# Stochastic Gradient Descent Captures
How Children Learn About Physics

Luca M. Schulze Buschoff Eric Schulz Marcel Binz

MPRG Computational Principles of Intelligence
Max Planck Institute for Biological Cybernetics
Tübingen, Germany Corresponding author<EMAIL_ADDRESS>
###### Abstract

As children grow older, they develop an intuitive understanding of the
physical processes around them. They move along developmental trajectories,
which have been mapped out extensively in previous empirical research. We
investigate how children’s developmental trajectories compare to the learning
trajectories of artificial systems. Specifically, we examine the idea that
cognitive development results from some form of stochastic optimization
procedure. For this purpose, we train a modern generative neural network model
using stochastic gradient descent. We then use methods from the developmental
psychology literature to probe the physical understanding of this model at
different degrees of optimization. We find that the model’s learning
trajectory captures the developmental trajectories of children, thereby
providing support to the idea of development as stochastic optimization.

##  1 Introduction

More than 70 years ago, Turing (1950) famously suggested that instead of
trying to produce a programme to simulate the adult mind, why not rather try
to produce one which simulates the child’s? If this were then subjected to an
appropriate course of education one would obtain the adult brain. If we want
to take Turing’s proposal seriously, we have to ask ourselves: how do children
learn?

The physical laws of nature are one of the earliest things that children learn
(Spelke & Kinzler, 2007; Lake et al., 2017). Therefore, they can serve as an
ideal testbed for investigating this question. There has already been a
substantial amount of work across different research areas to understand and
reproduce the human ability for physical reasoning. On the one hand, empirical
work in developmental psychology has provided us with a precise understanding
of the different developmental stages that children undergo during their
cognitive development (Baillargeon, 1996, 2004). On the other hand, machine
learning researchers have started to successfully apply tools from deep
learning to build models that mimic the intuitive physical understanding of
people (Battaglia et al., 2013; Lerer et al., 2016; Zhang et al., 2016; Piloto
et al., 2022; Smith et al., 2019, 2020).

Even though there are a number of models for physical reasoning, they
typically focus on reproducing adult-level performance. The goal of the
present paper is to instead – in the spirit of Turing – compare the learning
trajectories of artificial systems to the developmental trajectories of
children. We are particularly interested in examining the idea of _development
as stochastic optimization_ , which states that cognitive development results
from some form of stochastic optimization procedure (Gopnik et al., 2017;
Ullman & Tenenbaum, 2020; Giron et al., 2022).

Figure 1: A: Developmental trajectory for support events outlined by
(Baillargeon, 1996). The illustrations are taken from Baillargeon (1996) and
they show the physical rules acquired at the respective ages. B: Illustration
of our generative video prediction model.

To test this hypothesis, we train a deep generative model on video sequences
on a physical reasoning task. We then probe the knowledge of this model at
different training epochs using violation-of-expectation methods (Baillargeon,
2004; Piloto et al., 2018; Smith et al., 2019) and compare it to the knowledge
of children at different ages. We find that the acquisition order of concepts
in this model aligns with that of children, thereby providing support to the
idea of development as stochastic optimization.

##  2 Methods

###  2.1 Support events

Infants’ physical reasoning abilities have been investigated in many different
domains. Here, we use support events (such as the configurations of block
stacks shown in Figure 1A) as an exemplary physical reasoning task for
comparing the learning trajectories of artificial systems to the developmental
trajectories of children.

Baillargeon (1996) has shown that, as infants grow older, they make use of
increasingly complex rules to decide whether a specific configuration of
blocks is stable or not (see also (Baillargeon, 2002, 2004)). With $3$ months,
infants decide based on a simple contact or no contact rule. According to this
rule, a block configuration is considered to be stable if the two blocks touch
each other. At around $5$ months, infants understand that the type of contact
matters. Now, only configurations with blocks stacked on top of each other are
judged as stable. At $6.5$ months, they additionally consider the overlap to
determine the stability of a block configuration. Finally, at $12.5$ months
they are able to incorporate the block shapes into their judgement, relying
not only on the amount of contact but also on how the mass is distributed for
each block.

To assess the learning trajectories of artificial systems, we generated a data
set containing $100.000$ video sequences of support events using the Unity
game engine (Unity Technologies, 2005). The video sequences show stacks of two
coloured blocks in a gray room and they consist of $20$ frames with a size of
$64\times 64$ pixels (see Figure 2). We randomly varied a number of properties
to ensure sufficient variability in the training data (see Appendix A.1 for
further details).

Figure 2: The first row shows the surprise for the expected and violated test
sequences of the overlap rule. The second row shows the expected test
sequence. The third row shows the violated test sequence. The last row shows
the open-loop predictions from the model, given the first two frames of the
violated test sequence. It is important to note that the first three frames
were removed for this plot, as the uncertainty is very high when the model is
first given the sequence.

###  2.2 Modeling developmental trajectories

We investigate the development as stochastic optimization hypothesis by
training a generative video prediction model using gradient descent. To obtain
a learning trajectory of this model, we evaluate a snapshot of it in every
epoch.

We use the recurrent state space model (RSSM) (Hafner et al., 2019; Saxena et
al., 2021) as an exemplary model for our analysis. The RSSM can be seen as a
sequential version of a variational autoencoder (VAE). It maintains a latent
state at each time step, which is comprised of a deterministic component
$h_{t}$ and a stochastic component $s_{t}$. These components depend on the
previous time steps through a function $f(h_{t-1},s_{t-1})$, which is
implemented as a gated recurrent neural network. It is trained by optimizing
the following evidence lower bound:

$\displaystyle-\sum_{t=1}^{T}\mathbb{E}_{q(s_{t}\mid o_{\leq t})}[\textrm{ln
}p(o_{t}\mid s_{t})]+\mathbb{E}_{q(s_{t-1}\mid o_{\leq
t-1})}\big{[}\textrm{KL}(q(s_{t}\mid o_{\leq t})\mid\mid p(s_{t}\mid
s_{t-1}))\big{]}$ (1)

After training, the RSSM can be used to generate open-loop predictions. For
this, the model processes a number of initial observations to infer an
approximate posterior $q(s_{t-1}\mid o_{\leq t-1})$. The subsequent open-loop
predictions then result by decoding latent representations sampled from the
prior $p(s_{t}\mid s_{t-1})$. We refer the reader to Appendix A.2 for further
details about the model architecture and training procedure.

