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Question: What is the future direction mentioned in the conclusion?

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Context 1: Paper Info

Title: Is In-hospital Meta-information Useful for Abstractive Discharge Summary Generation?
Publish Date: 10 Mar 2023
Author List: Mamoru Komachi (from Tokyo Metropolitan University), Takashi Okumura (from Kitami Institute of Technology), Hiromasa Horiguchi (from National Hospital Organization), Yuji Matsumoto

Figure

Fig  1 Example of part of a discharge summary which is a dummy we created 
Fig  2  Overview of our proposed method A new feature embedding layer encoding hospital, physician, disease, and length of stay is added to the standard transformer architecture The figure shows an example of hospital embedding 
Statistics of our data for experiment 
of summarization models with different meta-information The best results are highlighted in bold Each score is the average of three models with different seeds The BS and BR indicate BERTScore and BLEURT, respectively 
Statistics on the number of cases handled by physicians C/P denotes Cases/Physician, which indicates how many cases an individual physician has Method of Grouping Physician IDs A most naive method of mapping physician IDs to features is without any grouping process The data contains 4,846 physicians, so |M | was set to 4,846 However it caused our model's training to be unstable This might be due to the many physician IDs appearing for the first time in the test time Table

abstract

During the patient's hospitalization, the physician must record daily observations of the patient and summarize them into a brief document called "discharge summary" when the patient is discharged  Automated generation of discharge summary can greatly relieve the physicians' burden, and has been addressed recently in the research community 
Most previous studies of discharge summary generation using the sequenceto-sequence architecture focus on only inpatient notes for input  However, electric health records (EHR) also have rich structured metadata (e g , hospital, physician, disease, length of stay, etc ) that might be useful  This paper investigates the effectiveness of medical meta-information for summarization tasks 
We obtain four types of meta-information from the EHR systems and encode each meta-information into a sequence-to-sequence model  Using Japanese EHRs, meta-information encoded models increased ROUGE-1 by up to 4 45 points and BERTScore by 3 77 points over the vanilla Longformer  Also, we found that the encoded meta-information improves the precisions of its related terms in the outputs 
Our results showed the benefit of the use of medical meta-information 

INTRODUCTION

Clinical notes are written daily by physicians from their consults and are used for their own decision-making or coordination of treatment  They contain a large amount of important data for machine learning, such as conditions, laboratory tests, diagnoses, procedures, and treatments  While invaluable to physicians and researchers, the paperwork is burdensome for physicians ,  
Discharge summaries, a subset of these, also play a crucial role in patient care, and are used to share information between hospitals and physicians (see an example in Figure )  It is created by the physician as a summary of notes during hospitalization at the time of the patient's discharge, which is known to be very time-consuming 
Researchers have begun to apply automatic summarization techniques to address this problem -   Previous studies used extractive or abstractive summarization methods, but most of them focused on only progress notes for inputs  Properly summarizing an admission of a patient is a quite complex task, and requires various meta-information such as the patient's age, gender, vital signs, laboratory values and background to specific diseases 
Therefore, discharge summary generation needs more medical meta-information, than similar but narrower tasks such as radiology report generation  However, what kind of meta-information is important for summarization has not been investigated, even though it is critical not only for future research on medical summarization but also for the policy of data collection infrastructure 
In this paper, we first reveal the effects of meta-information on neural abstractive summarization on admissions  Our model is based on an encoder-decoder transformer with an additional feature embedding layer in the encoder (Figure )  Hospital, physician, disease, and length of stay are used as meta-information, and each feature is embedded in the vector space 
For experiments, we collect progress notes, discharge summaries and coded information from the electronic health record system, which are managed by a largest multi-hospital organization in Japan  Our main contributions are as follows: • We found that a transformer encoding meta-information generates higher quality summaries than the vanilla one, and clarified the benefit of using meta-information for medical summarization tasks 
• We found that a model encoding disease information can produce proper disease and symptom words following the source  In addition, we found that the model using physician and hospital information can generate symbols that are commonly written in the summary  • We are the first to apply the abstractive summarization method to generate Japanese discharge summaries 
In the studies of summarization of medical documents, it is common to retrieve key information such as disease, examination result, or medication from EHRs -   Other researchs more similar to our study targeted to help physicians get the point of medical documents quickly by generating a few key sentences -  
Studies generating contextualized summaries can be categorized by the type of model inputs and architectures  Some studies produced a whole discharge summary using structured data for input - The sensitivity of the gram stain for bacterial meningitis is about 60%, and the sensitivity of the culture is not high either 
Also, the glucose in the cerebrospinal fluid would have been slightly lower  Although no definitive diagnosis could be made, bacterial meningitis was the most suspicious disease  The causative organism was assumed to be MRSA, and vancomycin and meropenem (meningitis dose) were used to cover a wide range of enteric bacteria 
a whole discharge summary from free-form inpatient records -   The free-form data is more challenging since it is noisier than structured data  In inputting of the free-form data, extractive summarization methods, which extract sentences from the source, are commonly used , -   On the other hands, an encoder-decoder model was used for abstractive summarization , , with a limited number of studies 
The various issues in the abstractive generation of discharge summary would be studied in the future  Studies using medical meta-information have long been conducted on a lot of tasks -   In abstractive summarization on discharge summary,  developed a model incorporating similarity of progress notes and information of the record author 
They presented an idea of integrating meta-information into the abstractive summarization model on medical documents, but did not reveal how meta-information would affect the quality of the summaries  Our method is based on the encoder-decoder transformer model  The transformer model is known for its high performance and has been widely used in recent studies, thus it is suitable for our purpose 
As shown in Figure , the standard input to a transformer's encoder is created by a token sequence T = [t 0 , t 1 ,    , t i ] and position sequence P = [p 0 , p 1 ,    , p i ], where i is the maximum input length  The token and position sequences are converted into token embeddings E T and positional embeddings E P by looking up the vocabulary tables 
The sum of E T and E P is input into the model  In this paper, we attempt to encode meta-information to feature embeddings  We follow the segment embeddings of BERT and the language embeddings of XLM , which provide additional information to the model  It is not a new idea but is suitable for our validation 
Our method is formulated as follows: Let M be feature type, M ∈ {Vanilla, Hospital, Physician, Disease, Length of stay}, since we set five types of features  Feature embeddings E M is created by looking up the feature table where m j is featue value (e g , pysician ID, disease code, etc ) and |M | is the maximum number of differences in a feature 
In our study, |M | is set to four different values depending on features  Specifically, they are as follows  a) Hospital: As shown in Table , the data includes five hospital records  They were obtained mechanically from the EHR system  b) Physician: Physicians are also managed by IDs in the EHR systems  We hashed the physician IDs into 485 groups containing 10 people each 
Specifically, as a naive strategy, we shuffled and listed the cases within each hospital, and hashed them into groups in the order of appearance of the physician IDs  So each group has the information about the relevance of the hospitals  The reason for employing a grouping strategy is described in Appendix A 
c) Disease: Two types of disease information exist in our EHRs: disease names and disease codes called ICD-10   We did not use any disease names in the inputs for our experiment  Instead, we encoded diseases with the first three letters of the ICD-10 code, because they represent well the higher level concept 
The initial three letters of the ICD-10 codes are arranged in the order of an alphabetic letter, a digit, and a digit, so there are a total of 2,600 ways to encode a disease  In our data, some ICD-10 codes were missing, although all disease names were systematically obtained from the EHR system  For such cases, we converted the disease names into ICD-10 codes using MeCab with the J-MeDic (MANBYO 201905) dictionary 
Also, diseases can be divided into primary and secondary diseases, but we only deal with the primary diseases  d) Length of stay: The length of stay can be obtained mechanically from the EHR system and the maximum value was set to 1,000 days  We set |M | for vanilla, hospital, physician, disease, and length of stay to 1, 5, 485, 2,600, and 1,000, respectively  
The vanilla embedding is prepared for the baseline in our experiment and to equalize the total number of parameters with the other models  The input to our model is the sum of E T , E P and E M   We also prepare an extra model with all features for our experiments  This takes all four feature embeddings (hospital, physician, disease, and length of stay) added to the encoder 

