductivity enjoys great interest as it significantly
influences the effectiveness of any nanoscale device [4, 5]. On the other
hand, room temperature non-Fourier heat conduction is not restricted to the
nanoscale exclusively, and it is observable in macroscopic bodies under
various conditions [6, 7, 8]. While the parallel diffusive and ballistic
propagation modes are present on a nanoscale, the macroscopic deviation is due
to the interaction of multiple parallel diffusive (and additional heat
transfer) channels. Typical examples are rocks [9] and foams [10, 11].
Although each component behaves according to Fourier’s law, the heterogeneous
material structure overall (effectively) leads to a more complex, non-Fourier
temperature history. That was the motivation for two-temperature models [12,
13, 14].

The presence of multiple time scales is the most visible by showing
experimental data obtained from a heat pulse experiment for an aluminum foam
(Figure 1) possessing multiple heat transfer channels. The response for a
short but finite single pulse ($0.01$ s), together with the best achievable
Fourier fit, shows that at least two heat conduction time scales are present
simultaneously. This is called over-diffusion [15, 6] and so far best modeled
with the continuum Guyer-Krumhansl (GK) heat equation [10]. Here, with the
word ”continuum” is an adjective, referring to the continuum thermodynamic
background of the GK equation [16], it is free from the usual kinetic theory
and phonon hydrodynamic assumptions, therefore it is valid on much larger
temperature and spatial scales, independently of the Knudsen number. It is
also worth noting that in Fig. 1, the time is re-scaled with respect to the
pulse length, i.e., the dimensionless time $\hat{t}=t/0.01$. The transients
become slow enough after about $7$ s, meaning that the heat transfer process
needs much more time to cancel out the effect of multiple time scales. In
other words, if thermal transients continuously occur in some particular
application, Fourier’s law might not apply. Otherwise, such an experiment
reveals the limitations on time scales, and the Guyer-Krumhansl equation
provides a refinement of the thermal parameters in order to cover the faster
processes as well. However, this property indeed scales with the size
(different heat capacities, heat transfer surfaces, time scales) and surface
(boundary conditions), and thus it is necessary to extend the experimental and
theoretical capabilities in this direction.

It is worth noting that recent heat exchanger applications exploit the
advantageous properties of a metal foam structure: having large heat transfer
surfaces, the matrix material is an excellent heat conductor, therefore such
solutions can essentially ease the realization of an effective thermal storage
method. One outstanding example is when an open-cell foam structure is filled
with phase change material [17, 18, 19, 20]. The phase change materials
usually have low thermal conductivity, significantly restricting their melting
or solidification properties. However, a surrounding foam structure can
notably enhance the thermal behavior, thus both the heating and cooling
processes can be much more efficient. This further motivates the present study
as there are currently no reliable thermal models which enable the resource-
friendly modeling of such complex structures. A non-Fourier model, however,
can be exceptional when the role of the parallel heat transfer channels is
understood correctly in such an approach. The present study aims to take a
step forward in this direction, deepening our understanding and extending our
modeling possibilities about the Guyer-Krumhansl heat equation. In the
following, let us briefly summarize the heat conduction models we consider
here.

Figure 1. Typical appearance of over-diffusion [10].

The well-known Fourier law is

$\displaystyle\mathbf{q}=-\lambda\nabla T,\quad\lambda\in\mathbb{R^{+}}$ (1)

together with the balance of internal energy ($e=c_{v}T$),

$\displaystyle\rho
c_{v}\partial_{t}T+\nabla\cdot\mathbf{q}=q_{v}(\mathbf{x},t),$ (2)

includes only one time scale, described by the thermal diffusivity
$\alpha=\lambda/(\rho c_{v})$, where $\lambda$, $\rho$ and $c_{v}$ are the
thermal conductivity, mass density, and isochoric specific heat, respectively.
Furthermore, $\mathbf{q}$ and $T$ stand for the heat flux and temperature
fields, and $q_{v}$ is an internal heat generation, which could be time and
space-dependent. We restrict ourselves to isotropic rigid materials.