###  2.3 Measuring surprise

In order to assess whether our model has understood a specific rule, we use
the violation of expectation paradigm (Piloto et al., 2018, 2022). The model
is presented with two video sequences: a violated sequence, which constitutes
a violation according to the rule, and an expected sequence, which is
consistent with the rule. If the model has successfully learned a specific
rule, it should show a larger degree of surprise for the violated compared to
the expected sequence. Following Piloto et al. (2018), we measure the model’s
surprise using the Kullback–Leibler (KL) divergence between the prior and
posterior over the latent representation (Baldi & Itti, 2010). This approach
closely resembles how developmental psychologists assess children’s
understanding of physical rules.

We constructed a violated and an expected test sequence with identical image
statistics for each of the four rules for support events outlined by
Baillargeon (1996). For example, according to the overlap rule, a block
configuration should only be stable if the blocks are stacked on top of each
other with enough overlap. The two test sequences for this rule therefore show
two blocks that only slightly overlap (see Figure 2). In the violated test
sequence, the block configuration nonetheless appears stable, with the top
block remaining on top of the bottom block. In contrast and consistent with
the rule, the expected test sequence shows the top block falling.

Figure 3: Linear regression fits for the difference between the surprise for
the expected and the violated sequence at every epoch and for each of the four
physical principles separately, as well as a 95% confidence interval around
the regression line. Additionally, the individual data points are binned in
bins of size 10 and displayed with a 95% confidence interval.

##  3 Results

Before investigating the learning trajectory of our model, we verified that it
is able to predict the given scenes accurately into the future. For this
purpose, we plotted the open-loop predictions of the fully trained model for
all test sequences. Figure 2 illustrates the result for the overlap test
sequences. We see that the predictions of the fully trained model closely
match the expected sequence, indicating that the model has learned the overlap
rule by the end of its training. This result also holds for the three other
physical rules, see Appendix A.3 for the corresponding plots.

Next, we validated that the fully trained model is surprised when it observes
a violation of a physical rule it has acquired. Figure 2 shows the KL
divergence over the frames for the overlap test sequences. We see that the KL
divergence for the violated sequence exceeds that for the expected sequence
from around the fifth frame, which is when the two sequences diverge.

To quantitatively assess whether the model has learned the four physical
rules, we computed the difference in KL divergence between the expected and
violated sequences summed up over all frames. Figure 3 shows this measure for
each of the four physical rules over the course of training. We find that the
difference in KL divergence is positive for all four physical rules in the
fully trained model, confirming that it has learned all of them by the end of
training.

Finally, we investigated whether the stochastic optimization hypothesis can
capture the developmental trajectories of children. For this purpose, we
fitted a simple linear regression for each of the rules, using the difference
in KL divergence as the dependent and training epochs as the independent
variable. This directly relates to the stochastic optimization hypothesis, as
the model becomes increasingly optimized over the epochs. The resulting
regression coefficients were $0.390\pm 0.024$, $0.261\pm 0.017$, $0.170\pm
0.011$, and $0.059\pm 0.007$ for the contact or no contact, type of contact,
overlap, and shape rules, respectively (all $p<0.001$). We can see that the
coefficients become smaller as the rules increase in complexity. The model
first acquires the contact or no contact, followed by the type of contact,
then the overlap, and finally the shape rule. This order of acquisition
matches how children acquire these rules, thereby providing support to the
idea of development as stochastic optimization.

##  4 Discussion

We have compared the learning trajectories of an artificial system to the
developmental trajectories of children in the domain of physical reasoning.
More specifically, we examined the idea of development as stochastic
optimization. For this purpose, we used the violation of expectation paradigm
to probe the knowledge of a modern deep generative neural network at different
stages of its training process. We found that the model’s learning trajectory
resembled the trajectories of children during their cognitive development.

In contrast to previous work in this domain (Binz & Endres, 2019; Giron et
al., 2022), our modeling approach employs high-dimensional visual stimuli
(i.e., video sequences) and solely relies on an unsupervised training
objective. It, therefore, more closely mirrors the actual learning processes
of children in the real world. Furthermore, the conducted experiments build on
a well-established paradigm from developmental psychology. Together, these
factors provide a high validity to our results.

Presently, the biggest mismatch between learning in our models and learning in
children is that the latter do not observe a large number of support events.
Instead, they simply witness the real world and generalize their acquired
knowledge to the given experimental setting. To capture this process, we
should ideally train our models in a similar way. This could, for example, be
accomplished by utilizing the SAYCam data set, which contains a large number
of longitudinal video recordings from infants’ perspectives (Sullivan et al.,
2021). Additionally, this data set includes time stamps indicating when a
child has encountered a particular scene, which could be used to investigate
how the nature of the training data influences development.

It also seems plausible that factors beyond stochastic optimization drive
human development. There is, for example, evidence suggesting that children
gain access to additional computational resources during their development,
allowing them to apply more complex strategies (Binz & Schulz, 2022). This
idea of _development as complexity increase_ could be readily incorporated in
our framework by replacing the evidence lower bound from Equation 2.2 with a
$\beta$-VAE objective (Higgins et al., 2016; Burgess et al., 2018).

To fully disentangle these hypotheses, a single paradigm will likely not be
sufficient. Instead, we will need to go beyond support events and investigate
developmental trajectories across a variety of different tasks. Findings that
hold across domains would further increase the reliability of our results and
have potential implications for cognitive psychology and artificial
intelligence alike.

## Acknowledgments and Disclosure of Funding

This work was funded by the Max Planck Society and the Volkswagen Foundation.

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##  Appendix A Appendix

###  A.1 Stimulus variations

The following variables were randomly varied in order to ensure sufficient
variability in the data set: lower block size, lower block color, upper block
color, lower block rotation, upper block rotation, upper block position
(offset) and camera angle. Additionally, the shape of the upper block was
varied: half of the trials featured a cube as an upper block, while the other
half featured an L-shaped block with randomly sampled side lengths.