Datasets and Metrics

We evaluated our proposed method on a subset of data from National Hospital Organization (NHO), the largest multiinstitutional organization in Japan  The statistics of our data are shown in Table  , which includes 24,630 cases collected from five hospitals  Each case includes a discharge summary and progress notes for the days of stay 
The data are randomly split into 22,630, 1,000, and 1,000 for train, validation, and test, respectively  Summarization performances are reported in ROUGE-1, ROUGE-2, ROUGE-L and BERTScore in terms of F1  In addition, we also employed BLEURT , which models human judgment 

Architectures and Hyperparameters

Due to our hardware constraints we need a model that is computationally efficient, so we employed the Longformer instead of the conventional transformer  Longformer can    In our model, number of layers, window size, dilation, input sequence length, output sequence length, batch size, learning rate and number of warmup steps are 8, 256, 1, 1024, 256, 4, 3e-5 and 1K, respectively 
Other hyperparameters are the same as in the original Longformer, except for the maximum number of epochs is not fixed and the best epoch  It is selected for each training using the validation data based on ROUGE-1  Also, the original Longformer imports pretrained-BART parameters to initial values, but we do not use pre-trained Japanese BART in this study 
We used three GeForce RTX 2080 TI for our experiments  Our vocabulary for preparing input to Longformer is taken from UTH-BERT , which is pre-trained on the Japanese clinical records  Since the vocabulary of UTH-BERT is trained by WordPiece , we also tokenize our data with WordPiece  However, the vocabulary does not include white space and line breaks, which cannot be handled, so we add those two tokens to the vocabulary, resulting in a total size of 25,002 
The vocabulary has all tokens in full characters, so we normalized full-wdith characters by converting all alphanumeric and symbolic characters to half-width for byte fallback  , we found that all the models with encoded medical meta-information perform better in ROUGE-1, ROUGE-L and BLEURT than the vanilla Longformer 
However, in BERTScore, only hospital and disease models outperform the vanilla  Specifically, disease information is most effective, improving ROUGE-1, ROUGE-2, ROUGE-L, BERTScore and BLEURT by 4 45, 0 73, 3 12, 3 77 and 0 21 points over the vanilla model, respectively  This seems to be because disease information and the ICD-10 ontology efficiently cluster groups with similar representations 
In contrast, in ROUGE-2 and ROUGE-L, the model with physician embedding is inferior to the vanilla model  This seems to be a negative effect of grouping physicians without any consideration of their relevance  It would be better to cluster them by department, physician attributes, similarity of progress notes, etc  Regarding low ROUGE-2 scores in all models, a previous study using the English data set also reported a low ROUGE-2 score of about 5%, which may indicate an inherent difficulty in discharge summary generation 
In BERTScore, the models with the physician and the length of stay did not reach the performance of the vanilla model, suggesting that the system's outputs are semantically inferior  The model with all features performed the lowest of all models in BERTScore  The reason for the low score of the model with all features seems to be that its number of parameters in feature embedding was four times larger than that of the model with the individual feature, and the amount of training data was insufficient 
In BLEURT, all models with meta-information outperform vanilla, which suggests that they are more natural to humans  To analyze the influence of encoded meta-information on the outputs, we evaluate the precisions of the generated text  Specifically, we measure the probability that the generated words are included in the gold summary to investigate if the proper words are generated 
Some previous studies on faithfulness, which also analyze the output of summarization, have employed words or entities -   In this study, we focused on words, not entities, because we wanted to visualize expressions that are not only nouns  The words were segmented by MeCab with the J-MeDic  For each segmented word, the numeral and symbol labels were assigned as parts of speech by MeCab, the morphological analyzer, while the disease and symptom were assigned by the J-Medic dictionary 
The results, shown in Figure , indicate that the encoded disease information leads to generate more proper disease and symptom words  This indicates that the meta-information successfully learns disease-related expressions  The encoded hospital or physician information also improved the precision of symbols generation 
This suggests that different hospitals and physicians have different description habits (e g , bullet points such as "•", "*" and "-", punctuation such as "。" and " ", etc ), which can be grouped by meta-information  In this paper, we conducted a discharge summary generation experiment by adding four types of information to Longformer and verified the impact of the meta-information 
The results showed that all four types of information exceeded the performance of the vanilla Longformer model, with the highest performance achieved by encoding disease information  We found that meta-information is useful for abstractive summarization on discharge summaries  Our limitations are that we used Japanese EHR, the limited number of tested features and not performing human evaluations 
As for the efficacy of the meta-information, we believe that our results are applicable to non-Japanese, but it is left as Fig    The precisions of words in the generated summaries  The vertical axis shows the probability that the words exist in the gold summary  a future work  Other meta-information may be worth verifying such as the patient's gender, age, race, religion and used EHR system, etc 
It is hard to collect a large amount of medical information and process it into meta-information, so we may need to develop a robust and flexible research infrastructure to conduct a more large scale cross-sectional study in the future  In the discharge summary generation task, which demands a high level of expertise, the human evaluation requires a lot of physicians' efforts and it is a very high cost which is unrealistic 
This is a general issue in tasks dealing with medical documents, and this study also could not perform human evaluations  On this research, informed consent and patient privacy are ensured in the following manner  Notices about their policy and the EHR data usage are posted at the hospitals  The patients who disagree with the policies can request opt-out and are excluded from the archive 
In case of minors and their parents, followed the same manner  In the case of minors and their parents are same  To conduct a research on the archive, researchers must submit their research proposals to the institutional review board  After the proposal is approved, the data is anonymized to build a dataset for analysis 
The data is accessible only in a secured room at the NHO headquarters, and only statistics are brought out of the secured room, for protection of patients' privacy  In the present research, the analysis was conducted under the IRB approval (IRB Approval No : Wako3 2019-22) of the Institute of Physical and Chemical Research (RIKEN), Japan, which has a collaboration agreement with the National Hospital Organization 
This data is not publicly available due to privacy restrictions  shows the detailed number of cases handled by physicians  In all hospitals, there is a large difference between the median and the maximum of cases/physician  This indicates that a few physicians handle a large number of cases and many physicians handle fewer cases 
It is impossible to avoid physician IDs first seen at test time without some process that averages the number of cases a physician holds 



ground_truth: Verifying other meta-information such as patient's gender, age, race, etc.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How does the framework capture the reduced-order dynamics?

Read the following text and answer briefly.

