# Multi-task transfer learning for finding actionable information from crisis-
related messages on social media

###### Abstract

The Incident streams (IS) track is a research challenge aimed at finding
important information from social media during crises for emergency response
purposes. More specifically, given a stream of crisis-related tweets, the IS
challenge asks a participating system to 1) classify what the types of users’
concerns or needs are expressed in each tweet, known as the information type
(IT) classification task and 2) estimate how critical each tweet is with
regard to emergency response, known as the priority level prediction task. In
this paper, we describe our multi-task transfer learning approach for this
challenge. Our approach leverages state-of-the-art transformer models
including both encoder-based models such as BERT and a sequence-to-sequence
based T5 for joint transfer learning on the two tasks. Based on this approach,
we submitted several runs to the track. The returned evaluation results show
that our runs substantially outperform other participating runs in both IT
classification and priority level prediction.

##  1 Introduction

Social media platforms such as Twitter have made it possible for users to
report on an ongoing event in their vicinity in a timely manner Fraustino et
al. (2012). This has motivated researchers to explore the potential of social
media platforms for finding actionable information from this user-generated
content during a crisis event Caragea et al. (2011); Imran et al. (2015);
McCreadie et al. (2019). Finding this type of information is especially
important for emergency response agencies to enable them to take immediate
actions to help those who are posting for help, which is known as situational
awareness Vieweg (2012); Vieweg et al. (2010). This naturally raises the
question: how can the process of finding the actionable information
effectively be automated, given the fact that the messages posted during a
crisis on social media are usually noisy and numerous?

The Incident streams (IS) track McCreadie et al. (2019, 2020) is proposed by
the Text REtrieval Conference (TREC) as a research challenge for this purpose.
Since it was introduced in 2018, the IS track has conducted two major tasks
regarding crisis short message processing. Given a stream of tweets from
crisis events, the foremost task is that it asks a participating system to
classify the information types (ITs) for each tweet. The ITs are simply a pre-
defined set of classes in relation to something that a user is likely to post
during a crisis. The ITs can be something important such as requesting
research and rescue, call for moving people, reporting goods available, etc.,
as well as something less important such as reporting weather or location,
expressing sentiment, etc.111There are 6 important ITs known as “actionable”
ITs pre-defined by the IS track and 19 are considered to be “non-actionable”.
For details, see McCreadie et al. (2019). In addition to the ITs
classification task, the IS track also asks the participating systems to
estimate the priority level for each tweet, indicating how important the tweet
is in taking immediate emergency response actions. The IS track pre-defines
four priority levels: critical, high, medium and low, which are ordered from
the highest to lowest priority.

The IS track was run once in 2018 and twice in each subsequent year, so it has
accumulated five editions as of 2020. For each edition, an annotated
collection of tweets from previous editions is used as the training data for
the community, and unseen tweets (non-annotated) are released as the test
tweets for official evaluation. The two most recent editions, conducted in
2020, are named 2020A and 2020B respectively. Slightly different from previous
editions, the two editions introduce a reduced set of ITs as well as a set of
test tweets related to the COVID-19 pandemic, resulting in three tasks
described as follows.

  * •

Task 1: This task remains the same as the editions before 2020, it uses all 25
ITs for classification and four priority levels for estimation.

  * •

Task 2: Different from Task 1, this task only asks the participating systems
to classify one or more of 12 IT classes. The 12 ITs include 11 that are
closely related to emergency response and the remaining as “Other-Any” 222For
full details, refer to
http://dcs.gla.ac.uk/~richardm/TREC_IS/2020/participate.html.

  * •

Task 3: Unlike Task 1 and 2 that relate to general crises such as earthquakes,
explosions or hurricanes, this task focuses on the COVID-19 domain. It
provides a stream of COVID-related tweets from different locations for IT
classification using only a subset of 9 ITs suitable for COVID-19 and priority
estimation using the same four priority levels as used in Task 1 and 2.

In this paper, we describe our system’s approach in the three tasks of the IS
track from our participation in both 2020A and 2020B. For different tasks, we
submitted different runs but all were based on the multi-task transfer
learning approach that we utilised in our system. Given the recent success of
transformers Wolf et al. (2020) in transfer learning for various language
tasks such as sentence classification, question answering, etc., we leverage
them in the IS challenge. We explored transformer encoder based models such as
BERT Devlin et al. (2019) and a sequence-to-sequence model - T5 Raffel et al.
(2020) for their potential in this challenge. By doing so, we fine-tune them
in a multi-task learning fashion (i.e, joint fine-tuning of the IT
classification and priority estimation). With this approach, we submitted five
runs to the IS track. The evaluation results show that our runs substantially
outperform other participating runs in both IT classification and priority
level prediction.

##  2 Related Work

To improve emergency response, the community has seen many works on exploring
computational techniques for knowledge acquisition from crisis messages on
social media. Caragea et al. (2011) applied traditional machine learning
algorithms including LDA and SVM to find important information such as people
trapped or food shortage from the 2010 Haiti Earthquake. As neural network
(NN) approaches have gained popularity in recent years, many deep learning
approaches have been applied to this domain. For example, Nguyen et al. (2017)
applied a convolution neural network (CNN) for classifying informative tweets
from general disasters such as the 2015 Nepal Earthquake, Typhoon Hagupit,
etc., whereas Alam et al. (2018) leveraged a CNN with adversarial training for
identifying whether a tweet is relevant to a certain crisis event.