###  A.2 Model implementation

The recurrent state space model (RSSM) can be seen as a sequential VAE (Hafner
et al., 2019; Saxena et al., 2021). The deterministic component $h_{t}$ and
stochastic component $s_{t}$ depend on the deterministic and stochastic
components at the previous time steps through $f(h_{t-1},s_{t-1})$, which is
implemented through a gated recurrent unit cell. As outlined by Hafner et al.
(2019), the individual components of the RSSM are:

Deterministic state model: $\displaystyle h_{t}$
$\displaystyle=f(h_{t-1},s_{t-1})$ Stochastic state model: $\displaystyle
s_{t}$ $\displaystyle\sim p(s_{t}\mid h_{t})$ Observation model:
$\displaystyle o_{t}$ $\displaystyle\sim p(o_{t}\mid h_{t},s_{t})$

The models were implemented in PyTorch (Paszke et al., 2019). For all models,
the size of the stochastic hidden dimension $s_{t}$ was kept at 20, while the
size of the deterministic hidden dimension $h_{t}$ was set to 200, as in
previous implementations of the RSSM (Hafner et al., 2019; Saxena et al.,
2021).

We used the encoder and decoder from Dittadi et al. (2020). The encoder
consists of three blocks. The first block consists of a convolutional layer
with a kernel of size 5 and a stride of 2 and a padding of 2, followed by a
leaky ReLU activation function, followed by two residual blocks. The second
block consists of a convolutional layer with a kernel of size 1 and a stride
of 1 and no padding, followed by average pooling with a kernel of size 2,
followed by two blocks residual blocks. The third block consists of average
pooling with a kernel of size 2, followed by two blocks residual blocks. The
fourth block consists of a convolutional layer with a kernel of size 1 and a
stride of 1 and no padding, followed by average pooling with a kernel of size
2, followed by two blocks residual blocks. The fifth block consists of average
pooling with a kernel of size 2, followed by two blocks residual blocks.

The decoder consists of five blocks. The first block consists of two residual
blocks, followed by upsampling with a scale factor of 2. The second block
consists of two residual blocks, followed by a deconvolutional layer with a
kernel size of 1 and a stride of 1, followed by upsampling with a scale factor
of 2. The third block again consists of two residual blocks, followed by
upsampling with a scale factor of 2. The fourth block consists of two residual
blocks, followed by a deconvolutional layer with a kernel size of 1 and a
stride of 1, followed by upsampling with a scale factor of 2. The fifth block
consists of two residual blocks, followed by upsampling with a scale factor of
2, a leaky ReLU activation funktion, followed by a deconvolutional layer with
a kernel size of 5 and a stride of 1 and a padding of 2.

The models were trained for 200 epochs using a batch size of 32. The stimulus
sets were randomly split into 99.000 training sequences and 1000 validation
sequences. The loss function was optimized using the Adam optimiser with a
learning rate of 0.001 (Kingma & Ba, 2014), which was divided by 10 every 50
epochs. The models were trained on a NVIDIA Quadro RTX 5000 for roughly 7
days. Our implementation of the RSSM borrows from a previous implementation on
GitHub. The complete code for this project, including our model
implementation, is available upon request.

###  A.3 Figures

Figure 4: The first row shows the surprise for the expected and violated test
sequences of the contact or no contact rule. The second row shows the expected
test sequence. The third row shows the violated test sequence. The last row
shows the open-loop predictions from the model, given the first two frames of
the violated test sequence. It is important to note that the first three
frames were removed for this plot, as the uncertainty is very high when the
model is first given the sequence. Figure 5: The first row shows the surprise
for the expected and violated test sequences of the type of contact rule. The
second row shows the expected test sequence. The third row shows the violated
test sequence. The last row shows the open-loop predictions from the model,
given the first two frames of the violated test sequence. It is important to
note that the first three frames were removed for this plot, as the
uncertainty is very high when the model is first given the sequence. Figure 6:
The first row shows the surprise for the expected and violated test sequences
of the shape rule. The second row shows the expected test sequence. The third
row shows the violated test sequence. The last row shows the open-loop
predictions from the model, given the first two frames of the violated test
sequence. It is important to note that the first three frames were removed for
this plot, as the uncertainty is very high when the model is first given the
sequence.


11institutetext: Center for Computational Engineering and Networking (CEN),
Amrita School of Engineering, Coimbatore,
Amrita Vishwa Vidyapeetham, India
11email: nskamal007<EMAIL_ADDRESS>
v_sowmya<EMAIL_ADDRESS>
22institutetext: RBG.AI, Resilience Business Grids LLP,
SREC Incubation Center, Coimbatore, Tamil Nadu, India
22email<EMAIL_ADDRESS>

# Geometry based Machining Feature Retrieval with Inductive Transfer Learning

N S Kamal 11 H B Barathi Ganesh 22 V V Sajith Variyar 11
V Sowmya 11 K P Soman 11

###### Abstract

Manufacturing industries have widely adopted the reuse of machine parts as a
method to reduce costs and as a sustainable manufacturing practice.
Identification of reusable features from the design of the parts and finding
their similar features from the database is an important part of this process.
In this project, with the help of fully convolutional geometric features, we
are able to extract and learn the high level semantic features from CAD models
with inductive transfer learning. The extracted features are then compared
with that of other CAD models from the database using Frobenius norm and
identical features are retrieved. Later we passed the extracted features to a
deep convolutional neural network with a spatial pyramid pooling layer and the
performance of the feature retrieval increased significantly. It was evident
from the results that the model could effectively capture the geometrical
elements from machining features.

###### Keywords:

Machining Feature RecognitionInductive LearningCADGeometrical FeaturesSpatial
Pyramid Pooling

##  1 Introduction

Computer-aided process planning (CAPP) helps determine the processing steps
required to manufacture a product or its parts. It serves as the connecting
link between Computer Aided Design (CAD) and Computer Aided Manufacturing
(CAM). Automated machining feature recognition is a critical component in the
detection of manufacturing information from CAD models. When it comes to
machine parts, features are semantically higher level geometric elements such
as a hole, passage, slot etc. Feature recognition is the process of
identifying these features from an image or a 3D model of these machine parts.