Context 2: Paper Info

Title: Interpretable reduced-order modeling with time-scale separation Interpretable reduced-order modeling with time-scale separation
Publish Date: 7 March 2023
Author List: Sebastian Kaltenbach (from CSE-Lab, ETH Zurich, Harvard SEAS), Phaedon-Stelios Koutsourelakis (from CSE-Lab, ETH Zurich, Harvard SEAS), Petros Koumoutsakos (from CSE-Lab, ETH Zurich, Harvard SEAS), Harvard Seas (from CSE-Lab, ETH Zurich, Harvard SEAS)

Figure

FIG  5  Comparison between the phase-space of the reference solution (left) and the phase-space of the predictions
FIG  7  Comparison between predictions and reference solutions for a new initial condition fort = 1 25, 3 75, 7 5, 12 5, 20, 30  (from left to right and top to down) We note that with longer prediction time the uncertainty bounds increases Despite the chaotic nature of the KS equation, the predictive posterior mean is close to the reference solution for t ≤ 12 5

abstract

Partial Differential Equations (PDEs) with high dimensionality are commonly encountered in computational physics and engineering  However, finding solutions for these PDEs can be computationally expensive, making model-order reduction crucial  We propose such a data-driven scheme that automates the identification of the time-scales involved and, can produce stable predictions forward in time as well as under different initial conditions not included in the training data 
To this end, we combine a non-linear autoencoder architecture with a time-continuous model for the latent dynamics in the complex space  It readily allows for the inclusion of sparse and irregularly sampled training data  The learned, latent dynamics are interpretable and reveal the different temporal scales involved 
We show that this data-driven scheme can automatically learn the independent processes that decompose a system of linear ODEs along the eigenvectors of the system's matrix  Apart from this, we demonstrate the applicability of the proposed framework in a hidden Markov Model and the (discretized) Kuramoto-Shivashinsky (KS) equation 
Additionally, we propose a probabilistic version, which captures predictive uncertainties and further improves upon the results of the deterministic framework 

INTRODUCTION

High-fidelity simulations of critical phenomena such as ocean dynamics and epidemics have become essential for decision-making  They are based on physically-motivated PDEs expressing system dynamics that span multiple spatiotemporal scales and which necessitate cumbersome computations   In recent years there is increased attention to the development of data-driven models that can accelerate the solution of these PDEs as well as reveal salient, lower-dimensional features that control the long-term evolution 
In most cases, data-driven reduced-order models are not interpretable  In particular, models based on neural networks despite good predictive capabilities , they offer a black-box description of the system dynamics  A possible remedy is applying a symbolic regression to the learned neural network representation , but this adds additional computational cost due to the two-step procedure 
A number of frameworks such as SINDy allows to learn interpretable dynamics but it relies on the a-priori availability of lower-dimensional descriptors and of time-derivatives which can be very noisy for both simulation and experimental data  Other frameworks are tailored to specific problems such as molecular dynamics  
Here, we present a framework that only needs the value of the observables, and not their derivatives, as training data and is capable of identifying interpretable latent dynamics  The deployment of interpretable latent dynamics ensures that conservation of important properties of that are reflected in the reduced-order model  
The present method is related to approaches based on the Koopman-operator extended Dynamic Mode Decomposition (eDMD) but uses continuous complex-valued latent space dynamics and only requires one scalar variable per latent dimension to describe the latent space dynamics  Therefore we do not have to enforce any parametrizations on the Koopman matrix  
The time-continuous formulation moreover allows to incorporate sparse and irregularly sampled training data and fast generation of predictions after the training phase  By using a complex-valued latent space we can also incorporate harmonic effects and reduce the number of latent variables needed  Linear and non-linear autoencoders are used to map the observed, high-dimensional time-series to the lower-dimensional, latent representation and we identify simultaneously the autoencoder as well as the latent dynamics by optimizing a combined loss function 
Hence the to tasks of dimensionality reduction and discovery of the reduced dynamics are unified while other frameworks treat the two parts separately   Apart from using an architecture based on autoencoders to identify the latent space, projection-based methods could also be employed   We are also proposing a probabilistic version of our algorithm ) that makes use of probabilistic Slow Feature Analysis  
This allows for a latent representation that arart from being time-continuous, can quantify the predictive uncertainty and hierarchically decompose the dynamics into their pertinent scales while promoting the discovery of slow processes that control the system's evolution over long time horizons  The rest of the paper is structured as follows: We introduce the methodological framework as well as algorithmic details in section II 
Particular focus is paid on the interpretability of the inferred lower-dimensional dynamics  In section III we present three numerical illustrations, i e  a system of linear ODEs, a hidden Markov Model and the discretized KS-equation  We then present in section IV the probabilistic extension of the framework and apply it to the KS-equation 
We conclude with a summary and a short discussion about possible next steps  We introduce the autoencoders deployed in this work, followed by the interpretable latent space dynamic and discuss the training process  We consider data from high-dimensional time series x n ∈ R f with n = 1,    , T   We remark that the intervals between the different states do not need to be uniformly spaced 

Autoencoder

A core assumption of the method is that each high-dimensional state x n can be compressed to a lower-dimensional representation z n ∈ C c with c << f   We identify this lower-dimensional representation by an autoencoder consisiting of a parameterized encoder and decoder  The encoder maps the high-dimensional representation to the latent space as:
The latent space is complex-valued  The decoder reconstructs the high-dimensional representation based on the latent variables as: We denote the parameters of the encoder as well as the decoder by θ  As discussed later in Section II C, both set of parameters are optimized simultaneously during training and therefore there is no need for differentiating them 

Interpretable Latent Space Dynamics

We employ a propagator in the latent space to capture the reduced-order dynamics of the system  In contrast to other time-extended variational autoencoder frameworks, our representation uses complex valued latent variables  In addition the latent variables are treated independently  The latter feature enables us to have an interpretable latent dynamics as well as a model that is especially suitable for being trained in the Small Data regime due to the small number of required parameters 
This is in contrast to temporal propagators such as LSTMs   For each dimension i of the latent variable z we are using the following continuous ODE in the complex plane: By solving this ODE, we can define the operator: Interpretable reduced-order modeling with time-scale separation Here, λ is a vector containing all the individual λ's and ∆t n indicates the time-step between the latent states 
The symbol is used to indicate a component-wise multiplication  We remark that the latent variables and the parameter governing the temporal evolution are complex numbers and their role in describing the system dynamics is similar to that of an eigenvalue  The real part is associated with growth and decay whereas the imaginary part is representing the periodic component 
This approach has similarities with the Koopman-operator based methods and the extended dynamic mode decomposition   In contrast to the methods mentioned before we are using a continuous formulation in the latent space that allows us to incorporate scarce and irregularly sampled training data and directly rely on complex numbers in the latent space 

Training and Predictions

We optimize a loss function that combines both a reconstruction loss as well as a loss associated with the error of our learned propagator in the latent space: (5) We note that we could directly incorporate mini-batch training by only taking the summation over a subset of the N available training data 
For new predictions of unseen states, we use the encoder to generate a latent representations which is then advanced in time by the learned propagator  At a designated time step we are using the decoder to reconstruct the high-dimensional solution  We applied our algorithm to three systems  First, we show that the algorithm is capable of exactly reproducing the solution of a linear ODE and to identify its eigenvalues 
Afterwards we are applying the framework to a high-dimensional process generated by a complex latent dynamics, which is correctly identified  As a final test case, we are applying the algorithm to a Kuramoto Shivashinski (KS) equation  Interpretable reduced-order modeling with time-scale separation

Linear ODE

We are considering a two-dimensional ODE system for x = y 1 y 2 : Based on the obtained training data we run our algorithm using a linear encoder and decoder structure as well as two latent variables z  The loss function was optimized using the Adam algorithm   As we consider a linear ODE we can analytically compute the eigenvalues involved and compare it with the parameters λ identified by our algorithm 
We observe in Figure that the algorithm was able to recover the correct values, i e  the eigenvalues 7 and 3 of the given linear ODE  The system does not have a periodic component and the two imaginary parts correctly go to zero, whereas the real parts converge to the reference value  Moreover we are also able to identify for the linear mapping between our latent variables z and the training data a matrix consisting of a multiple of the eigenvectors (1,1) and (1,-1) and thus the correct solution 
This example was chosen to show that the algorithm is able to quickly identify the exact solution of a linear ODE in terms of its linearly independent components 