In recent years, since the attention-based transformer model was introduced
Vaswani et al. , several variations have been proposed such as BERT Devlin et
al. (2019), ELECTRA Clark et al. (2020) and T5 Raffel et al. (2020),
collectively known as the transformers Wolf et al. (2020), achieving state-of-
the-art performance in many language tasks with transfer learning. It is
common that the transformers are first pre-trained on a large general text
corpus and then are fine-tuned on specific downstream language tasks such as
text classification. Given the strong transfer capability of transformers,
they have been widely studied for crisis messages processing also. Liu et al.
(2020) fine-tuned BERT for crisis identification and detection tasks and Wang
and Lillis (2020b) applied T5 for extracting useful information such as who
tested positive/negative or cannot get test from COVID-related tweets by
treating it as a question-answering task. Our approach in the IS track is
similar to this line of work, which applies the transformers with transfer
learning for finding actionable information in the tasks as proposed by the IS
track. However, our approach is different in the way it fine-tunes the
transformers by multi-task learning, aiming to make use of shared model
weights between different tasks.

Since the IS track has been run for several years, the participating systems
have proposed various techniques specifically for this track. Such approaches
can broadly be summarised in three categories. First, traditional machine
learning algorithms have been used with careful pre-processing steps and
handcrafted input features. For example, Wang et al.  applied models including
Naïve Bayes, SVM, Random Forest, and the ensemble of these models. To train
these models, they used hand-crafted features such as the length, sentiment
polarity of a tweet, number of followers of the user, combining with context-
free GloVe and FastText embeddings as well as context-aware BERT embeddings as
the input features. The second category uses deep learning approaches that
pre-date the widespread adoption of transformers. For instance, Miyazaki et
al. (2019) proposed the method using label embedding with a BiLSTM model in
this track while Wang and Lillis (2020a) applied a BiLSTM network along with
pre-trained ELMo embeddings and trainable embeddings as the input features for
crisis tweet categorisation. The last category encompasses transformer-based
fine-tuning approaches. One example is that Zahera et al. (2019) fine-tuned
BERT for the multi-label ITs classification task using the training tweets
after preprocessing.

##  3 Method

Our approach is based on multi-task transfer learning through fine-tuning both
transformer encoder-based models such as BERT and sequence-to-sequence
transformers such as T5. The following details the process of the two types of
models used, which we name the encoders scenario and sequence-to-sequence
scenario respectively. Each type of model was used for both the IT
classification task and priority prediction task.

Encoders scenario: This scenario simply adds two linear projection layers on
top of transformer encoders such as BERT. Our architecture is agnostic as to
the specific transfer encoder used. One projection layer transforms the
encoder’s pooled output (namely, the [CLS] output vector of BERT) to a vector
representing the IT classes. The IT representation is then passed to the
sigmoid function that calculates the probability distribution for every IT
class. The other projection layer is used to transform the encoder’s output to
a vector representing the four priority levels. Similarly, it is then passed
to the sigmoid function, which calculates a score indicating the priority
levels as follows.

$(0.75,1]\longrightarrow\mathbf{Critical}$ (1)
$(0.5,0.75]\longrightarrow\mathbf{High}$
$(0.25,0.5]\longrightarrow\mathbf{Medium}$
$[0.0,0.25]\longrightarrow\mathbf{Low}$

In order to achieve the joint learning of both tasks, the encoder model is
fine-tuned with the loss function linearly combining the binary cross entropy
loss between the IT probability distribution and ground truths (a multi-label
classification problem) as well as the mean squared error between the
importance scores and priority ground truths (a regression problem).

Sequence-to-sequence scenario (seq2seq): This scenario is mostly motivated by
the work that applies T5 for COVID-related event extraction by treating it as
a multi-choice question answering task Wang and Lillis (2020a). We adapt it to
the IS track for multi-task transfer learning using seq2seq transformers such
as T5. Basically, the seq2seq model takes a sequence of text as the input,
known as the source sequence, and outputs the target sequence conditional on
the source sequence. Under this mechanism, the template used to construct the
source and target sequences in both tasks of the IS track is presented as
follows.

Source: context: T question: IQ/PQ choices: IC/PC

Target: I/P

  * •

T refers to the raw tweet text without any reprocessing except for being
lower-cased.

  * •

IQ/PQ refers to the IT classification and priority estimation task-specific
ad-hoc question texts, which are “what type of information does the tweet
convey relating to a crisis?” and “what level of urgency is likely expressed
in this tweet relating to a crisis?” respectively.

  * •

IC/PC implies the flatted texts concatenating all IT and priority levels
respectively. For example, IC is something like “call for donations, call to
move people, …” which varies in different IT classification tasks. The PC is
simply “critical, high, medium, low”.

  * •

I/P indicates the generated predictions for ITs and priority level, which are
direct textual predictions from IC/PC respectively.

runtag | scenario | task target | submission type | training data
---|---|---|---|---
run1 | Encoders | Task 1 & 2 | one-hot | prior to 2020B excluding COVID
run2 | Encoders | Task 1 & 2 | probability | prior to 2020B excluding COVID
run3 | seq2seq | Task 1 & 2 | one-hot | prior to 2020B excluding COVID
run4 | seq2seq | Task 3 | one-hot | prior to 2020B including COVID
run5 | seq2seq | Task 1 & 2 | one-hot | prior to 2020B including COVID
Table 1: The summary of our submitted runs for TREC-IS 2020-B. Run1, 2, 3, 5
submitted to task 1 are also submitted to task 2 for evaluation.