Approaching Machining Feature Recognition (MFR) as a supervised problem is not
feasible for real-time applications in industries. This is because, different
industries have their own data lakes to store the huge pool of 3D CAD models
that include different combinations with respect to their requirements.
Preparing a supervised corpus and training them to predict large number of
classes is a complex and tedious process to achieve. By observing this, we
propose an approach that includes inductive transfer learning for geometrical
feature extraction[1]. CAD models of machining features[2] were converted to
point cloud data and Minkowski engine[19] was used to handle sparse
convolutions. Frobenius norm was used for measuring similarity between the
extracted geometrical features and later a spatial pyramid pooling layer was
added for making the feature matrices to span the same number of spaces. We
have detailed the relevant work associated with machining feature retrieval in
section 2, the proposed approach is given in section 3 and observed results
with the experiments conducted are given in section 4.

##  2 Literature Review

The attempts at integrating computer-aided process planning (CAPP) with CAD
began decades ago[3]. Several approaches to development of feature recognition
techniques followed [4,5]. Major approaches being graph-based methods[11,15],
hint based methods[12], convex decomposition, volume decomposition[13] and
free form features[14]. One common problem addressed by all these approaches
was an increase in computational complexity. Introduction of Neural Networks
revolutionized feature recognition as it was excellent at recognizing patterns
and was able to tolerate noise in input data.

Some of the most influential recent works on 3D data representation and
feature recognition are as 3D ShapeNets[6], VoxNet[7] and PointNet [8].
However, the most recent and important work on machining feature recognition
is FeatureNet [2]. The authors have used Deep 3D Convolutional Neural Networks
and it could recognize manufacturing features from the low-level geometric
data such as voxels with a very high accuracy. To train FeatureNet, a large-
scale dataset of 3D CAD models with labeled machining features was constructed
by the authors. They achieved a feature recognition accuracy of 96.7 % for the
recognition of single features.

The recent break-through in geometric feature extraction was using Fully
Convolutional Geometric Features by C. Choy et al[1]. The authors present a
method to extract geometric features computed in a single pass by a 3D fully-
convolutional network. The predictions have a one-to-one correspondence with
input image in spatial dimension. The authors have used a UNet structure with
skip connections and residual blocks to extract sparse fully-convolutional
features. They used the Minkowski Engine to handle sparse convolutions.

Majority of the existing works on structuring 3D models of machining features
are based on low-level geometric characteristics, which makes the model
localized and domain specific. Extracting high level features will result in a
generalized model that can be applied in various domains. Utilizing fully
convolutional neural networks for feature extraction is proven to be
significantly faster than conventional methods[1].

##  3 Materials and Methods

Figure 1: The proposed method for feature extraction and retrieval

For benchmarking purpose we have taken the synthetic data that includes 24
machining features[2]. Geometrical features were extracted from the CAD models
using an inductive transfer learning technique using a model pre-trained with
fully convolutional geometric features for the purpose of image
registration[1]. Point cloud registration is the underlying source task of
this process. The number of extracted geometrical feature vectors from this
process was varying with respect to the CAD models. In order to get the same
number of feature spaces,later an SPP layer was introduced as the target task
of inductive transfer learning process. Frobenius norm was computed to measure
the similarity between the CAD models. Based on the obtained similarity score,
3D models were assigned to the respective classes. The overall proposed
approach is given in Figure 1.

###  3.1 3D CAD Models

The Dataset used for feature retrieval is adopted from the synthetic dataset
generated for FeatureNet [2]. It consists of 24 commonly found machining
features in the manufacturing industry (e.g., O-ring, Rectangular passage).
Each of the 24 classes has 1000 3D CAD models of varying dimensions and
orientations. The features are present in the CAD models as portions removed
from a solid cube of dimensions 10cm x 10 cm x10cm (fig:1). The 3D models are
available in stereolithography(STL) format, which is a widely used format in
the field of computer-aided manufacturing. STL files describe only the surface
geometry of a three-dimensional object without any representation of color or
texture.

###  3.2 Methodology

####  3.2.1 Feature extraction with Inductive Transfer Learning

The 3D files in STL format were converted into point cloud data with the help
of Open3D library[16]. Point cloud data was given as the input to the neural
network with ResUNet architecture pre-trained on 3D Match dataset which
extracts the high level geometric features [1].The extracted feature vectors
were observed to be 32 Dimensional. The number of feature vectors extracted
were found to be different, even for 3D models of the same family. The
features were projected to three dimensions with t-Distributed Stochastic
Neighbor Embedding(t-SNE)[17] for visualization purposes. A sample 3D model of
a through hole with its extracted features laid over it (See Fig: 2).

(a)

(b)

Figure 2: (a)A Sample 3D model from class ’through hole’, (b) Extracted
feature vectors projected to 3D space and laid over the model

####  3.2.2 Frobenius Norm

The extracted feature vectors supposedly should capture the geometrical
elements of the machining features. To test this, the extracted features
needed to be compared with a similarity measure. For this purpose all the
24000 models were paired with random samples from each class and the Frobenius
norm of their feature matrices were found. The Frobenius Norm of a matrix ‘A’
of dimension m x n is defined as the square root of the sum of the squares of
the elements of the matrix.

$\left\|{A}\right\|_{F}=\sqrt{\sum_{i=1}^{m}\sum_{j=1}^{n}|a_{ij}|^{2}}$ (1)

The difference in frobenius norm obtained for each pair was sorted in
descending order along with their corresponding labels. The position of the
true label on this list is found out and then the accuracy is calculated. The
accuracy and top-five accuracy obtained were 39% and 86% respectively. The
Top-five accuracy should be satisfactory for retrieval applications, since a
set of similar features are returned. From the result it was evident that the
FCGF could capture some geometric information from the 3D models. But the
accuracy was not satisfactory enough for real world application.