Hidden multiscale dynamics

We consider eight-dimensional synthetic time series data produced by an underlying twodimensional complex valued process  In particular, the data points x are generated by first solving for the temporal evolution for the two complex-valued processes p 1 and p 2 and than mapping to the eight-dimensional space by using a randomly sampled linear mapping W  
One of the two processes used to generate the data is chosen to be much slower than the other one and both processes have a periodic component  dp 2 dt = (−0 9 + 1 5i)p 2 (8) As training data we consider 40 time series with 150 data points each, obtained by simulating the described processes for a maximum of t = 15 s and then sampling from the obtained data points 
Hence the training data consists of: • 40 time-series • with each consisting 150 observations of the x at a uniform time-step ∆t = 0 0025 The autoencoder obtained consists of one linear layer for both the decoder as well as the encoder  The model is trained for 5000 iterations using the Adam optimizer and a learning rate of 10 −3  
The results for the convergence of the parameters λ 1 and λ 2 can be found in Figure   We note that the process which is slower decaying and thus more responsible for the long-term evolution of the system has a higher convergence rate than the faster process  With the obtained parameters λ as well as the trained autoencoder, we compute predictions based on the last time step used for training, i e  we apply the encoder to obtain our latent representation and than use the latent dynamics to advance the latent representation in time 
Afterwards, we employ the decoder to reconstruct the full high-dimensional system  The results can be found in Figure and show very good agreement between predictions and reference data  This example shows that our model is successfully able to carry out dimensionality reduction and moreover indicates that the convergence rate between latent processes can be different 
The latter is relevant when training models as for accurate predictions all latent processes and their dynamics should be converged 

Kuramoto-Sivashinsky

Finally, we applied our algorithm to the KS equation and aim to identify a reduced-order model for the solution u(y, t): We employed periodic boundary conditions, µ = 1 and a domain size y ∈ [0, 22]  For this domain-size, the KS-equation exhibits a structurally stable chaotic attractor as discussed in The black lines divides the area for which training data was given from the area without raining data 
;   The equation is discretized in space using a discretization step of 22 64 resulting in a state vector x of dimension 64 and a nonlinear system of coupled ODEs  This is solved using a stiff fourth-order solver  We employed a non-linear encoder and decoder with four fully-connected layers each and ReLU-activation functions as well as Dropout Layers between the fully-connected layers 
We trained the model for 200000 iterations using Adam and a learning rate of 5 • 10 4 and assuming a five-dimensional latent space  We obtained the λ's in Figure   Four latent variables have λ's close to zero and thus a slow temporal dynamic that is responsible for the long-term evolution whereas one latent variable is quickly decaying 
Based on the obtained parameters, we do predictions based on an unseen initial condition not contained in the training data  We are able to reconstruct the correct phase space based on our predictions despite only using a very limited amount of training data  The results for the phase space can be seen in Figure  
Although the small-scale fluctuations in the temporal dynamics are not well captured, the model identifies the correct manifold which has a good accuracy compared to the reference solution  All phase-spaces were obtained by using a finite-difference operator on the data or predictions  These results are in accordance Interpretable reduced-order modeling with time-scale separation with whose LSTM-based temporal dynamic model was also able to find the correct phase space but not to track the actual dynamics for long-term predictions 
Our model is not able to account for noise in the temporal evolution and thus dealing with chaotic, small-scale fluctuations is challenging  We believe that a probabilistic version of our algorithm could be advantageous here  This section contains a fully probabilistic formulation for the deterministic model discussed before 
We replace the Autoencoder with a Variational Autoencoder and the ODE in the latent space with a SDE  The loss function which we optimize is the Evidence-Lower Bound (ELBO) 

Model Structure

We postulate the following relations for our probabilistic model using an Ornstein-Uhlenbeck (OU) for each dimension i of the latent space and a Wiener process W t in the latent space: We again assume that the latent variables z t are complex-valued and a priori independent  Complex variables were chosen as their evolution includes a harmonic components which are observed in many physical systems 
We assume an initial conditions z 0,i ∼ CN (0, σ 2 0,i )  The total parameters associated with the latent space dynamics of our model are thus {σ 2 0,i , σ 2 i , λ i } c i=1 and will be denoted by θ together with all parameters responsible for the decoder mapping G (see next section)  These parameters along with the state variables z t have to be inferred from the data x t  
Based on probabilistic Slow Feature Analysis (SFA) , we set σ 2 i = 2; (λ j ) and σ 2 0,i = 1  As a consequence, a priori, the latent dynamics are stationary  A derivation and reasoning for this choice can be found in Appendix A  Hence the only independent parameters are the λ i , the imaginary part of which can account for periodic effects in the latent dynamics 

Variational Autoencoder

We employ a variational autoencoder to account for a probabilistic mappings from the lower-dimensional representation z n to the high-dimensional system x n   In particular we are employing a probabilistic decoder The encoder is used to infer the state variables z based on the given data and thus defined in the inference and learning section 

Inference and Learning

Given the probabilistic relations , our goal is to infer the latent variables z 0:T as well as all model parameters θ  We follow a hybrid Bayesian approach in which the posterior of the state variables is approximated using amortized Variational Inference and Maximum-A-Posteriori (MAP) point-estimates for θ are computed 
The application of Bayes' rule for each data sequence x 0:T leads to the following posterior: where p(θ) denotes the prior on the model parameters  In the context of variational inference, we use the following factorization of the approximate posterior i e  we infer only the mean µ and variance σ for each state variable based on the given data points 
This conditional density used for inference is the encoder-counterpart to the probabilistic decoder defined in the section before  It can be readily shown that the optimal parameter values are found by maximizing the Evidence Lower Bound (ELBO) F(q φ (z 0:T ), θ) which is derived in Appendix B  We compute Monte Carlo estimates of the gradient of the ELBO with respect to φ and θ with the help of the reparametrization trick and carry out stochastic optimization with the ADAM algorithm  

Results for the probabilistic extension

We applied our probabilistic version to the KS-equation  We used the same settings as for the deterministic approach but considered up to 10 complex latent variables  The obtained λ's are in Figure   The probabilistic model allows us to quantify the uncertainty in predictions  In Figure predictions for various time-steps and the respective uncertainty bounds are shown for an unseen initial condition 
Due to the chaotic nature of the KS-equation and the small amount of training data, the underlying linear dynamic of our model is only able to capture the full dynamics for a limited time horizon  Fortunately, due to the probabilistic approach the model is capable of capturing chaotic fluctuations with increasingly wide uncertainty bounds 
We also computed the phase space representation for the KS-equation based on the predictions obtained by our model and compare it with the reference solution  The probabilistic model identifies the correct manifold with a better accuracy than the deterministic model  As some of the small-scale fluctuations are accounted as noise, the resulting manifold is more concentrated at the origin and the obtained values are slightly smaller than the reference manifold although their shape is very similar 



ground_truth: By using a propagator in the latent space.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How does the conduction gap depend on the strain direction?

Read the following text and answer briefly.