Using this template, each tweet in the training set is converted to an IT-
specific source-target pair and a priority-specific source-target pair. In
order to achieve the joint learning of both tasks, the sequence-to-sequence
model is fine-tuned on batches of training sequences that contain both the IT
pairs and priority pairs.

##  4 Experiments

This section describes the details of our system’s runs submitted to the
latest 2020B edition of the IS track. Since our system was developed based on
our previous experience in this track, the method we described in Section 3
also covers our approach to the 2020A edition (actually the encoders
scenario). Our baseline run (run1) for 2020B, which is an ensemble run under
the encoders scenario from 2020A that we consider as a strong baseline. In
2020B, we submitted a total of five runs to Task 1, 2 and 3 as mentioned in
Section 1 and they are summarised in Table 1 and described as follows.

  * •

run1: This is a baseline with techniques initially developed in 2019A. In
2020A, we proposed the encoders scenario, achieving strong performance as
compared to other participating techniques. To further make it a strong
baseline, we used a simple ensemble approach combining the predictions made by
the fine-tuned individual models333The individual models that were used in
this run included fine-tuned bert-base-uncased, electra-base-discriminator,
albert-basev2 and distilbert-base-uncased, which are all available in the
transformers library Wolf et al. (2020). under the encoders scenario. The
ensemble run simply predicts the final IT predictions for each tweet to be the
union of individual IT predictions and the final priority level to the highest
of the individual priority predictions. Per the guideline of 2020B, both the
IT and priority levels are expected to be numeric instead of being categorical
as required prior to 2020B. Hence, we transform the final IT predictions to
one-hot encodings and map the priority level prediction to its importance
score by: Critical: 1.0, High: 0.75, Medium: 0.5, Low: 0.25.

  * •

run2: Similar to run1, the difference is that for run2, the final ITs
predictions are the highest probability values among the predictions by
individual models. The final priority predictions are simply the highest of
the individual models’ outputs without applying the conversion as defined in
Equation 1.

  * •

run3: For this run, the seq2seq scenario is conducted for multi-task transfer
learning. We follow the T5 base architecture initialised with t5-base weights
and fine-tune it on the training tweets prior to 2020B (excluding the COVID-
related tweets from the 2020A edition). Since the seq2seq model outputs the
generated texts as the predictions for both priority and ITs, we convert the
IT predictions to one-hot encodings and priority level to the importance score
before they are submitted.

  * •

run4: With a similar setup to run3, run4 is submitted for Task 3 and thus it
includes the training tweets prior to 2020B including the COVID-related tweets
from 2020A.

  * •

run5: With a similar setup to run3, run5 is submitted for Task 1 & 2 and it
uses all previous training tweets including the COVID tweets for fine-tuning
the T5 model.

Run | nDCG@100 |  Info-Type F1 [Actionable] |  Info-Type F1 [All] |  Info-Type Accuracy |  Priority F1 [Actionable] |  Priority F1 [All]
---|---|---|---|---|---|---
BJUT-run | 0.4346 |  0.0266 |  0.0581 |  0.8321 |  0.1744 |  0.0905
njit.s1.aug | 0.4480 |  0.2634 |  0.3103 |  0.8655 |  0.2029 |  0.1518
njit.s2.cmmd.t1 | 0.4475 |  0.1879 |  0.2223 |  0.8475 |  0.2029 |  0.1518
njit.s3.img.t1 | 0.4222 |  0.1879 |  0.2223 |  0.8475 |  0.1959 |  0.1417
njit.s4.cml.t1 | 0.4164 |  0.1712 |  0.1465 |  0.8445 |  0.1054 |  0.1064
ufmg-sars-test | 0.3634 |  0.0001 |  0.0493 |  0.8337 |  0.1285 |  0.1378
ucd-run1 (ours) | 0.5033 |  0.3215 |  0.3810 |  0.8520 |  0.2582 |  0.2009
ucd-run2 (ours) | 0.5022 |  0.3078 |  0.3692 |  0.8316 |  0.2582 |  0.2016
ucd-run3 (ours) | 0.5038 |  0.3001 |  0.3448 |  0.8653 |  0.2803 |  0.3046
ucd-run5 (ours) | 0.5252 |  0.3036 |  0.3444 |  0.8601 |  0.2801 |  0.3126
Table 2: Evaluation results of participating runs at TREC-IS 2020-B Task 1. Highest in columns are bold. Run | nDCG@100 | Info-Type F1 [All] | Info-Type Accuracy | Priority F1 [All]
---|---|---|---|---
| Task-1 Systems
BJUT-run | 0.4350 | 0.0472 | 0.7977 | 0.1337
njit.s1.aug | 0.4487 | 0.3480 | 0.8846 | 0.1838
njit.s2.cmmd.t1 | 0.4467 | 0.2494 | 0.8612 | 0.1838
njit.s3.img.t1 | 0.4215 | 0.2494 | 0.8612 | 0.1708
njit.s4.cml.t1 | 0.4176 | 0.1278 | 0.8360 | 0.1162
ufmg-sars-test | 0.3630 | 0.0127 | 0.8419 | 0.1480
ucd-run1 (ours) | 0.5020 | 0.4036 | 0.8913 | 0.2320
ucd-run2 (ours) | 0.5027 | 0.3961 | 0.8364 | 0.2322
ucd-run3 (ours) | 0.5032 | 0.3689 | 0.8932 | 0.2867
ucd-run5 (ours) | 0.5240 | 0.3674 | 0.8845 | 0.3003
| Task-2 Systems
njit.s1.aug.t2 | 0.4478 | 0.2548 | 0.8656 | 0.1838
njit.s2.cmmd.t2 | 0.4478 | 0.2548 | 0.8656 | 0.1838
njit.s3.img.t2 | 0.4213 | 0.2548 | 0.8656 | 0.1708
njit.s4.cml.t2 | 0.4189 | 0.1713 | 0.8327 | 0.1162
ufmg-sars-test-t2 | 0.3637 | 0.0127 | 0.8419 | 0.1480
Table 3: Evaluation results of participating runs at TREC-IS 2020-B Task 2.
The Task-1 systems refer to the runs from Task 1 re-evaluated under Task 2
while Task-2 systems are the submitted runs specific to Task 2.