The reason behind the low performance was later found not to be the extracted
geometrical features but the inability of Frobenius norm to serve as a good
similarity measure in this particular case. The machining features varied
significantly in size and orientation even within the same family. Frobenius
norm could only capture the energy of the feature matrix which was not good
enough for comparison.

####  3.2.3 Spatial Pyramid Pooling Layer

The next target was to improve the results by finding out a better method to
capture the similarity between feature matrices of different shapes. A deep
neural network with a Spatial pyramid pooling(SPP) layer was used to solve
this issue[9,18]. SPP maintains spatial information in local spatial bins of
fixed size and removes the fixed-size constraint of the network by giving an
output of a fixed dimension irrespective of input size. The dataset of 24000
models belonging to 24 classes, was split into three subsets – training,
validation,testing at 70%, 15%, and 15%, respectively. A CNN architecture with
97,272 learnable parameters was used to learn the extracted features and in
the pre-final layer, Spatial pyramid pooling was introduced as shown in (fig
:3). The Frobenius norm is equivalent to the Euclidean norm generalised to
matrices instead of vectors. So the Euclidean distance between the output
vectors from SPP layer was taken as the measure of similarity between
features. Adam optimizer and categorical cross entropy loss was used for
training the model for a mere 30 epochs and the loss and accuracy is shown in
the graphs (See Fig :4).

Figure 3: The architecture with SPP layer used for learning extracted features

##  4 Results and Discussion

The inclusion of SPP layer imporved the performance of feature retrieval.The
testing accuracy obtained from the current model was 86% and the top-5
accuracy was 95 % for 30 epochs. Pyramid pooling is robust to object
deformations and is suitable for data of higher dimensions. Hence it was able
to handle machining features with varying physical dimensions and it proved to
be a better method than taking matrix norm of the extracted features directly.
As the dimensionality of the data increases, the less useful Euclidean
distance becomes. This might be one of the reasons why it didn’t work directly
on the feature matrices but worked well on the lower dimensional output from
the SPP layer.

(a)

(b)

Figure 4: (a).Training and Validation Loss Curves for 30 epochs, (b). Training
and Validation Accuracy for 30 epochs

One major application of this work in real life would be the retrieval of CAD
models from the database identical to an input CAD model given by the user. To
test the feasibility of this application, some CAD models belonging to
families of machining features with different geometrical properties were
taken from the test set and we performed a retrieval of the most similar
features. The search returned other features from the same family first and
then returned features from other geometrically similar families. The
categories of the top features returned by sample files from four families of
distinct geometrical properties are listed in 1. Family 1 is the most
identical one to the family of test feature and family 2 is the second most
identical one.

The models retrieved were rendered and inspected for geometric similarities.
Searching with circle based features returned CAD models from other circle
based families(e.g., O-ring returned circular end pocket and blind hole). The
same was the case with rectangular,triangular and 6 sided features. From the
output, it was clear that the extracted feature vectors could successfully
capture the geometric properties of the machining features. It became evident
that the similarity in size or orientation was preceded by the similarity in
geometrical features. Each CAD model in the dataset was given a numerical
identification number for convenience. The indices of the top retrieved
features along with the corresponding Euclidean distances for a test file from
the family ’circular end blind slot’ is shown in Table 2. We can observe that
all the top 5 features belong to the same family as the test file in this
case. The test file and the top retrieved CAD model are rendered in Figure 5.
We can observe that the two features are of different size but the network
could still capture the similarity between them.

Table 1: Sample files from four families chosen for testing and the top 2 families retrieved from database  Family of Test file | Family 1 | Family 2
---|---|---
O-ring | Circular end pocket | Blind hole
Rectangular passage | Rectangular pocket | Rectangular blind step
Triangular pocket | Triangular passage | rectangular blind step
6 Sides passage | 6 Sides pocket | Circular blind step
Table 2: Top-5 models retrieved for a sample 3D model of the family circular end blind spot (ID: 1990) No. | Model ID | Family | Euclidean Distance
---|---|---|---
1 | 1214 | Circular end blind spot | 8.17
2 | 1561 | Circular end blind spot | 8.28
3 | 1326 | Circular end blind spot | 8.49
4 | 1853 | Circular end blind spot | 8.89
5 | 1726 | Circular end blind spot | 8.94

(a)

(b)

Figure 5: [a].Test model - ID: 1990, Family: Circular end blind slot, [b].
Result no.1 - ID: 1214, Family: Circular end blind slot

##  5 Conclusion and Future work

The results indicate that the inductive transfer learning model performed with
fully convolutional neural networks, could capture and learn the geometric
similarity between 3D models of machining features. Even though the network
was pre-trained on 3D image data, it performed well on the 3D CAD model data.
Spatial pyramid pooling proved to be very efficient in handling feature
vectors of varying sizes. The model was good enough to perform feature
retrieval with 95 % top-5 accuracy and the retrieval of CAD files from
databases proved to be successful enough for practical applications. This work
could be extended for recognition and retrieval of multiple features present
in a single CAD model. This work could also be extended to a generalised shape
search application.

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# Bright Purcell enhanced single-photon source in the telecom O-band based on
a quantum dot in a circular Bragg grating

Sascha Kolatschek<EMAIL_ADDRESS>Cornelius Nawrath
Stephanie Bauer  Jiasheng Huang  Julius Fischer  Robert Sittig  Michael Jetter
Simone L. Portalupi  Peter Michler  [

###### Abstract

The combination of semiconductor quantum dots (QDs) with photonic cavities is
a promising way to realize non-classical light sources with state-of-the-art
performances in terms of brightness, indistinguishability and repetition rate.
In the present work we demonstrate the coupling of an InGaAs/GaAs QDs emitting
in the telecom O-band to a circular Bragg grating cavity. We demonstrate a
broadband geometric extraction efficiency enhancement by investigating two
emission lines under above-band excitation, inside and detuned from the cavity
mode, respectively. In the first case, a Purcell enhancement of 4 is attained.
For the latter case, an end-to-end brightness of 1.4 % with a brightness at
the first lens of 23 % is achieved. Using p-shell pumping, a combination of
high count rate with pure single-photon emission (g(2)(0) = 0.01 in
saturation) is achieved. Finally a good single-photon purity (g(2)(0) = 0.13)
together with a high detector count rate of 191 kcps is demonstrated for a
temperature of up to 77 K.