Context 3: \section{Introduction}

In spite of being an attractive material with excellent electronic properties \cite{ahcn09},  practical applications of graphene as in conventional semiconductor devices are still questionable due to its gapless nature  In particular, the ON/OFF current ratio is low while the saturation of current is poor in pristine graphene transistors \cite{schw10}  Many efforts of bandgap engineering in graphene \cite{yhan07,khar11,lher13,jbai10,zhan09} have been made to solve these issues  The pioneer technique proposed \cite{yhan07} is to cut 2D graphene sheets into 1D narrow nanoribons  In 2D graphene sheets, some options as Bernal-stacking of graphene on hexagonal boron nitride substrate \cite{khar11}, nitrogen-doped graphene \cite{lher13}, graphene nanomesh lattice \cite{jbai10,berr13} and Bernal-stacking bilayer graphene \cite{zhan09} have been explored  However, the possibility to open a sizable bandgap in graphene as large as those of standard semiconductors is still very unlikely  In particular, it requires a very good control of lattice geometry and edge disorder in narrow graphene nanoribbons (GNRs) \cite{quer08} and in graphene nanomesh lattices \cite{hung13}, while the bandgap opening in bilayer graphene by a perpendicular electric field may not be large enough for realistic applications \cite{fior09}  Other methods should be further verified by experiments 

\begin{figure}[!t]
\centering
\includegraphics[width=2 8in]{Fig01 pdf}
\caption{Schematic of unstrained/strained graphene junctions investigated in this work }
\label{fig_sim1}
\end{figure}

On the other hand, graphene was experimentally demonstrated to be able to sustain a much larger strain than conventional semiconductors, making it a promising candidate for flexible electronics (see in a recent review \cite{shar13})  Indeed, strain engineering has been suggested to be an alternative approach to modulating efficiently the electronic properties of graphene nanomaterials  In particular, the bandgap has periodic oscillations in the armchair GNRs \cite{ylu210} while the spin polarization at the ribbon edges (and also the bandgap) can be modulated by the strain in the zigzag cases  In 2D graphene sheets, a finite gap can open under large strains, otherwise, it may remain close to zero but the Dirac points are displaced \cite{cocc10,per209,pere09,huan10}  Many interesting electrical, optical, and magnetic properties induced by strain in graphene have been also explored, e g  see in \cite{bunc07,pere09,kuma12,per010,pell10,guin10,tlow10,zhai11} 

Besides, local strain is a good option to improve the electrical performance of graphene devices \cite{pere09,ylu010,fuji10,juan11,baha13}  For instance, it has been shown to enhance the ON current in a GNR tunneling FET \cite{ylu010} and to fortify the transport gap in GNR strained junctions \cite{baha13}  In a recent work \cite{hung14}, we have investigated the effects of uniaxial strain on the transport in 2D unstrained/strained graphene junctions and found that due to the strain-induced shift of Dirac points, a significant conduction gap of a few hundreds meV can open with a small strain of a few percent  This type of strained junction was then demonstrated to be an excellent candidate to improve the electronic operation of graphene transistors  It hence motivates us to further investigate the properties of this conduction gap so as to optimize the performance of graphene devices  On the one hand, the effects of strain should be, in principle, dependent on its applied direction  On the other hand, because the appearance of conduction gap is a consequence of the shift of Dirac points along the $k_y$-axis, it is predicted that this gap should also depend on the transport direction  Note that here, Oy (Ox) - axis is assumed to be perpendicular (parallel) to the transport direction  The effects of both strain and transport directions will be clarified systematically in the current work 

\section{Model and calculations}

In this work, the $\pi$-orbital tight binding model constructed in \cite{per209} is used to investigate the electronic transport through the graphene strained junctions schematized in Fig  1  The Hamiltonian is ${H_{tb}} = \sum\nolimits_{nm} {{t_{nm}}c_n^\dag {c_m}}$ where $t_{nm}$ is the hopping energy between nearest neighbor \emph{n}th and \emph{m}th atoms  The application of a uniaxial strain of angle $\theta$ causes the following changes in the $C-C$ bond vectors:
\begin{eqnarray}
 {{\vec r}_{nm}}\left( \sigma \right) &=& \left\{ {1 + {M_s}\left( \sigma, \theta \right)} \right\}{{\vec r}_{nm}}\left( 0 \right) \\
 {M_s}\left( \sigma, \theta \right) &=& \sigma \left[ {\begin{array}{*{20}{c}}
 {{{\cos }^2}\theta  - \gamma {{\sin }^2}\theta }&{\left( {1 + \gamma } \right)\sin \theta \cos \theta }\\
 {\left( {1 + \gamma } \right)\sin \theta \cos \theta }&{{{\sin }^2}\theta  - \gamma {{\cos }^2}\theta }
 \end{array}} \right] \nonumber
\end{eqnarray}
where $\sigma$ represents the strain and $\gamma \simeq 0 165$ is the Poisson ratio \cite{blak70}  The hopping parameters are defined as $t_{nm} \left( \sigma \right) = t_0 \exp\left[-3 37\left(r_{nm} \left( \sigma \right) /r_0 - 1\right)\right]$, where the hopping energy $t_0 = -2 7$ $eV$ and the bond length $r_{nm} \left( 0 \right) \equiv r_0 = 0 142$ $nm$ in the unstrained case  Therefore, there are three different hoping parameters $t_{1,2,3}$ corresponding to three bond vectors ${\vec r}_{1,2,3}$, respectively, in the strained graphene part of the structure (see Fig  1)  Here, we assume a 1D profile of applied strain, i e , the strain tensor is a function of position along the transport direction Ox while it is constant along the Oy-axis  The transport direction, $\phi$, and strain direction, $\theta$, are determined as schematized in Fig  1  Based on this tight binding model, two methods described below can be used to investigate the conduction gap of the considered strained junctions 

\textbf{Green's function calculations } First, we split the graphene sheet into the smallest possible unit cells periodically repeated along the Ox/Oy directions with the indices $p/q$, respectively (similarly, see the details in \cite{hung12})  The tight-binding Hamiltonian can therefore be expressed in the following form:
\begin{eqnarray}
{H_{tb}} = \sum\limits_{p,q} {\left( {{H_{p,q}} + \sum\limits_{{p_1},{q_1}} {{H_{p,q \to p_1,q_1}}} } \right)}
\end{eqnarray}
where $H_{p,q}$ is the Hamiltonian of cell $\{p,q\}$, and $H_{p,q \to p_1,q_1}$ denotes the coupling of cell $\{p,q\}$ to its nearest neighbor cell $\{p_1,q_1\}$  We then Fourier transform the operators in Eq  (2) as follows:
\begin{eqnarray}
 {c_{p,q}} = \frac{1}{{\sqrt {{M_{cell}}} }}\sum\limits_{{\kappa_y}} {{e^{i{q\kappa_y}}}} {{\hat c}_{p,{\kappa_y}}},
\end{eqnarray}
where $M_{cell}$ is the number of unit cells and $\kappa_y \equiv k_y L_y$ with the size $L_y$ of unit cells along the Oy direction  The Hamiltonian (2) is finally rewritten as a sum of $\kappa_y$-dependent 1D-components:
\begin{eqnarray}
{H_{tb}} &=& \sum\limits_{{\kappa_y}} {\hat H\left( {{\kappa_y}} \right)} \\
\hat H\left( {{\kappa_y}} \right) &=& \sum\limits_p {{{\hat H}_{p \to p - 1}}\left( {{\kappa_y}} \right) + {{\hat H}_p}\left( {{\kappa_y}} \right) + {{\hat H}_{p \to p + 1}}}\left( {{\kappa_y}} \right) \nonumber
\end{eqnarray}
With this Hamiltonian form, the Green's function formalism can be easily applied to compute transport quantities in the graphene strained junction with different transport directions  In particular, the conductance at zero temperature is determined as:
\begin{eqnarray}
 \mathcal{G} \left( \epsilon \right) = \frac{{e^2 W}}{{\pi h L_y}}\int\limits_{BZ} {d{\kappa_y} \mathcal{T}\left( {\epsilon, {\kappa_y}} \right)}
\end{eqnarray}
where $\mathcal{T}\left( {\epsilon,{\kappa_y}} \right)$ is the transmission probability computed from the Green's functions  The integration over $\kappa_y$ is performed in the whole first Brillouin zone  As in ref  \cite{hung13}, the gap of conductance (conduction gap) is then measured from the obtained data of conductance 