###  4.1 Training Details

As described, our runs mainly focus on fine-tuning several transformer encoder
models and a t5-base sequence-to-sequence model in a multi-task learning way.
For the fine-tuning of t5-base, we follow the same hyper-parameter
configuration as used in Wang and Lillis (2020b). For fine-tuning each of the
transformer encoder models, we use the same set of the hyper-parameters that
are configured with reference to a similar work in this domain Liu et al.
(2020). For training, we sample around 10% of the training data as the
validation set first. Then, we fine-tune each model with a batch size of 32,
learning rate of 5e-5, linear warm-up ratio of 0.1 with Adam optimizer Kingma
and Ba (2015). For the input length, we set the maximum input length to be 256
since we found few examples has length beyond this number. All training
examples in our experiments are not pre-processed but used in raw texts.

###  4.2 Results

(a) F1 Scores by Information Type

(b) Priority level prediction F1 per information type

Figure 1: Performance visualisation by information types of ucd-run1 in Task 1. Run | nDCG@100 |  Info-Type F1 [Actionable] |  Info-Type F1 [All] |  Info-Type Accuracy |  Priority F1 [Actionable] |  Priority F1 [All]
---|---|---|---|---|---|---
njit.s1.aug.t3 | 0.4322 |  0.1629 |  0.1450 |  0.8593 |  0.2551 |  0.1499
njit.s2.cmmd.t3 | 0.4329 |  0.1590 |  0.1184 |  0.8586 |  0.2551 |  0.1499
njit.s3.img.t3 | 0.3986 |  0.1590 |  0.1184 |  0.8586 |  0.2544 |  0.1562
njit.s4.cml.t3 | 0.4249 |  0.0210 |  0.0650 |  0.8626 |  0.1375 |  0.1502
ucd-run4 | 0.4497 |  0.1425 |  0.1817 |  0.8541 |  0.3443 |  0.2867
Table 4: Evaluation results of participating runs at TREC-IS 2020-B Task 3.

Having submitted the five runs as described in Table 1 to the track, they were
officially evaluated and the results are reported in Tables 2, 3 and 4. The
tables show the performance of participating runs in Task 1, 2 and 3
respectively. The columns are the official metrics used to evaluate different
aspects of a run’s performance, which are described briefly as follows.

  * •

Information type classification: There are two types of information type (IT)
F1. The “Actionable IT” F1 reflects a run’s performance in classifying
actionable ITs 444They are Request-GoodsService, Request-SearchAndRescue,
Report-NewSubEvent, Report-ServiceAvailable, CallToAction-MovePeople, and
Report-EmergingThreats.. The “All IT” F1 measures a run’s performance across
all information types (25 in Task 1, 12 in Task 2 and 9 in Task 3). The IT
accuracy is the overall accuracy in IT classification.

  * •

Prioritisation: Similarly, the Actionable priority F1 measures a run’s
performance in priority level prediction for only the tweets that are labeled
as actionable ITs while the All F1 measures the performance for all test
tweets. Moreover, the nDCG@100 is used to measure a run’s average performance
in ranking top 100 test tweets per event by priority.

As seen from Table 2, in Task 1, our runs substantially outperform other
participating runs in both IT classification and prioritisation 555The
exception if accuracy, where only a small difference is observed accuracy
across the participating runs: our results are substantially higher than other
participating runs in the remaining metrics.. In particular, our runs are
effective in classifying actionable ITs. For example, our run1 and run3
achieve the top actionable IT F1 score of 0.3215 and the best actionable
priority F1 of 0.2803 respectively. This is further evidenced by the runs’
performance in Task 2, as in Table 3. All the runs overall perform well in IT
classification and prioritisation in Task 2 (the condensed more emergency
response related 12 ITs).

In Task 1 and 2, run1 and run2 perform similarly across the metrics since both
are based on the encoder scenario and only differ in the final submission
type. It is interesting that run5 performs similarly to run3 across the
metrics except for being better in nDCG@100: 0.5252 versus 0.5038. The two
runs are both based on the seq2seq scenario and only difference is in their
training data. This indicates that adding the COVID data (similar domain) to
the general crisis data for training can be helpful in the priority-centric
ranking performance. To compare between the four runs, it is found that no one
run dominates the other runs across all the metrics. This indicates that the
multi-task transfer learning approach using either the transformer encoder or
the seq2seq as the base model is likely to bring similar performance.