###### keywords:

quantum dot, single-photon source, telecom O-band, Purcell

These authors contributed equally to this work These authors contributed
equally to this work These authors contributed equally to this work Stuttgart
University] Institut für Halbleiteroptik und Funktionelle Grenzflächen (IHFG),
Center for Integrated Quantum Science and Technology (IQST) and SCoPE,
University of Stuttgart, Allmandring 3, 70569 Stuttgart, Germany

In the emerging fields of quantum communication and quantum cryptography one
main requirement is the use of bright non-classical light sources. In recent
years semiconductor quantum dots (QDs) have become of great significance in
the field of quantum optics 1, due to their outstanding photon emission
properties in terms of single-photon emission 2, indistinguishability 3 and
(pair) entanglement fidelity 4. Although state-of-the-art performances have
been achieved in the near-infrared region 5, 6, 7, 8, 9, 10, the versatility
of QDs also allows for single-photon emission in the telecom O-band and C-band
both in the InAs/InP 11, 12, 13, 14 and InGaAs/GaAs material system 15, 16,
17, 18. Especially in long-distance applications employing optical fibers,
sources in the telecom wavelength regime are desirable due to both, low photon
absorption and wave-packet dispersion. Furthermore, using the existing fiber
network, single-photons in the telecom O-band are favorable not only because
of zero wave-packet dispersion but also for wavelength division multiplexing
with classical communication signals in the telecom C-band 19. One of the main
challenges for QD single-photon sources is the high refractive index contrast
between the surrounding semiconductor material and the air. In the near-
infrared, numerous approaches with cavities like micropillars 6, 20 and
photonic crystal cavities 21, 22 have been realized to enhance the extraction
efficiency next to geometric approaches such as microlenses 23, 24, photonic
trumpets 25 and nanowires 26.

Recently, circular Bragg gratings 27, 28 have been given particular attention
due to a new proposed flip-chip layout 29, avoiding a fragile free standing
membrane 30. This allowed for the demonstration of entangled photon pair
emission with high brightness and indistinguishability for GaAs QDs emitting
around 780 nm 31 as well as In(Ga)As QDs at 900 nm 32.

However, for the telecom regime, most of the results have been demonstrated in
the InAs/InP material system 33, 34, 35, 36, 37, 38 with approaches in the
InGaAs/GaAs material system being limited to etched microlenses and mesas 39,
40, dielectric antennas 41 and micropillars 42.

The feasibility of circular Bragg gratings in the telecom regime was suggested
with simulations 43 as a promising approach, however, the realization of such
a structure was still outstanding.

Here, we investigate emission from self-assembled InGaAs/GaAs QDs in a
circular Bragg grating operating in the telecom O-band (Figure 1). Coupling
between emitter and cavity is achieved, yielding a Purcell enhancement by a
factor of around 4. Furthermore, bright single-photon emission is shown for
transitions under above-band and p-shell pumping. Finally, investigations
confirm the preservation of bright single-photon emission for elevated
temperatures up to 77 K.

Figure 1: (a) SEM top view of a processed circular Bragg grating. (b)
Schematic cross section of the cavity with an exemplary self-assembled QD. (c)
Top: Simulated Purcell factor and collection efficiency into NA=0.6. Bottom:
Cavity mode measured in $\upmu$-PL under strong cw above-band pumping. Figure
2: (a) Normalized $\mu$-PL spectra under above-band pumping (cavity position
as shaded area) with the investigated emission line marked by the black arrow.
(b) Pulsed g(2)($\tau$) measurement integrated over 5 minutes and displayed
with a binning of 50 ps. (c) Evaluation of the g(2)(0) and count rate for
different binning widths w. The binning width w is hereby defined by the width
symmetrically around the peak at each repetition were the counts are taken
into account (inset).

For optimal light extraction, a flip-chip process together with a gold
backside mirror is chosen as proposed by Yao and co-authors 29. In contrast to
previous realizations of this cavity at lower wavelengths 31, 32, here a 320
nm thick Al2O3 is used as the spacer layer between the GaAs membrane and the
gold mirror (Figure 1b). Thereby, the surface quality increased, minimizing
scattering of the light at defects.

To optimize the device design, finite-difference time-domain (FDTD)
simulations are performed. With the investigated cavity, a theoretical Purcell
enhancement of 18.6 combined with a Q-factor of around 200 can be expected
(Figure 1c top). Unique for this cavity design is the broadband collection
with an efficiency of over 80 % into an NA of 0.6 for a wavelength range of
more than 60 nm. Furthermore misplacing the emitter from the center by up to
400 nm has no effect on the high collection efficiency, however, strongly
reduces the Purcell factor (not shown). Hence, this cavity structure exhibits
high tolerance to spatial and spectral mismatch while maintaining bright
single-photon emission. The measured cavity mode under strong continuous-wave
(cw) above-band pumping shown in the bottom of Figure 1c and the simulated
cavity mode are in good agreement. The strong pumping allows for a uniform
feeding of the cavity mode via background emission. Due to a small ellipticity
of the structure, the fundamental cavity mode exhibits a polarization
splitting of 1 nm. Small fabrication imperfections, e.g. not perfectly
vertical sidewalls, lead to a measured Q-factor of around 130 in comparison to
a simulated Q-factor under ideal conditions of 200. Taking the measured
Q-factor into account, the maximally achievable Purcell factor for perfect
spatial and spectral overlap is reduced to 12.

For the optical characterization of a non-deterministically placed cavity with
at least 2 QDs in the central disk, a confocal micro-photoluminescence
($\upmu$-PL) setup is used with the sample mounted inside a helium flow
cryostat and cooled down to 4 K. To excite the QD under non-resonant
excitation above the bandgap of the barrier material (above-band excitation),
a 775 nm pulsed laser with a repetition rate of 76 MHz and a pulse width of 3
ps is used. A bright excitonic line can be seen at 1293 nm, around 9 nm blue-
shifted with respect to the cavity mode (Figure 2a).