For a non-Fourier heat conduction model, the constitutive equation, Eq. (1),
is exchanged with a more general expression, usually consisting of additional
time and space derivatives. In the present paper, we consider the following
two constitutive equations among the various models. First, we study the
Guyer-Krumhansl equation,

$\displaystyle\tau\partial_{t}\mathbf{q}+\mathbf{q}=-\lambda\nabla
T+\eta_{1}\Delta\mathbf{q}+\eta_{2}\nabla\nabla\cdot\mathbf{q},\quad\lambda,\tau,\eta_{1},\eta_{2}\in\mathbb{R^{+}},$
(3)

in which $\tau$ is the relaxation time; $\eta_{1}$ and $\eta_{2}$ are
independent intrinsic length scales, not associated with a propagation
mechanism in a continuum model [16]. We note that in the conventional
treatment of the GK equation, $\eta_{1}=l^{2}$ with $l$ being the mean free
path of phonons, and $\eta_{2}/\eta_{1}=2$ for the particular approximations
performed by Guyer and Krumhansl [21]. We emphasize that for a macroscopic
room temperature problem, the phonon approach is not valid anymore, while the
continuum model, although possessing the same structure, is free from any
prior specific assumptions on the propagation mechanism, hence extending the
model’s domain of validity.

Second, we will continue our analysis with the Jeffreys equation (JE), i.e.,

$\displaystyle\tau_{q}\partial_{t}\mathbf{q}+\mathbf{q}=-\lambda\nabla
T-\lambda\tau_{T}\partial_{t}\nabla
T,\quad\lambda,\tau_{q},\tau_{T}\in\mathbb{R^{+}}$ (4)

where, instead of introducing further spatial derivatives, two time lags
appear similarly to the popular dual-phase-lage (DPL) concept [15]. However,
while no thermodynamic background is behind the DPL model, Eq. (4) can be
derived on a thermodynamic basis [22]. Although we remain in the linear
regime, it is worth noting that the coefficients are not completely
independent of each other (also for Eq. (3)) in a sense that the
$T$-dependence of $\lambda$ would influence all the other parameters, too
[23]. For the Jeffreys equation, $\tau_{q}$ and $\tau_{T}$ can be adjusted
almost independently, and the only exception is that when $\tau_{q}=0$,
$\tau_{T}=0$ follows immediately (but not vice versa). The GK and JE models
consist of two time scales in different ways, and they share the same
$T$-representation in a one-dimensional setting. However, their physical basis
is quite different, and the GK equation fits much better into the systematic
structure of non-Fourier models.

In this paper, we choose to study large surface bodies since the typical
heterogeneous materials also show strong size-dependent behavior [9]. In other
words, extending the existing flash experiment for much larger bodies will be
necessary as the usual thickness limit is about $3-6$ mm for standardized
equipment. This can be much smaller than the representative sample size for a
heterogeneous material, especially for foams with large (3-5 mm) open-cell
structures. Furthermore, we aim to investigate both models using an analytical
solution for a two-dimensional setting, which will emphasize the structural
differences between these equations. We note that even the book of Carslaw and
Jaeger [24] has limitations towards the problem setting we discuss in the
following.

For simplicity and clarity, we start and present our method on the example of
the Fourier heat equation. This will provide an insight into the problem
setting and the solution method. We will continue with the GK and JE
equations, demonstrating how the solution method is applied to more
complicated models. Such analytical solutions, especially for the GK equation,
cannot be found in the literature. Moreover, as there is also missing a
reliable two or three-dimensional numerical method, we offer a good starting
point for future studies in this direction. Finally, we will compare the
temperature histories to the Fourier equation and investigate whether we find
the transient behavior similar (or even the same) compared to the one-
dimensional room temperature experiments on small samples.

##  2\. Problem statement

Let us consider a plane wall constantly heated on one the left side in a
circular area ($r<r_{h}$) with $q_{w}$ such as Figure 2 (left side)
prescribes. In fact, the book of Carslaw and Jaeger [24] offers a solution for
the Fourier heat equation for constant heating in the domain $r<r_{h}$,
however, we are looking for the temperature history further away from $r_{h}$
such as how the blue dots showing in Fig. 2. Additionally, since the boundary
condition is space-dependent on the left side, it is challenging and difficult
to apply the findings for non-Fourier heat equations. Therefore, we decided to
reduce this original problem to a simpler one and substitute the surface
heating with a space-dependent, surface-concentrated internal heat generation,
as the characteristics present in Fig. 3. Hence the boundary conditions remain
homogeneous but still applicable to the original problem. Furthermore, we
assume that at a large enough distance from the heat source, the temperature
remains constant, thus we prescribe constant temperature boundary conditions
on the right side and on the top.

Figure 2. Schematic problem setting, presenting the boundary conditions and
the characteristics of internal heat generation.