To further examine our runs’ performance at every IT level, we report the IT
F1s and priority F1s per IT of the run1 in Task 1, as presented in Figure 1.
Figure 1(a) shows that the run performs well in categorising some actionable
ITs, such as “CallToAction-MovePeople” and “Report-EmergingThreats” while not
the best in actionable ITs such as “Request-GoodsService”, as compared to the
non-actionable ITs. However, taking a look at the priority F1s per IT in
Figure 1(b), we found that the run performs relatively better in priority
level prediction for actionable ITs than non-actionable ITs, where
“CallToAction-MovePeople”, “Request-GoodsService” and “Report-
ServiceAvailable” are the top 3 the runs achieve in priority F1.

Apart from the four runs to Task 1 and Task 2, we submitted run4 to Task 3 and
the results are reported in Table 4. We see that the run is competitive with
other participating runs, particularly in prioritisation. Unlike our other
four runs in Task 1 and 2, this run achieves 0.1425 in actionable IT F1, next
to the best 0.1629. Since Task 3 is COVID-related and newly introduced, we
expect our run to be improved in future iterations of this track as more data
accumulates.

##  5 Conclusion

This paper introduces University College Dublin’s (UCD) participation in the
2020 TREC-IS track. The IS track was run twice in 2020: namely 2020A and
2020B. Based on our experience from previous editions, we describe our multi-
task transfer learning approach using pre-trained encoder-based and sequence-
to-sequence transformers. With these approaches, we submitted five runs to the
track’s 2020-B edition - four for Task 1 and Task 2, and one for Task 3. The
results show that our runs to Task 1 and Task 2 substantially outperform other
participating runs in both information type classification and priority level
prediction. In addition, our runs are effective in finding some actionable
information types in Task 1 and Task 2 and the run to Task 3 performs
competitively with other participating runs. Regarding future work, we expect
to explore the incorporation of knowledge graphs to enhance the model’s
identification of the crisis-related tweets.

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# Vortex Lattices in Active Nematics with Periodic Obstacle Arrays

Cody D. Schimming<EMAIL_ADDRESS>Theoretical Division and Center for
Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico,
87545, USA  C. J. O. Reichhardt  Theoretical Division and Center for Nonlinear
Studies, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA
C. Reichhardt  Theoretical Division and Center for Nonlinear Studies, Los
Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA

###### Abstract

We numerically model a two-dimensional active nematic confined by a periodic
array of fixed obstacles. Even in the passive nematic, the appearance of
topological defects is unavoidable due to planar anchoring by the obstacle
surfaces. We show that a vortex lattice state emerges as activity is
increased, and that this lattice may be tuned from “ferromagnetic” to
“antiferromagnetic” by varying the gap size between obstacles. We map the rich
variety of states exhibited by the system as a function of distance between
obstacles and activity, including a pinned defect state, motile defects, the
vortex lattice, and active turbulence. We demonstrate that the flows in the
active turbulent phase can be tuned by the presence of obstacles, and explore
the effects of a frustrated lattice geometry on the vortex lattice phase.

Active nematics are a class of active fluids whose microscopic constituents
are anisotropic and exert dipolar forces [1, 2]. Examples of such systems
include cytoskeletal filaments with molecular motors [3], cellular tissues
[4], suspensions of bacteria in nematic liquid crystals [5, 6], and soil
bacteria [7]. One of the key features of active nematics is that in many cases
the dynamics can be described by the motion of defects with different
topological charges [3, 8, 9, 10]. There is growing interest in developing
methods to control the defect dynamics and flows produced by active nematics,
such as by applying external fields [11], introducing anisotropic substrates
[12, 13] or imposing a geometric confinement of the material [14, 15].

In systems with well-defined length scales such as the average distance
between topological defects, new ordered phases can arise upon coupling to a
periodic substrate [16], as observed for vortices in superconductors [17] and
Bose-Einstein condensates [18] coupled to periodic arrays, ordering of
surfaces [19], cold atom systems on optical substrates [20] and colloids
coupled to ordered substrates [21, 22]. Active nematics are another system in
which commensuration effects can arise; however, due to their nonequilibrium
nature, it should be possible for dynamical commensuration effects to appear
as well. Here, we examine the effects of geometric confinement on active
nematics induced by a periodic array of fixed obstacles for a varied range of
activity levels and obstacle sizes. Previous theoretical and experimental
studies of active nematic confinement have typically employed an external
boundary as a confining structure, resulting in a channel, circular disk, or
annular geometry [23, 24, 25, 26, 14, 27, 28, 29, 30]. As the system size and
active force magnitude is varied, such systems can exhibit anomalous flow
states such as dancing topological defects or system-wide circulation [24, 25,
26, 14]. Relatively few studies have addressed the effects of fixed, embedded
obstacles on the active nematic flow state; however, recent experiments have
successfully produced active nematics in obstacle laden environments [31, 32].