To investigate the single-photon purity of the emission, the excitonic line is
spectrally filtered (full width at half maximum (FWHM) = 0.04 nm) and send to
a fiber-based Hanbury-Brown and Twiss setup reaching a total count rate of up
to 1.3 Mcps. A second-order autocorrelation is integrated for 5 minutes and
shown in Figure 2b. At zero time delay, a clear antibunching dip can be
observed. In order to evaluate the g(2)(0), a binning width as large as the
repetition is applied to account for all the measured counts on the detector.
Comparing the area around zero time delay to the mean area of the outer peaks
(for this measurement, no blinking of the emission line was observed) yields a
g(2)(0) = 0.262 $\pm$ 0.001. Due to a relatively long decay time, an overlap
between the neighboring peaks can be observed. The small peaks in between the
laser repetition are a measurement artifact due to electronic reflections in
the setup. Correcting the measured count rate for the non-zero g(2)(0) results
in a true single-photon count rate of 1.1 Mcps on the detectors, corresponding
to an end-to-end brightness of 1.4 % (the correction is analogues to Ref.38).
Considering the setup efficiency of 6.4 %, a brightness at the first lens of
23 % is achieved. Here, it is worth noting that due to the applied above-band
pumping scheme multiple transitions are excited. Therefore the stated
brightness gives only a lower bound as no competing transitions e.g. trions of
the same QD, are taken into account 44. To assess the performance of the
cavity for time-gated applications, different binning widths w are applied for
the g(2)(0) evaluation (Figure 2c). The count rate is hereby directly
proportional to the square root of the mean area under the outer peaks
integrated over the binning width while g(2)(0) shows a linear dependence with
the binning width. Therefore by time-gating, an improved g(2)(0) value can be
achieved with only a minor decrease in the count rate, e.g. an increase of the
single-photon purity by a factor of two can be gained at the cost of only 15 %
in the count rate.

Due to the layout of the cavity, the central disk is located on an Al2O3
layer, completely isolated from the surrounding GaAs. As a consequence, this
may lead to a more sensitive charge carrier environment, e.g. after strongly
pumping the cavity mode a change in the relative probability of different QD
transitions due to changes in the charge dynamics and surface states can be
observed under above-band pumping. In between the measurements, the sample was
stored for some days outside the cryostat under a nitrogen atmosphere. This
resulted in a substantial permanent change of the emission probabilities.

Figure 3: (a) Normalized $\mu$-PL spectra under above-band pumping (cavity
position as shaded area) with the investigated emission line marked by the
black arrow. (b) Decay time measurements of the emission line inside the
cavity and an exemplary reference line in the membrane outside the cavity. (c)
Measured count rate (black) as a function of the excitation power and
corrected for the non-perfect g(2)(0) values (red) as well as the
corresponding g(2)(0) values (blue squares). The filled squares denote
exemplary g(2)($\tau$) for saturation power (d) and low power (e).

While the excitonic line shown in Figure 2a required stronger pumping than
before, a second excitonic emission line from a different QD almost resonant
with the cavity mode emerged for low pumping powers (Figure 3a). For this
line, only a small spectral mismatch with regard to the cavity mode can be
seen. A conclusive way to probe the spatial overlap between the QD and the
cavity mode is by measuring the Purcell enhancement of the transition. The
decay time measurement of the considered emission line together with an
exemplary reference line is shown in Figure 3b. With respect to measurements
of decay times from multiple QDs outside the cavity
($\tau_{mean}=1209.4\,\mathrm{ps}$ with a standard deviation of
$226.3\,\mathrm{ps}$) a strongly reduced decay time of
$\tau_{Purcell}=300.8\pm 0.8\,\mathrm{ps}$ is observed. By comparing the decay
times, a Purcell enhancement of 4.0 $\pm$ 0.7 is deduced. The deviation from
the optimal theoretical Purcell factor of 12 is attributed mainly to the non-
perfect spatial overlap between emitter and cavity mode due to the non-
deterministic approach. The measured Purcell enhancement not only improves the
photon indistinguishability but could in the future also allow for higher
repetition rates 45. In order to evaluate the count rate of the emission line
a power series is performed (Figure 3c black circles). However, no clear
saturation behavior is observed. Therefore, in a next step power dependent
second-order correlation measurements are carried out (Figure 3c blue
squares). For fitting, the central cw-dip as well as blinking are taken into
account in the form of an envelope function 46, 47. The good agreement with
the measured data given by the fitting model is verified for high and low
excitation powers as shown in Figure 3d and 3e respectively. In saturation,
the evaluated g(2)(0) = 0.70 $\pm$ 0.07 shows that a strong multi-photon
probability is present. At high excitation power, a strong refilling of the
QD, presumably from nearby charge carrier traps can be observed (Figure 3d).
However, the cw g(2)(0) value can be estimated by the depth of the central dip
yielding g${}^{(2)}_{cw}$(0) = 0.10 $\pm$ 0.04. This shows that the measured
counts stem from one emitter and the g(2)(0) is mainly deteriorated by the
strong refilling. To deduce the true single-photon count rate on the detector,
the measured count rate is corrected for the non-perfect g(2)(0) yielding at
saturation a count rate of 550 kcps.

In order to avoid refilling, a quasi-resonant excitation scheme via the
discrete energy state (p-shell) is applied 16, 38. Around 1200 nm, p-shell
resonances of multiple transitions can be found. In the following, an
exemplary line with its p-shell at 1207 nm is investigated. This allows for an
easy filtering of the excitation laser with off-the-shelf 1250 nm long-pass
filters. With the excitation laser at 1207 nm, this bright emission line,
again blue-shifted with respect to the cavity mode, is observed (Figure 4(a)).