It is more convenient to formulate our models in a cylindrical coordinate
system using dimensionless quantities. For the length scale, we use $R$, the
radius of the whole domain, and we set the thickness to be the same, i.e.,
$L=R$. The usual Fourier number is introduced using the thermal diffusivity
$\alpha=\lambda/(\rho c_{v})$ for the time scale. The temperature field is
homogenized and normalized with the initial temperature $T_{0}$. Overall,
these lead to the following set of dimensionless quantities,

$\displaystyle\hat{r}=\frac{r}{R},\quad\hat{z}=\frac{z}{R},\quad\hat{t}=\frac{\alpha
t}{R^{2}},\quad\hat{T}=\frac{T-T_{0}}{T_{0}},\quad\hat{q}_{r,z}=q_{r,z}\frac{R}{\lambda
T_{0}},\quad\hat{q}_{v}=q_{v}\frac{R^{2}}{\lambda T_{0}},$ (5)

and thus, the non-Fourier parameters read

$\displaystyle\hat{\eta}_{1,2}=\eta_{1,2}\frac{1}{R^{2}},\quad\hat{\tau}=\frac{\alpha\tau}{R^{2}},\quad\hat{\tau}_{T}=\frac{\alpha\tau_{T}}{R^{2}}.$
(6)

In the following, we leave the hat notation for simplicity and show the units
wherever necessary.

Figure 3. Internal heat generation characteristics (with dimensionless
parameters), the heat source is concentrated on the surface with respect to
the scaling.

Taking account that it is a two-dimensional problem for $r$ and $z$ in a
cylindrical coordinate system, the balance of internal energy

$\displaystyle\partial_{t}T+\partial_{r}q_{r}+\frac{1}{r}q_{r}+\partial_{z}q_{z}=q_{v}(r,z),\quad
t\in[0,\infty),\quad(r,z)\in[0,1]\times[0,1],$ (7)

and the constitutive equations are

Fourier: $\displaystyle q_{r}=-\partial_{r}T,$ (8) $\displaystyle
q_{z}=-\partial_{z}T,$ (9) GK:
$\displaystyle\tau\partial_{t}q_{r}+q_{r}=-\partial_{r}T+(\eta_{1}+\eta_{2})\left[\partial_{rr}-\frac{1}{r^{2}}+\frac{1}{r}\partial_{r}\right]q_{r}+\eta_{1}\partial_{zz}q_{r}+\eta_{2}\partial_{rz}q_{z},$
(10)
$\displaystyle\tau\partial_{t}q_{z}+q_{z}=-\partial_{z}T+(\eta_{1}+\eta_{2})\partial_{zz}q_{z}+\eta_{1}\left[\partial_{rr}+\frac{1}{r}\partial_{r}\right]q_{z}+\eta_{2}\left[\frac{1}{r}\partial_{z}+\partial_{rz}\right]q_{r},$
(11) JE:
$\displaystyle\tau\partial_{t}q_{r}+q_{r}=-\partial_{r}T-\tau_{T}\partial_{tr}T,$
(12)
$\displaystyle\tau\partial_{t}q_{z}+q_{z}=-\partial_{z}T-\tau_{T}\partial_{tz}T,$
(13)

accompanying the $T=0$ initial condition, and $q=0$ and $T=0$ boundary
conditions with respect to Fig. 2. We are looking for the corresponding heat
flux and temperature fields.

##  3\. Solution method

Although the problem seems complicated, there is quite efficient method to
handle such a complicated set of partial differential equations. Here is the
strategy we follow. First, we exploit that the problem can be separated into
two parts, viz., we can solve the homogeneous ($q_{v}=0$) transient case
($T_{h}(r,z,t)$) separately and the inhomogeneous ($q_{v}\neq 0$) steady-state
case ($T_{\textrm{st}}(r,z)$), and we find the solution as their
superposition, $T(r,z,t)=T_{\textrm{st}}(r,z)+T_{h}(r,z,t)$. Second, we start
with the Fourier heat equation, not merely because we want to compare the non-
Fourier solutions to Fourier’s, but because we can exploit Fourier’s steady-
state solution in solving the non-Fourier models. The steady temperature field
remains the same for both Fourier and non-Fourier cases. However, the heat
flux fields can differ, hence we also include this aspect. In the case of
Fourier’s heat equation, we can quickly determine the proper eigenfunctions
and eigenvalues through the Sturm-Liouville problem. It is worth noting that
even the non-Fourier models do not introduce higher-order spatial derivatives
for the temperature field beyond the Laplacian. Therefore what eigenfunctions
we find can also be applied to the GK and JE models. Third, we solve the non-
Fourier models exploiting the ansatz that their solutions can be represented
in the same set of eigenfunctions with time-dependent coefficients, this is a
sort of Galerkin method. That approach simplifies the complicated system of
partial differential equations to a set of ordinary differential equations for
the time-dependent coefficients.