Motivated by the recent experimental work, we use a minimal, active nematic
continuum model to simulate an array of obstacles that separates the system
into interacting circular domains where the obstacle shape makes the formation
of topological defects unavoidable. We show that a variety of phases appear,
including a low active force state where the defects remain pinned to the
obstacles, a state where the defects are motile, and an intermediate activity
state where the flow organizes into a lattice of vortices. By varying the size
of the obstacles, we can tune the vortex lattice from “ferromagnetic,” in
which the vortices are all of the same chirality, to “antiferromagnetic,” in
which each nearest neighbor vortex pair is of opposite chirality. To our
knowledge, this is the first report of a ferromagnetic vortex lattice state in
active nematics. We compare the active turbulent phase in systems with and
without obstacles and find that the fluid flow slows as the obstacles increase
in size, while the directional distribution of the fluid velocity becomes
peaked along diagonal lattice directions. Finally, we explore the effects of
lattice frustration by simulating the active nematic on a triangular lattice.
Our findings provide an experimentally viable method for controlling and
tuning vortex lattices and flows in active nematics.

We model a two-dimensional active nematic using a well-documented continuum
model that has been shown to capture the key features of experimental active
nematics [33, 2]. In our dimensionless equations, presented in detail in the
supplemental material [34], lengths are measured in units of the nematic
correlation length $\xi$ and times are measured in units of the nematic
relaxation time $\sigma$. The nematic state is captured by the tensor order
parameter
$\mathbf{Q}=S\left[\mathbf{n}\otimes\mathbf{n}-\left(1/2\right)\mathbf{I}\right]$
where $S$ is the scalar order parameter indicating the local degree of
alignment and $\mathbf{n}$ is the director, giving the local direction of
orientation. The evolution of $\mathbf{Q}$ is given by

$\frac{\partial\mathbf{Q}}{\partial
t}+\left(\mathbf{v}\cdot\nabla\right)\mathbf{Q}-\mathbf{S}=-\frac{\delta
F}{\delta\mathbf{Q}}$ (1)

where $F$ is the Landau-de Gennes free energy with one elastic constant [35],
$\mathbf{S}$ is a generalized tensor advection [36], and $\mathbf{v}$ is the
fluid velocity. The free energy is such that the passive liquid crystal is in
the nematic phase. We assume low Reynolds number flow, so the fluid velocity
is given by the Stokes equation:

$\nabla^{2}\mathbf{v}=\nabla
p+\alpha\nabla\cdot\mathbf{Q},\qquad\nabla\cdot\mathbf{v}=0$ (2)

where $p$ is the fluid pressure, and the last term is an addition to the usual
Stokes equation that models an active force of dimensionless strength
$\alpha$. The divergence free condition models incompressible flows.

Equations (1) and (2) are discretized in space and time and solved using the
MATLAB/C++ package FELICITY [37, 34]. We consider domains with an array of
astroid shaped obstacles, as shown in Fig. 1(a). The obstacles may be thought
of as separating circular domains that overlap over a distance $d$ between
obstacles. In this study, we vary $d$ from $1$ to $10$ while fixing the
distance between centers of obstacles to $a=14$ so that the overall system
size remains constant. The limit $d=0$ gives individual, non-interacting
circular domains, while $d=a=14$ indicates a system with no obstacles. We
employ periodic boundary conditions on the side edges of the domain, while for
the obstacles we impose strong planar anchoring for $\mathbf{Q}$ and no-slip
conditions for $\mathbf{v}$. The shape of the inclusions and the strong planar
anchoring necessitate the formation of topological defects, or points where
the nematic director is singular. Specifically, there must be a total
topological charge (winding number) of $+N$ for $N$ obstacles because each
obstacle carries a charge of $-1$ that must be accounted for in the nematic.
Therefore, topological defects inherently exist in the system even in the
passive state with $\alpha=0$.

Figure 1: (a) Schematic of the computational domain with a periodic array of
astroid shaped obstacles summarizing the boundary conditions on the obstacles
and domain edges. (b,c) “Ferromagnetic” vortex lattice state at $d=2$ and
$\alpha=1.8$. (b) Nematic scalar order parameter $S$ where white lines
indicate the director $\mathbf{n}$. The points where $S=0$ are topological
defects. (c) Fluid vorticity with black arrows showing the velocity field. (d)
$S$ and (e) $\omega/\omega_{\rm max}$ for an “antiferromagnetic” vortex
lattice state at $d=6$ and $\alpha=1.5$.

Upon increasing $\alpha$ for various obstacle sizes, we typically find three
qualitative transitions. At zero and small activity, the topological defects
are pinned to the obstacles. As the activity is increased, the defects begin
to unpin and move from obstacle to obstacle with little to no unbinding of new
defects. When activity is further increased, a central vortex forms in each
circular domain with two $+1/2$ topological charge defects encircling one
another, as has been shown previously for individual circular domains [26, 14,
30]. At higher activities the $+1/2$ defects merge and a stable $+1$ spiral
defect forms in the domain center. Figure 1(b–e) shows examples of the nematic
order parameter and velocity field in the vortex lattice state, which can have
either ferromagnetic or antiferromagnetic order. At still higher activities
the central vortex in each domain is no longer stable and an active turbulent
phase persists in which defects are constantly unbinding and annihilating (see
Supplemental Movie 1).

To better quantify the transitions discussed above, we map our system to a
lattice of vortex “spins” by measuring the average vorticity in a circle of
diameter $a/2$ at the center of each circular domain. The spins are then
indexed by their lattice position $s_{i}$. Figure 2(a–c) shows plots of
$\langle|s_{i}|\rangle_{i,t}$, where $\langle\cdot\rangle_{i,t}$ denotes an
average over lattice sites and time, as a function of activity for obstacle
gaps $d=4$, $d=6$, and $d=8$. The transition to the vortex lattice state is
marked by a jump followed by a linear increase in
$\langle|s_{i}|\rangle_{i,t}$. A second transition is marked by an abrupt
decrease in $\langle|s_{i}|\rangle_{i,t}$ which remains roughly constant. This
is the active turbulent phase.