\textbf{Bandstructure analyses } To determine the conduction gap of strained junctions, we find that another simple way based on the analysis of graphene bandstructures could be efficiently used  It is described as follows  Since the conductance is computed from Eq  (5), the appearance of conduction gap is essentially governed by the gaps of transmission probability, which is determined from the energy gaps in the unstrained and strained graphene sections  These energy gaps can be defined directly from the graphene bandstructures  Therefore, our calculation has two steps, similar to that in \cite{hung14}  From the graphene bandstructures obtained using the tight-binding Hamiltonian above, we first look for the energy gaps $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ for a given $\kappa_y$ of two graphene sections  The maximum of these energy gaps determines the gap $E_{junc}^{gap}\left( {{\kappa_y}} \right)$ of transmission probability through the junction  Finally, the conduction gap $E_{cond gap}$ is obtained by looking for the minimum value of $E_{junc}^{gap}\left( {{\kappa_y}} \right)$ when varying $\kappa_y$ in the whole Brillouin zone 

In particular, the energy bands of strained graphene are given by
\begin{eqnarray}
 E\left( {\vec k} \right) =  \pm \left| {{t_1}{e^{i\vec k{{\vec a}_1}}} + {t_2}{e^{i\vec k{{\vec a}_2}}} + {t_3}} \right|
\end{eqnarray}
where the plus/minus sign corresponds to the conduction/valence bands, respectively  For a given direction $\phi$ of transport, in principle, the vectors $\vec L_{x,y}$ defining the sizes of unit cell along the Ox and Oy directions, respectively, can be always expressed as ${\vec L_x} = {n_1}{\vec a_1} + {n_2}{\vec a_2}$ and ${\vec L_y} = {m_1}{\vec a_1} + {m_2}{\vec a_2}$ with $\cos \phi = \frac{{{{\vec L}_x}\vec L_x^0}}{{{L_x}L_x^0}}$ and $\sin \phi = \frac{{{{\vec L}_x}\vec L_y^0}}{{{L_x}L_y^0}}$ while $\vec L_{x,y}^0 = {\vec a_1} \pm {\vec a_2}$  Note that $n_{1,2}$ and $m_{1,2}$ are integers while $\frac{{{m_1}}}{{{m_2}}} =  - \frac{{{n_1} + 2{n_2}}}{{{n_2} + 2{n_1}}}$, i e , ${\vec L_{x}} {\vec L_{y}} = 0$  In other words, we have the following expressions
\begin{eqnarray}
{{{\vec a}_1} = \frac{{ - {m_2}{{\vec L}_x} + {n_2}{{\vec L}_y}}}{{{n_2}{m_1} - {n_1}{m_2}}};\,\,\,{{\vec a}_2} = \frac{{{m_1}{{\vec L}_x} - {n_1}{{\vec L}_y}}}{{{n_2}{m_1} - {n_1}{m_2}}}}
\end{eqnarray}
On this basis, the energy bands can be rewritten in terms of $\kappa_{x, y} = \vec k \vec L_{x,y} \left( { \equiv {k_{x,y}}{L_{x,y}}} \right)$ by substituting Eqs  (7) into Eq  (6)  This new form of energy bands is finally used to compute the conduction gap of strained junctions 

As a simple example, in the case of $\phi = 0$ (armchair direction), we calculate the conduction gap as follows  First, Eq  (6) is rewritten in the form
\begin{eqnarray}
 E_{\phi = 0}\left( {\vec \kappa} \right) =  \pm \left| {{t_1}{e^{i\kappa_y/2}} + {t_2}{e^{ - i\kappa_y/2}} + {t_3}{e^{ - i\kappa_x/2}}} \right|
\end{eqnarray}
with the vectors $\vec L_{x,y} \equiv \vec L_{x,y}^0$  Using this new form, the energy gap of strained graphene for a given $\kappa_y$ is determined as
\begin{equation}
{E_{strain}^{gap}}\left( {{\kappa_y}} \right) = 2 \left| {\sqrt {{{\left( {{t_1} - {t_2}} \right)}^2} + 4{t_1}{t_2}{{\cos }^2}\frac{{{\kappa_y}}}{2}}  + {t_3}} \right|
\end{equation}
while ${E_{unstrain}^{gap}}\left( {{\kappa_y}} \right)$ is given by the same formula with $t_1$ = $t_2$ = $t_3$ $\equiv$ $t_0$  The gap of transmission probability through the junction is then determined as ${E_{junc}^{gap}}\left( {{\kappa_y}} \right) = \max \left[ {E_{unstrain}^{gap}\left( {{\kappa_y}} \right),E_{strain}^{gap}\left( {{\kappa_y}} \right)} \right]$ and, finally, the conduction gap is given by ${E_{cond gap}} = \min \left[ {E_{junc}^{gap}\left( {{\kappa_y}} \right)} \right]$ for $\kappa_y$ in the whole Brillouin zone 

We would like to notice that the Green's function calculations and the banstructure analyses give the same results of conduction gap in the junctions where the transition region between unstrained and strained graphene sections is long enough, i e , larger than about 5 to 6 nm  In the case of short length, as discussed in \cite{baha13,hung14}, this transition zone can have significant effects on the transmission between propagating states beyond the energy gaps and hence can slightly enlarge the gap of conductance, compared to the results obtained from the bandstructure calculations 