###  3.1. Fourier heat equation

In the case of the Fourier equation, it is easier to start with its
$T$-representation, i.e.,

$\displaystyle\partial_{t}T=\partial_{r}^{2}T+\frac{1}{r}\partial_{r}T+\partial_{z}^{2}T+q_{v}(r,z),$
(14)

and applying the standard separation of variable technique for the homogeneous
part (where $q_{v}$ is absent), $T_{h}(r,z,t)=\varphi(t)\xi(r,z)$, one obtains

for time:
$\displaystyle\frac{\textrm{d}\varphi}{\textrm{d}t}+\beta^{2}\varphi=0,$ (15)
for space:
$\displaystyle\frac{1}{r}\partial_{r}\xi+\partial_{r}^{2}\xi+\partial_{z}^{2}\xi+\beta^{2}\xi=0,\quad\Rightarrow\quad\xi(r,z)=\rho(r)\zeta(z),\quad\Rightarrow$
(16) for $r$:
$\displaystyle\frac{1}{r}\frac{\textrm{d}\rho}{\textrm{d}r}+\frac{\textrm{d}^{2}\rho}{\textrm{d}r^{2}}+\mu^{2}\rho=0,$
(17) for $z$:
$\displaystyle\frac{\textrm{d}^{2}\zeta}{\textrm{d}z^{2}}+\gamma^{2}\zeta=0,$
(18)

thus $\mu^{2}+\gamma^{2}=\beta^{2}$. Applying the boundary conditions

$\displaystyle q_{r}(r=0,z,t)=\partial_{r}T|_{r=0}=0,\quad T(r=1,z,t)=0,\quad
q_{z}(r,z=0,t)=\partial_{z}T|_{z=0}=0,\quad T(r,z=1,t)=0,$ (19)

we find the following eigenfunctions and eigenvalues,

$\displaystyle\rho(r)=J_{0}(\mu_{n}r),\quad\mu_{n}:\
J_{0}(\mu_{n})=0,\quad\zeta(z)=\cos(\gamma_{m}z),\quad\gamma_{m}=\frac{\pi}{2}+m\pi,$
(20)

and hence the solution is

$\displaystyle
T_{h}(r,z,t)=\sum_{n=1}^{\infty}\sum_{m=0}^{\infty}K_{nm}e^{-\beta_{nm}^{2}t}J_{0}(\mu_{n}r)\cos(\gamma_{m}z),\quad\beta_{nm}^{2}=\mu_{n}^{2}+\gamma_{m}^{2}.$
(21)

Consequently, we can use Eq. (20) to construct the steady-state solution
$T_{\textrm{st}}$, including the heat generation as well, i.e.,

$\displaystyle
T_{\textrm{st}}(r,z)=\sum_{n=1}^{\infty}\sum_{m=0}^{\infty}C_{nm}J_{0}(\mu_{n}r)\cos(\gamma_{m}z),\quad
q_{v}(r,z)=\sum_{n=1}^{\infty}\sum_{m=0}^{\infty}B_{nm}J_{0}(\mu_{n}r)\cos(\gamma_{m}z),$
(22) $\displaystyle
B_{nm}=\int\displaylimits_{0}^{1}\int\displaylimits_{0}^{1}rq_{vr}(r)q_{vz}(z)J_{0}(\mu_{n}r)\cos(\gamma_{m}z)\textrm{d}r\textrm{d}z.$
(23)

Substituting Eq. (22) into Eq. (14) (with $\partial_{t}T=0$), we can find the
relation between the known $B_{nm}$ and the unknown $C_{nm}$,

$\displaystyle
C_{nm}\left(-\frac{1}{r}\mu_{n}J_{1}(\mu_{n}r)-\frac{\mu_{n}^{2}}{2}\Big{(}J_{0}(\mu_{n}r)-J_{2}(\mu_{n}r)\Big{)}-J_{0}(\mu_{n}r)\gamma_{m}^{2}\right)\cos(\gamma_{m}z)=B_{nm}J_{0}(\mu_{n}r)\cos(\gamma_{m}z).$
(24)