Figure 2: (a–c) Average vortex “spin” $\langle|s_{i}|\rangle_{i,t}$ as a
function of activity $\alpha$ for obstacle gaps (a) $d=4$, (b) $d=6$, and (c)
$d=8$. Colors indicate the phase identity. (d) Phase diagram as a function of
$d$ vs $\alpha$, where dot color indicates the value of the time-averaged
spin-spin correlation function $\langle\chi\rangle_{t}$.
$\langle\chi\rangle_{t}$ is not well-defined in the pinned defect or motile
defect phases.

We first focus on the vortex lattice phase. As shown in Fig. 1(b–e), vortex
lattices appear with either “ferromagnetic” or “antiferromagnetic” order. To
quantify this order we measure the spin-spin correlation function

$\chi=\frac{\sum_{\left\langle i,j\right\rangle}s_{i}s_{j}}{\sum_{\left\langle
i,j\right\rangle}|s_{i}s_{j}|}$ (3)

where $\sum_{\left\langle i,j\right\rangle}$ denotes a sum over nearest
neighbor pairs. For perfectly ferromagnetic order $\chi=1$ while for perfectly
antiferromagnetic order $\chi=-1$. In Fig. 2(d) we plot the boundaries of the
phases as well as $\langle\chi\rangle_{t}$ as a function of activity and
obstacle gap for systems in both the vortex lattice and active turbulence
regimes. We do not define $\chi$ for small activities since the central
vortices, and hence spins, are not well-established in the pinned defect and
motile defect phases. As the obstacle gap increases, there is a window of
activity values where the vortex lattice abruptly transitions from
ferromagnetic to antiferromagnetic. In this window, the obstacle size can be
used to tune the state of the vortex lattice. For large obstacle gaps the
vortex lattice phase disappears and only active turbulence occurs. At the
transition from vortex lattice to active turbulence, $\langle\chi\rangle_{t}$
decreases in size but typically maintains the same sign, indicating that
vestigial ferromagnetic or antiferromagnetic order persists until $\alpha$
becomes large enough that $\langle\chi\rangle_{t}\to 0$.

In previous work on active nematics, a one-dimensional antiferromagnetic
vortex lattice was observed in channel confinement geometries [24], while
numerical predictions indicate that a two-dimensional antiferromagnetic vortex
lattice should appear in systems with large enough substrate friction [38]. In
our system, we assume zero substrate friction so the vortices are stabilized
purely by geometric confinement, but we expect the vortex lattice phase to be
stable against inclusion of some substrate friction based on recent work on
circularly confined systems [30]. We find both ferromagnetic and
antiferromagnetic vortex lattices depending on the values of $d$ and $\alpha$,
and to our knowledge, a ferromagnetic vortex lattice has not previously been
observed or predicted in active nematics.

Vortex lattices can also form in bacterial suspensions confined by pillar
arrays, but in that system, hydrodynamic interactions are the dominant
ordering mechanism, and therefore the vortex lattice typically has
antiferromagnetic ordering [39, 40]. In contrast, in the active nematic
considered here, elastic forces play the dominant role in ordering. As shown
in Fig. 1(b), since the central $+1$ defect that forms is of spiral type, the
elastic energy between circular domains is minimized if each domain has the
same chirality. Thus, elastic interactions promote ferromagnetic order.

The transitions to antiferromagnetic order and active turbulence may be
explained by a competition between elastic and active forces and a hierarchy
of length scales. The active nematic length scale $\xi_{a}\propto
1/\sqrt{\alpha}$ sets the defect density of a bulk active nematic system [2].
For a given obstacle size, there is an effective length scale associated with
the circular domains $R_{\rm eff}$. If $\xi_{a}\lesssim R_{\rm eff}$, there is
enough space to nucleate defects and reach the optimal defect density, so the
system transitions to an active turbulent state. There is, however, another
length scale associated with the size of the obstacles: the obstacle gap $d$.
In Fig. 1(d) the antiferromagnetic vortex lattice contains extra $\pm 1/2$
defect pairs that sit in the obstacle gaps and mediate the change in chirality
between central vortices. If $\xi_{a}\gtrsim d$, a defect pair will not be
stable in the gap. On the other hand, if $d\gtrsim\xi_{a}\gtrsim R_{\rm eff}$,
the vortex lattice phase is stable since an extra defect pair can nucleate in
the gaps to mediate the antiferromagnetic vortex order. We note that if
$d<R_{\rm eff}$, this hierarchy of length scales cannot occur, explaining why
we do not observe antiferromagnetic vortex order for small $d$.

We now turn our attention to the active turbulent phase that occurs for large
$\alpha$. As mentioned above, the active turbulent regime is associated with a
sharp decrease in $\langle|s_{i}|\rangle_{i,t}$ (Fig. 2). We can also measure
the average number of defects $\langle N_{D}\rangle_{t}$ to detect the
transition (see the Supplemental Material for details on how this and other
measures are computed [41, 42, 34]). In the vortex lattice phase, $\langle
N_{D}\rangle_{t}$ is roughly constant, while in active turbulence $\langle
N_{D}\rangle_{t}$ grows linearly with $\alpha$ (Fig. S1). Both measures are
consistent with one another in marking the transition and we use them to
determine the boundary shown in Fig. 2 between the motile defect phase and
active turbulence for large $d$.