Figure 4: (a) Normalized $\mu$-PL spectra under p-shell pumping (cavity
position as shaded area) with the investigated emission line marked by the
black arrow. (b) Measured count rate on the detector (black) and power-
dependent g(2)(0) values (blue squared). The filled square denotes the value
g(2)(0) = 0.0115 $\pm$ 0.0002, obtained for saturation (c) g(2)($\tau$) for
saturation power.

In contrast to the previous two investigated emission lines, here no fine
structure splitting is observed hinting strongly towards the presence of a
charged transition. To assess the brightness and purity of this emission line,
a power series in combination with power dependent second-order correlation
measurements is performed. In saturation a count rate of 355 kcps is detected
(Figure 4b). The fitted g(2)(0) values are mostly power independent and range
between 0.01 and 0.02 with a value of 0.012 under saturation power. The small
increase for low excitation powers can probably be attributed to a slightly
different background behavior for high and low excitation powers. Further
measurements on different excitonic as well as charged transitions show
similar g(2)(0) values. The excellent g(2)(0) values prove that indeed by
adapting the pumping scheme, a strong suppression of the refilling mechanism
can be achieved. This is also reflected in the fitting function, where the
amplitude of the secondary decay is strongly reduced with respect to the
above-band excitation case.

Furthermore, a strong blinking of the emitter on a timescale of around 100 ns
is observed as seen in Figure 4c which is taken into account when normalizing
the g(2)($\tau$) measurements. The resulting long off-time of the transition
significantly limits the maximal attainable count rate to around 25 % in
comparison to the case without blinking.

Final measurements show the relevance of the described structure in real-world
implementations. Indeed, for practical applications, it is advantageous to
achieve bright single-photon emission at elevated temperatures making the
source compatible with cryocoolers or liquid nitrogen. Therefore the
temperature is increased to 40 K and 77 K respectively and the charged
transition seen in Figure 4a is investigated under saturation power in terms
of count rate and single-photon purity. As higher temperatures lead to a
phonon broadening of the emission line, the spectral selection window is
increased to FWHM = 0.1 nm to avoid spectral filtering with a width smaller
than the zero-phonon line 48. To verify that this is not the case for the
approach for the previous measurements at 4 K with the smaller spectral
selection window of 0.04 nm, the corresponding measurement are repeated using
the larger spectral selection window. The absence of a significant degradation
regarding the detected count rate as well as g(2)(0) confirms that at 4 K the
FWHM of the emission line is much smaller than the spectral selection window.

Figure 5: (a) Normalized $\mu$-PL spectra under p-shell pumping at (a) 40 K
and (c) 77 K with the investigated emission line marked by the black arrow.
g(2)($\tau$) in saturation power at (b) 40 K with g(2)(0) = 0.026 $\pm$ 0.001
and (d) 77 K with g(2)(0) = 0.127 $\pm$ 0.004.

The spectrum obtained at 40 K is shown in Figure 5a. An expected red-shift of
the emission line of around 3 nm is observed 48. With a combined count rate of
262 kcps on the detectors, a g(2)(0) of 0.026 $\pm$ 0.001 is obtained (Figure
5b). Further increasing the temperature to 77 K results in an additional red-
shift and line broadening (Figure 5c). Here a decrease of the count rate in
saturation to 191 kcps on the detectors is observed, which corresponds to
around 50 % of the count rate at 4 K. Despite the relatively high temperature,
a g(2)(0) of 0.127 $\pm$ 0.004 is measured. This value, clearly below 0.5,
proves the good single-photon purity of the investigated emission line inside
the circular Bragg grating even at liquid-nitrogen temperature. Furthermore,
the reported g(2)(0) values shows, that almost no deterioration of the single-
photon purity is present up to 40 K. The small deviation at 77 K can most
probably be attributed to background emission from neighboring transitions as
suggested in Figure 5c, spectrally overlapping with the investigated
transition. Comparing Figure 5b and 5d, a decrease of the FWHM of the peaks
with increasing temperature is seen. This points towards an increase in non-
radiative charge carrier loss channels which results in a decrease of the
count rate as observed for higher temperatures.

In conclusion, we realized a circular Bragg grating in the telecom O-band with
bright transition lines exhibiting single-photon emission, both spectrally
resonant and blue-shifted with respect to the cavity mode. Using above-band
excitation a true single-photon count rate with an end-to-end brightness of
1.4 % and first-lens brightness of 23 % was achieved in combination with a
g(2)(0) of 0.262 $\pm$ 0.001, clearly below 0.5. A decay time reduction of 4.0
$\pm$ 0.7 due to the Purcell effect was observed for a bright line spectrally
resonant with the cavity mode. This will in the future allow for higher
repetition rates further increasing the amount of usable photons. However,
despite a pronounced central dip, strong refilling prevented even better
g(2)(0) values. Therefore a p-shell pumping scheme was applied. This allowed
for a strong reduction of the refilling processes resulting in saturation in a
g(2)(0) of 0.012 combined with a count rate of 355 kcps on the detector. In
the future, by applying electric fields to stabilize the charge environment,
the observed blinking could be reduced, enabling even higher count rates.
Finally, single-photon emission with strong multi-photon suppression was shown
up to 77 K (g(2)(0) = 0.127). The capability of emitting on-demand single
photons even at elevated temperature, i.e. 40K and 77K, without the need for
complex schemes to filter the excitation laser light is very appealing for
future technological implementations. Indeed, the achieved results prove that
such kind of telecom QD-cavity structures can be operated in combination with
compact cryocoolers (which typically can reach temperatures around 40K) or
even in nitrogen based processes. This would allow for a drastic reduction of
the overall setup footprint and power consumption making the realized sources
appealing for real-world implementations as quantum information and satellite
based quantum cryptography.

The authors gratefully acknowledge the funding by the German Federal Ministry
of Education and Research (BMBF) via the project Q.Link.X (No. 16KIS0862) and
the European Union’s Horizon 2020 research and innovation program under Grant
Agreement No. 899814 (Qurope). The work reported in this paper was partially
funded by Project No. EMPIR 17FUN06 SIQUST and Project No. EMPIR 20FUN05
SEQUME. This project has received funding from the EMPIR programme co-financed
by the Participating States and from the European Union’s Horizon 2020
research and innovation program.

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