Then Eq. (24) is multiplied with $rJ_{0}(\mu_{n}r)\cos(\gamma_{m}z)$ and
integrated from $0$ to $1$ with respect to both $r$ and $z$, following the
Galerkin procedure. The $z$-direction is straightforward as both sides are
multiplied with $\cos(\gamma_{m}z)$, the non-trivial part originates from the
$r$ direction, and results in

$\displaystyle C_{nm}=B_{nm}\frac{1}{\beta_{nm}^{2}},$ (25)

which holds for any internal heat generation $q_{v}(r,z)$. Due to the
separation $T(r,z,t)=T_{\textrm{st}}(r,z)+T_{h}(r,z,t)$, the initial condition
for the homogeneous part reads $T_{h}(r,z,t=0)=-T_{\textrm{st}}(r,z)$ as
$T(r,z,t=0)=0$, and thus $K_{nm}=-C_{nm}$.

###  3.2. Guyer-Krumhansl heat equation

Although we follow the same technique here, we also separate the homogeneous
and inhomogeneous parts, but it is more advantageous to determine the steady-
state heat flux field first. Since the steady temperature field
$T_{\textrm{st}}(r,z)$ is inherited, we can substitute
$\partial_{r}T_{\textrm{st}}(r,z)$ and $\partial_{z}T_{\textrm{st}}(r,z)$ into
Eqs. (10)-(11). Furthermore, we suppose that each term inherits the
corresponding set of eigenfunctions and eigenvalues as the boundary conditions
remain, and thus

$\displaystyle
q_{r}=\sum_{n=1}^{\infty}\sum_{m=0}^{\infty}D_{nm}J_{1}(\mu_{n}r)\cos(\gamma_{m}z),\quad
q_{z}=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}E_{nm}J_{0}(\mu_{n}r)\sin(\gamma_{m}z),$
(26)

respecting the corresponding derivatives, too. In that steady-state, we
consider that $\partial_{t}q_{r}=\partial_{t}q_{z}=0$ in Eqs. (10)-(11), and
after substituting Eq. (26) into Eqs. (10)-(11) and integrating, we obtain a
set of algebraic relations among the coefficients,

$\displaystyle c_{1}D_{nm}=\mu_{n}C_{nm}-c_{2}E_{nm},\quad
c_{3}E_{nm}=\gamma_{m}C_{nm}-c_{4}D_{nm},$ (27)

with

$\displaystyle
c_{1}=1+(\eta_{q}+\eta_{2})(2+\mu_{n}^{2})+\eta_{1}\gamma_{m}^{2},\quad
c_{2}=\eta_{2}\mu_{n}\gamma_{m},\quad
c_{3}=1+\gamma_{m}^{2}(\eta_{1}+\eta_{2})+\mu_{n}^{2}\eta_{1},$ $\displaystyle
c_{4}=2\gamma_{m}\eta_{2}\left(\frac{1}{\mu_{n}J_{1}(\mu_{n})^{2}}+\mu_{n}\right).$
(28)

Since $C_{nm}$ is known from Eq. (25), Eq. (27) can be solved for $D_{nm}$ and
$E_{nm}$, and it holds for any heat sources. Then the steady-state solution is
given by Eqs. (22) and (26).

The transient (homogeneous) solution is constructed similarly, however, the
coefficients are now time-dependent, viz., we assume that

$\displaystyle
T_{h}(r,z,t)=\sum_{n=1}^{\infty}\sum_{m=0}^{\infty}\tilde{C}_{nm}(t)J_{0}(\mu_{n}r)\cos(\gamma_{m}z),\quad
q_{r}(r,z,t)=\sum_{n=1}^{\infty}\sum_{m=0}^{\infty}\tilde{D}_{nm}(t)J_{1}(\mu_{n}r)\cos(\gamma_{m}z),$
(29) $\displaystyle
q_{z}(r,z,t)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\tilde{E}_{nm}(t)J_{0}(\mu_{n}r)\sin(\gamma_{m}z),$
(30)

are still valid. Furthermore, now we need to exploit the energy balance as
well to obtain $\tilde{C}_{nm}(t)$,

$\displaystyle\frac{\textrm{d}}{\textrm{d}t}\tilde{C}_{nm}(t)+\mu_{n}\tilde{D}_{nm}(t)+\gamma_{m}\tilde{E}_{nm}(t)=0.$
(31)

After following the same procedure, we obtain almost the same set of equations
except that the time derivative terms appear. Consequently, the set of PDEs is
reduced to a set of ODE,