Figure 3: Flow velocity distributions in the active turbulent phase with
$\alpha=1.5$ and various obstacle gap sizes. (a–c) Distributions of
$|\mathbf{v}|$ for (a) a bulk system with no obstacles, (b) obstacle gap
$d=10$, and (c) obstacle gap $d=4$. (d–e) Velocity direction distributions
$p(\theta_{v})$ for (d) a bulk system with no obstacles, (e) obstacle gap
$d=10$, and (f) obstacle gap $d=4$.

It is instructive to compare the active turbulent phase of the obstacle array
system with the $d=a$ bulk system free of obstacles. While the flows in all
systems become decorrelated over long time scales (Fig. S2), the flow velocity
distributions $p(|{\bf v}|)$ vary. In Fig. 3(a–c) we plot $p(|\mathbf{v}|)$
for systems with $\alpha=1.5$ at $d=14$ (the bulk system), $d=10$, and $d=4$.
Figure 3(a) indicates that $p(|{\bf v}|)$ has a two-dimensional Maxwell-
Boltzmann distribution with a maximum weight that shifts toward $|{\bf v}|=0$
as the obstacle size increases and $d$ becomes smaller. This is a natural
consequence of the fact that the obstacles have no-slip conditions and their
surface area increases as $d$ decreases. The corresponding velocity direction
distributions $p(\theta_{v})$ in Fig. 3(d–f), where
$\theta_{v}=\tan^{-1}({v_{y}/v_{x}})$, show an isotropic distribution for
small obstacles ($d=10$) that is nearly identical to $p(\theta_{v})$ for the
bulk system. For larger obstacles, $p(\theta_{v})$ becomes anisotropic and
peaks along the lattice diagonals. These velocity statistics suggest that
immersed obstacles can provide control over the flows even in the active
turbulent phase, which could contribute to the development of novel
microfluidic devices composed of active fluids.

Finally, we simulate the active nematic in a domain constrained by a honeycomb
lattice of concave, triangular obstacles similar to those used in recent
experiments [31]. Here, the obstacles introduce a triangular lattice of
circular domains. An example nematic configuration and flow profile is shown
in Fig. 4(a,b). In order to promote antiferromagnetic ordering, we place the
system just outside the ferromagnetic vortex lattice phase for this geometry:
$a=8$, $d=4$, and $\alpha=2$; however, the frustration in the lattice prevents
the emergence of antiferromagnetic vortex order. Instead we find a primarily
ferromagnetic state in which the competition between elastic and active forces
results in constantly flipping spins and the formation of a dynamical state
similar to active turbulence. In Fig. 4(c) we plot $\chi(t)$ over the course
of a simulation. There are multiple time intervals where $\chi=1$, indicating
a perfectly ferromagnetic vortex lattice; however, spin flips generated by the
unbinding of new defects constantly reduce $\chi$, which sometimes becomes
negative. While the dynamics resemble active turbulence, we argue that they
are actually closer to those of the vortex lattice state in the square lattice
with $d=6$ and $\alpha=0.9$, in which the underlying vortex lattice order is
ferromagnetic but defect unbinding pushes the system towards antiferromagnetic
order. We show in Fig. S3 that the $\chi$-$\chi$ temporal autocorrelation
function for the frustrated triangular lattice is similar to that of the
square lattice with $d=6$ and $\alpha=0.9$. In both systems, but unlike an
active turbulent system, $\chi$ is correlated over long times, indicating that
spins and spin flips are also correlated over time. In Fig. 4(d) we show that
$p(\theta_{v})$ peaks at the six diagonals of the lattice. We expect the peaks
to become more prominent as the obstacle size increases.

Figure 4: Active nematic system in a honeycomb lattice of concave triangular
obstacles. (a) Time snapshot of the nematic configuration at $a=8$, $d=4$, and
$\alpha=2$ with color given by the scalar order parameter $S$ and white lines
indicating the nematic director $\mathbf{n}$. (b) The corresponding vorticity
(color) and flow velocity (black arrows). (c) Spin-spin correlation function
$\chi$ vs time $t$. (d) Velocity direction distribution $p(\theta_{v})$.

Conclusion— We have numerically studied the effects on active nematics of
fixed periodic astroid shaped obstacles with planar nematic anchoring. As a
function of activity and obstacle size, we find a wide variety of phases,
including a pinned defect phase, motile defect regime, and a vortex lattice
phase that can be tuned from ferromagnetic to antiferromagnetic. There is an
active turbulent phase that displays unique anisotropic velocity distributions
at high activities, suggesting a new method to control active turbulence with
obstacles. We also find that an antiferromagnetic vortex lattice on a
triangular lattice exhibits an active frustrated state. Our system should be
experimentally realizable using existing approaches for creating obstacles in
active nematic systems [31, 32]. Future directions are to consider obstacle
shapes that stabilize other kinds of topological defects. Also, different
lattices may yield even more exotic flow states, which opens the prospect of
flow control in active nematics using obstacles.

###### Acknowledgements.

This work was supported by the U.S. Department of Energy through the Los
Alamos National Laboratory. Los Alamos National Laboratory is operated by
Triad National Security, LLC, for the National Nuclear Security Administration
of the U.S. Department of Energy (Contract No. 89233218CNA000001).

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