\section{Results and discussion}

\begin{figure}[!t]
\centering
\includegraphics[width=3 0in]{Fig02 pdf}
\caption{Dependence of graphene bandgap (in the unit of eV) on the applied strain and its direction: tensile (a) and compressive (b)  The radius from the central point indicates the strain strength ranging from 0 (center) to 30 $\%$ (edge of maps) while the graphene lattice is superimposed to show visibly the strain direction  The orange circle corresponds to the strains of $\sigma = 23 \%$ }
\label{fig_sim2}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=3 4in]{Fig03 pdf}
\caption{Conductance ($G_0 = e^2W/hL_y$) as a function of energy in graphene strained junctions for $\sigma = 4 \%$ with different strain directions  The transport along the armchair direction ($\phi = 0$) is considered  The data obtained in a uniformly strained graphene is displayed for the comparison }
\label{fig_sim6}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=5 8in]{Fig04 pdf}
\caption{Local density of states (left panels) and corresponding transmission coefficient (right panels) for three different wave-vectors $k_y$ obtained in an unstrained/strained graphene junction of $\sigma = 4 \%$, and $\theta \equiv \phi = 0$  On the top is a schematic of graphene bandedges illustrating the strain-induced shift of Dirac points along the $k_y$-direction }
\label{fig_sim4}
\end{figure*}
\begin{figure*}[!t]
\centering
\includegraphics[width=5 6in]{Fig05 pdf}
\caption{Maps of conduction gap in unstrained/strained graphene junctions: tensile (a,c) and compressive cases (b,d)  The transport is along the armchair $\phi = 0$ (a,b) and zigzag $\phi = 30^\circ$ directions (c,d)  The strain strength ranges from 0 (center) to 6 $\%$ (edge of maps) in all cases }
\label{fig_sim4}
\end{figure*}
First, we re-examine the formation of the bandgap of graphene under a uniaxial strain  From Eq  (9), it is shown that a strain-induced finite-bandgap appears only if ${E_{strain}^{gap}}\left( {{\kappa_y}} \right) > 0$ for all $k_y$ in the first Brillouin zone, i e , ${k _y} \in \left[ { - \frac{\pi}{L_y}, \frac{\pi}{L_y}} \right]$, otherwise, the bandgap remains zero  Hence, the condition for the bandgap to be finite is either
\begin{equation*}
 \left| {{t_1} - {t_2}} \right| > \left| {{t_3}} \right|\,\,\,\,\,{\rm{OR}}\,\,\,\,\,\left| {{t_3}} \right| > \left| {{t_1} + {t_2}} \right|
\end{equation*}
and the corresponding values of bandgap are
\begin{equation*}
 {E_{gap}} = 2\left( {\left| {{t_1} - {t_2}} \right| - \left| {{t_3}} \right|} \right)\,\,\,\,\,{\rm{OR}}\,\,\,\,\,2\left( {\left| {{t_3}} \right| - \left| {{t_1} + {t_2}} \right|} \right)
\end{equation*}
This result was actually reported in \cite{per209,hase06}  We remind as displayed in Fig  2(a) that a finite bandgap opens only for strain larger than $\sim 23 \%$ and the zigzag (not armchair) is the preferred direction for bandgap opening under a tensile strain \cite{per209}  We extend our investigation to the case of compressive strain and find  (see in Fig  2(b)) that (i) the same gap threshold of $\sigma \simeq 23 \%$ is observed but (ii) the preferred direction to open the gap under a compressive strain is the armchair, not the zigzag as the case of tensile strain  This implies that the properties of graphene bandstructure at low energy should be qualitatively the same when applying strains of $\left\{ {\sigma ,\theta } \right\}$ and of $\left\{ {-\sigma ,\theta + 90^\circ} \right\}$  This feature can be understood by considering, for example, strains of $\left\{ {\sigma , \theta = 0} \right\}$ and of $\left\{ {-\sigma , \theta = 90^\circ} \right\}$  Indeed, these strains result in the same qualitative changes on the bond-lengths, i e , an increased bond-length $r_3$ and reduced bond-lengths $r_{1,2}$  However, for the same strain strength, because of the exponential dependence of hoping energies on the bond-lengths, the compressive strain generally induces a larger bandgap than the tensile one, as can be seen when comparing the data displayed in Figs  2(a) and 2(b)  To conclude, we would like to emphasize that a large strain is necessary to open a bandgap in graphene  This could be an issue for practical applications, compared to the use of graphene strained junctions explored in \cite{hung14} 

We now go to explore the properties of conduction gap in the graphene strained junctions  In Fig  3, we display the conductance as a function of energy computed from Eq  (5) using the Green's function technique  As discussed above, a small strain of a few percent (e g , 4 $\%$ here) can not change the gapless character of graphene, i e , there is no gap of conductance in the case of uniformly strained graphene  However, similar to that reported in \cite{hung14}, a significant conduction-gap of a few hundreds meV can open in the unstrained/strained graphene junctions  The appearance of this conduction gap, as mentioned previously, is due to the strain-induced shift of Dirac points and is explained as follows  Actually, the strain causes the lattice deformation and can result in the deformation of graphene bandstructure  Therefore, the bandedges as a function of wave-vector $k_y$ in unstrained and strained graphene can be illustrated schematically as in the top panel of Fig  4  As one can see, the shift of Dirac points leads to the situation where there is no value of $\kappa_y$, for which the energy gaps $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ are simultaneously equal to zero  This means that the transmission probability always shows a finite gap for any $\kappa_y$  For instance, the energy gap is zero (or small) in the unstrained (resp  strained) graphene section but finite in the strained (resp  unstrained) one in the vicinity of Dirac point $k_y = K_{unstrain}$ (resp  $K_{strain}$)  Accordingly, as illustrated in the pictures of LDOS in the left panels of Fig  4 and confirmed in the corresponding transmissions in the right panels, clear gaps of transmission are still obtained  Far from these values of $k_y$, $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ are both finite (e g , see the LDOS plotted for $k_y = K_{gap}$) and hence a finite gap of transmission also occurs  On this basis, a finite gap of conductance is achieved  More important, Fig  3 shows that besides the strength of strain, the strain effect is also strongly dependent on the applied direction  For instance, the conduction gap takes the values of $\sim$ 295, 172 and 323 meV for $\theta = 0$, $30^\circ$ and $90^\circ$, respectively 

Below, we will discuss the properties of the conduction gap with respect to the strain, its applied direction, and the direction of transport  Note that due to the lattice symmetry, the transport directions $\phi$ and $\phi + 60^\circ$ are equivalent while the applied strain of angle $\theta$ is identical to that of $\theta + 180^\circ$  Hence, the data obtained for $\phi$ ranging from $-30^\circ$ to $30^\circ$ and $\theta  \in \left[ {0^\circ ,180^\circ } \right]$ covers the properties of conduction gap in all possible cases 

In Fig  5, we present the maps of conduction gap with respect to the strain and its applied direction in two particular cases: the transport is either along the armchair ($\phi = 0$) or the zigzag ($\phi = 30^\circ$) directions  Both tensile and compressive strains are considered  Let us first discuss the results obtained in the armchair case  Figs  5(a,b) show that (i) a large conduction gap up to about 500 meV can open with a strain of 6 $\%$ and (ii) again the conduction gap is strongly $\theta$-dependent, in particular, its peaks occur at $\theta = 0$ or $90^\circ$ while the gap is zero at $\theta \approx 47^\circ$ and $133^\circ$ for tensile strain and at $\theta \approx 43^\circ$ and $137^\circ$ for compressive strain  In principle, the conduction gap is larger if the shift of Dirac points in the $\kappa_y$-axis is larger, as discussed above about Figs  3-4  We notice that the strain-induced shifts can be different for the six Dirac points of graphene \cite{kitt12} and the gap is zero when there is any Dirac point observed at the same $\kappa_y$ in the two graphene sections  From Eq  (9), we find that the Dirac points are determined by the following equations:
\begin{eqnarray*}
  {\cos}\frac{\kappa_y}{2} &=& \pm \frac{1}{2}\sqrt{\frac{{t_3^2 - {{\left( {{t_1} - {t_2}} \right)}^2}}}{{{t_1}{t_2}}}}, \\
  \cos \frac{{\kappa_x}}{2} &=& \frac{{{t_1} + {t_2}}}{{\left| {{t_3}} \right|}}\cos \frac{{\kappa_y}}{2},\,\,\,\sin \frac{{\kappa_x}}{2} = \frac{{{t_2} - {t_1}}}{{\left| {{t_3}} \right|}}\sin \frac{{\kappa_y}}{2},
\end{eqnarray*}
which simplify into ${\cos}\frac{\kappa_y}{2} = \pm \frac{1}{2}$ and, respectively, $\cos \left( {\frac{{{\kappa _x}}}{2}} \right) = \mp 1$ in the unstrained case  Hence, the zero conduction gap is obtained if
\begin{equation*}
  \frac{{t_3^2 - {{\left( {{t_1} - {t_2}} \right)}^2}}}{{4{t_1}{t_2}}} = \frac{1}{4}
\end{equation*}
Additionally, it is observed that the effects of a strain $\{\sigma,\theta\}$ are qualitatively similar to those of a strain $\{-\sigma,\theta+90^\circ\}$, i e , the peaks and zero values of conduction gap are obtained at the same $\theta$ in these two situations  To understand this, we analyze the strain matrix $M_s \left(\sigma,\theta\right)$ and find that in the case of small strains studied here, there is an approximate relationship between the bond lengths under these two strains, given by \[{r \left( \sigma, \theta \right)} - {r \left( -\sigma, \theta + 90^\circ\right)} \simeq \sigma \left( {1 - \gamma } \right) r_0,\] which is $\theta$-independent for all \emph{C-C} bond vectors  It implies that there is a fixed ratio between the hopping energies $t_i \left( \sigma, \theta \right)$ and $t_i \left( -\sigma, \theta + 90^\circ\right)$ and hence there is the similar shift of Dirac points in these two cases 
\begin{figure}[!t]
\centering
\includegraphics[width=3 4in]{Fig06 pdf}
\caption{Map showing the dependence of conduction gap on the directions ($\theta,\phi$) for $\sigma = 4 \%$  The top is a diagram illustrating the rotation of Dirac points in the \emph{k}-space with the change in the transport direction $\phi$ }
\label{fig_sim6}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=5 5in]{Fig07 pdf}
\caption{Maps of conduction gap obtained in tensile/compressive strained junctions  The transport along the armchair/zigzag directions is considered in (a,b)/(c,d), respectively  The strains $\sigma_c = -2 \%$ and $\sigma_t = 2 \%$ are applied in (a,c) while $\sigma_c = -1 \%$ and $\sigma_t = 3 \%$ in (b,d) }
\label{fig_sim4}
\end{figure*}