$\displaystyle\frac{\textrm{d}}{\textrm{d}t}\begin{bmatrix}\tilde{C}_{nm}(t)\\\
\tilde{D}_{nm}(t)\\\
\tilde{E}_{nm}(t)\end{bmatrix}=\begin{bmatrix}0&-\mu_{n}&-\gamma_{m}\\\
\frac{\mu_{n}}{\tau}&-\frac{c_{1}}{\tau}&-\frac{c_{2}}{\tau}\\\
\frac{\gamma_{m}}{\tau}&-\frac{c_{4}}{\tau}&-\frac{c_{3}}{\tau}\end{bmatrix}\begin{bmatrix}\tilde{C}_{nm}(t)\\\
\tilde{D}_{nm}(t)\\\ \tilde{E}_{nm}(t)\end{bmatrix},$ (32)

in which the coefficients are inherited from Eq. (28), and its solution can
easily be found in the form of a matrix exponential, $\exp(\mathbf{M}_{nm}t)$.
Let us recall that the initial condition
$T_{h}(r,z,t=0)=-T_{\textrm{st}}(r,z)$ is the same, and therefore we can
exploit the known coefficients $\tilde{C}_{nm}(t=0)=-C_{nm}$,
$\tilde{D}_{nm}(t=0)=-D_{nm}$, and $\tilde{E}_{nm}(t=0)=-E_{nm}$.

###  3.3. Jeffreys heat equation

Here, the solution is simpler as the steady-state heat flux field is identical
to Fourier’s case, hence we do not need to compute it separately. We repeat
the ansatz of Eq. (29)-(30) to determine the coefficient matrix
$\mathbf{M}_{nm}$, and following the same steps, that procedure results in

$\displaystyle\frac{\textrm{d}}{\textrm{d}t}\begin{bmatrix}\tilde{C}_{nm}(t)\\\
\tilde{D}_{nm}(t)\\\
\tilde{E}_{nm}(t)\end{bmatrix}=\begin{bmatrix}0&-\mu_{n}&-\gamma_{m}\\\
\frac{\mu_{n}}{\tau}&-\frac{1+\tau_{T}\mu_{n}^{2}}{\tau}&-\frac{\tau_{T}\mu_{n}\gamma_{m}}{\tau}\\\
\frac{\gamma_{m}}{\tau}&-\frac{\tau_{T}\mu_{n}\gamma_{m}}{\tau}&-\frac{1+\tau_{T}\gamma_{m}^{2}}{\tau}\end{bmatrix}\begin{bmatrix}\tilde{C}_{nm}(t)\\\
\tilde{D}_{nm}(t)\\\ \tilde{E}_{nm}(t)\end{bmatrix},$ (33)

together with the known coefficients from Fourier’s solution, the time
evolution for the Jeffreys case is obtained in the form of
$\exp(\mathbf{M}_{nm}t)$.

##  4\. Steady-state distributions

First, let us begin with the steady-states, primarily focusing on the
differences between the Fourier and GK equations. From a theoretical point of
view, and according to Alvarez et al. [25], GK’s steady heat flux field can
differ from Fourier’s. Here we have to discover in what sense and in what
measure they can differ from each other. From a practical point of view, it is
possible to measure the average local heat flux, usually on a minimal area of
$10\times 10$ mm2 up to about $80\times 80$ mm2. Furthermore, it is known that
such sensors can significantly distort the local heat flux field [26, 27].
Consequently, if one aims to observe the traces of non-Fourier heat conduction
at this level, such spatial scales must be included in the preliminary
analysis as smooth flux distributions (such as for the temperature field)
cannot be measured. These analytical calculations can ease such analysis as
well. Since the JE model has the same steady-state as Fourier’s, we leave this
analysis aside for that heat equation.

Figure 4. Two-dimensional steady-state temperature distribution, using
$N=M=200$ terms, showing only partial spatial domain.

Fig. 4 presents the temperature distribution for Fourier’s heat equation,
which remains the same for the Guyer-Krumhansl and Jeffreys equations.
Concerning the heat flux field, the situation becomes quite different. It is
worth studying the outcome of the GK equation closer, see Fig, 5 for the
details about $q_{r}(r,z=0)$ and $q_{z}(r,z=0.05)$. That characteristics is
preserved for any $q_{r}(r,z=\textrm{const.})$ distributions. The influence of
$\eta_{1}$ is clear, and it can significantly decrease the maximum. Similarly
to $\eta_{1}$, $\eta_{2}$ has the same effect on the heat flux field, being
more influential on $q_{z}$, see Fig. 6 for a particular solution.

Although these effects are clear and strong for such a parameter interval, the
situation of observing them in a steady-state is not that hopeful. Let us
consider the flash experiments on rocks and their GK-evaluation with a one-
dimensional model [10], we find that $\eta_{1}+\eta_{2}\approx 10^{-7}$ m2, in
general. Consequently, the most substantial effects could be observed for a
body with $R=0.01$ m or less, which would probably violate our initial
assumptions, and the boundary conditions would not be valid as well. For
larger bodies, e.g., with $R=0.1$ m or even larger ($R=1$ m), the effect
becomes small and difficult to detect. This is one reason why this scaling
property is most important for nanoscale objects.

Figure 5. Steady-state $q_{r}(r,z=0)$ (left) and $q_{z}(r,z=0.05)$ (right)
distributions with $\eta_{2}=0$, using $N=M=200$ terms, showing only partial
spatial domain. Figure 6. Steady-state $q_{r}(r,z=0)$ (left) and
$q_{z}(r,z=0.05)$ (right) distributions with $\eta_{2}=10^{-4}$, using
$N=M=200$ terms, showing only partial spatial domain.

##  5\. Transient distributions

###  5.1. Fourier’s heat equation.

Let us recall that we are using the conventional Fourier number for the time
scale. The characteristic size could be $1$ m as relatively large bodies are
considered. Consequently, we must choose small Fourier numbers since those can
express relatively large time instants. Figure 7 shows the temperature
distribution for Fourier’s heat equation, in which we can observe that the
characteristics of the distribution establish quickly and does not change
significantly for larger time intervals. The color scaling, however, changes,
showing how the equilibrium is approximated. At farther away from the heat
source, the temperature changes more slowly since the gradients are much
smaller. This is also presented in Fig. 8, showing the surface temperature
history at different radii, as it is denoted previously in Fig. 2. We use Fig.
8 for comparative purposes in case of non-Fourier models as temperature maps
(such as Fig. 7) would not highlight the characteristics of the non-Fourier
behavior.

Figure 7. Transient temperature distribution following from Fourier’s heat
equation, using $N=M=40$ terms, showing only partial spatial domain. Figure 8.
Transient temperature history on the surface at different radii from Fourier’s
equation, using $N=M=100$ terms.

###  5.2. Guyer-Krumhansl heat equation.

Although the GK equation can reproduce the heat wave solutions following the
Cattaneo equation with $\eta_{1}=\eta_{2}=0$, called second sound, it is not
meaningful in our situation. First, in order to generate such a heat wave, the
time scale of the excitation (e.g., a heat pulse) should match the material’s
characteristic properties, most importantly, its relaxation time $\tau$. Since
its values range from $10^{-10}--1$ s depending on the propagation phenomenon
we investigate [28, 29, 30], such effects become irrelevant for large bodies
on much larger time scales. Second, we use the GK equation as an effective
approach to model the parallel diffusive mechanisms instead of modeling heat
waves (or anything else related to phonon hydrodynamics). Fig. 9 presents two
cases for the $\eta_{1}=\eta_{2}=0$ setting, with $\tau=10^{-3}$ (left) and
$\tau=10^{-2}$ (right). Similarly to the Fourier number, the relaxation time
has the same scaling, therefore even $\tau=10^{-3}$ is so large, it still does
not show any difference from Fourier’s solution, unlike the second case with
unrealistically large relaxation time, but the effect is still weak. This
setting will not be relevant for such continuous heating in a large macroscale
body.

In fact, it is not easy to find parameters resulting in a remarkably different
solution. Figure 10 presents an example in which the radius we use is much
closer to the heat source, for farther away and longer time intervals, the
differences vanish. It is worth studying how we can recover a similar behavior
observed in heat pulse experiments (see Fig. 1 for the experimental
characteristics). Interestingly, being closer to the heat source, GK’s
solution is not faster than Fourier’s, however, it changes with the radius.
Figure 11 presents these characteristics, and thus it is best to measure the
temperature farther from the heat source to more reliably observe the non-
Fourier behavior.