We now go to analyze the properties of conduction gap shown in Figs  5(c,d) where the transport is along the zigzag direction $\phi = 30^\circ$  In fact, the conduction gap in this case can reach a value as high as that of the case of $\phi = 0$ but has different $\theta$-dependence  In particular, the conduction gap has peaks at $\theta \approx 47^\circ$ and $133^\circ$ for tensile strain and at $\theta \approx 43^\circ$ and $137^\circ$ for compressive strain, where it is zero in the case of $\phi = 0$  It is also equal to zero at $\theta = 0$ and $\theta = 90^\circ$ where the peaks of conduction gap occur in the latter case of $\phi = 0$  The relationship between these two transport directions can be explained as follows  On the one hand, based on the analyses above for $\phi = 0$, we find that for a given strength of strain, a maximum shift of Dirac points along the $k_y$-axis corresponds to a minimum along the $k_x$-one and vice versa when varying the strain direction $\theta$  On the other hand, as schematized in the top of Fig  6 below, the change in the transport direction results in the rotation of the first Brillouin zone, i e , the $k_x$ (resp  $k_y$) axis in the case of $\phi = 30^\circ$ is identical to the $k_y$ (resp  $k_x$) axis in the case of $\phi = 0$  These two features explain essentially the opposite $\theta$-dependence of conduction gap for $\phi = 30^\circ$, compared to the case of $\phi = 0$ as mentioned  Again, we found the same qualitative behavior of conduction gap when applying the strains of $\{\sigma,\theta\}$ and $\{-\sigma,\theta+90^\circ\}$ 

Next, we investigate the conduction gap with respect to different transport directions $\phi$  We display a ($\theta,\phi$)-map of conduction gap for $\sigma = 4 \%$ in Fig  6 and, in the top, an additional diagram illustrating the rotation of Dirac points in the $k-$space with the change in the transport direction  It is clearly shown that (i) a similar scale of conduction gap is obtained for all different transport directions, (ii) there is a smooth and continuous shift of $E_{cond gap}-\theta$ behavior when varying $\phi$, and (iii) the same behavior of $E_{cond gap}$ is also observed when comparing the two transport directions of $\phi$ and $\phi+30^\circ$, similarly to the comparison above between $\phi = 0^\circ$ and $30^\circ$  The data plotted in Fig  6 additionally shows that $E_{cond gap}$ takes the same value in both cases of $\{\phi,\theta\}$ and $\{-\phi,-\theta\}$ with a remark that the strains of $-\theta$ and $180^\circ-\theta$ are identical  Moreover, the values of $\theta$ and $\phi$, for which the conduction gap has a peak or is equal to zero, have an almost linear relationship  In particular, the relationship for conduction gap peaks is approximately given by $\theta = \theta_A - \eta_s \phi$  For tensile strains, $\eta_s$ takes the values of $\sim 1 5667$ and $1 4333$ for $\theta_A = 0$ and $90^\circ$, respectively  On the opposite, it is about $1 4333$ and $1 5667$ for $\theta_A = 0$ and $90^\circ$, respectively, for compressive strain cases  All these features are consequences of the rotation of Dirac points in the $k$-space with respect to the transport direction $\phi$ as illustrated in the diagram on the top and the lattice symmetry of graphene 

Finally, we investigate other junctions based on compressive and tensile strained graphene sections  The idea is that in this type of strained junction, the shifts of Dirac points are different in two graphene sections of different strains, which offers the possibilities to use smaller strains to achieve a similar conduction gap, compared to the case of unstrained/strained junction  In Fig  7, we display the maps of conduction gap with respect to the directions of compressive ($\theta_c$) and tensile ($\theta_t$) strains in two cases of transport direction $\phi = 0$ (armchair) and $30^\circ$ (zigzag) for given strain strengths  Indeed, as seen in Fig  7(a,b), with smaller strains $\left\{ {{\sigma _c},{\sigma _t}} \right\} = \left\{ { - 2\% ,2\% } \right\}$ or $\left\{ { - 1\% ,3\% } \right\}$, similar conduction gap of about 310 meV can be achieved (see Figs  7(a,b)) while it requires a strain of 4 $\%$ in the unstrained/strained junctions discussed above  However, since the shift of Dirac points is strongly dependent on the direction of applied strains and the transport direction, the properties of conduction gap are more complicated than in the latter case  In particular, our calculations show that the preferred transport directions to achieve a large conduction gap are close to the armchair one  Otherwise, the conduction gap is generally smaller, similarly to the data for $\phi = 30^\circ$ compared to $\phi = 0$, as shown in Fig  7  Additionally, it is shown that the preferred directions of applied strains in the case of $\phi = 0$ are close to ${\theta _c} \equiv {\theta _t} = 0$ or $90^\circ$ 

\section{Conclusion}

Based on the tight binding calculations, we have investigated the effects of uniaxial strain on the transport properties of graphene strained junctions and discuss systematically the possibilities of achieving a large conduction gap with respect to the strain, its applied direction and the transport direction  It has been shown that due to the strain-induced deformation of graphene lattice and hence of graphene bandstructure, a finite conduction gap higher than 500 meV can be achieved for a strain of only 6 $\%$  Moreover, as a consequence of the shift of Dirac points along the $k_y$-axis, the conduction gap is strongly dependent not only on the strain strength but also on the direction of applied strain and the transport direction  A full picture of these properties of conduction gap has been presented and explained  The study hence could be a good guide for the use of this type of unstrained/strained graphene junction in electronic applications 

\textbf{\textit{Acknowledgment }} This research in Hanoi is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103 02-1012 42  We also acknowledges the French ANR for financial support under the projects NANOSIM-GRAPHENE (Grant no  ANR-09-NANO-016) and MIGRAQUEL (Grant no  ANR-10-BLAN-0304) 





ground_truth: Peaks occur at certain strain directions, while the gap is zero at others.