Figure 9. Transient temperature history on the surface, comparing the Fourier
and GK equations with $\eta_{1}=\eta_{2}=0$ and $\tau=10^{-3}$ (left),
$\tau=10^{-2}$ (right), using $N=M=100$ terms. Figure 10. Transient
temperature history on the surface, comparing the Fourier and GK equations
with $\eta_{2}=0$, using $N=M=100$ terms. Figure 11. Transient temperature
history on the surface at different radii, comparing the Fourier and GK
equations, using $N=M=100$ terms.

###  5.3. Jeffreys heat equation.

The $\tau_{T}=0$ subcase coincides with the previous analysis using the GK
equation (see the right side of Fig. 9), therefore we do not repeat it again.
Instead, we focus on studying the effects induced by the extra time derivative
term in the constitutive relation with $\tau_{T}\neq 0$ and comparing our
findings with Fig. 11. Although $\tau_{T}$ acts analogously on the temperature
field, their heat flux fields differ. That difference is essential for
superfluids and further low-temperature modeling problems [31]. In engineering
practice, however, that difference could be negligible as we seek only an
effective model to provide a more accurate description of heterogeneous
materials. An effective description cannot simultaneously model the
temperature and heat flux fields. In that sense, neither approach is more
accurate than the other. Both can reproduce the same temperature history,
however, the GK equation can be much more complicated. On the one hand, for
the JE model, as simply the time derivative of Fourier’s equation is added, it
is easier to utilize the usual approaches for initial and boundary conditions.
On the other hand, although thermodynamically compatible, the JE model does
not fit into the systematic generalization of non-Fourier equations.

Figure 12. Transient temperature history on the surface at different radii,
comparing the Fourier and JE equations, using $N=M=100$ terms.

##  6\. Discussion

There is a need for an advanced heat conduction model to overcome the
difficulties emerging together with the use of complex heterogeneous
materials. Their effective thermal behavior can significantly differ from
Fourier’s prediction, even though the classical heat equation governs each
component. The interaction of parallel heat transfer channels results in a
non-Fourier behavior. Two promising extensions of the Fourier equation are the
Guyer-Krumhansl and the Jeffreys models. Both describe two heat conduction
time scales, even in an isotropic setting. These models are analytically
solved for a two-dimensional situation.

The analytical solution revealed that there is no need for additional boundary
conditions for these GK and JE models, the same set of eigenfunctions can be
used. That property notably eases the solution of these more complex models.
The resulting ordinary differential equations can be solved easily for the
linear situation; thus, unknown time-dependent coefficients are found.
Furthermore, we also exploited that the steady-state temperature distribution
given by Fourier’s law remains the same in the non-Fourier case, therefore, it
is advantageous to handle that heat conduction problem as a superposition of
the transient and steady distributions. It also makes it more straightforward
how to take into account the initial conditions.

Studying the time evolution of the surface temperature time histories $T(t)$,
we observed similarities compared to the one-dimensional heat pulse
experiments, however, these situations are not directly comparable. First, as
the surface temperature is the most straightforward to measure, it seems more
advantageous if the thermometer is situated further from the heated region to
observe the over-diffusive behavior possibly. This can change significantly in
space. Second, with such constant heating, no steep transients occur in a
large body due to its large heat capacity, thus, it is more challenging to
observe the non-Fourier behavior. Based on the analytical solutions, it is
clear that both models require unusually large parameters to obtain any
observable deviation from Fourier’s law. In the future, it would be worth
investigating the outcome of periodic heating, especially the effects of the
period time, and that could enhance the over-diffusive phenomenon. Third, for
the GK equation, while the steady heat flux field differs from Fourier’s, it
is not practically measurable due to the steep change in the heated region,
the available heat flux sensors are too large for such an application.
Moreover, as the most significant difference occurs under the heater, it
excludes this possibility. However, the analytical calculations also revealed
that if such constant heating does not induce notable non-Fourier effects, one
could design a measurement method to characterize such objects with Fourier’s
law remaining valid thermally.

## Funding

Project no. TKP-6-6/PALY-2021 has been implemented with the support provided
by the Ministry of Culture and Innovation of Hungary from the National
Research, Development and Innovation Fund, financed under the TKP2021-NVA
funding scheme. The research was funded by the Sustainable Development and
Technologies National Programme of the Hungarian Academy of Sciences (FFT NP
FTA), by the grant National Research, Development and Innovation Office-NKFIH
FK 134277, and supported by the ÚNKP-22-5-BME-312 New National Excellence
Program of the Ministry for Culture and Innovation from the source of the
National Research, Development and Innovation Fund.

## Declarations

Conflict of interest The author declares no competing interests.

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