# Task Arithmetic can Mitigate Synthetic-to-Real Gap in
Automatic Speech Recognition

Hsuan Su♡ Hua Farn♡ Fan-Yun Sun♢ Shang-Tse Chen♡ Hung-yi Lee♡

♡National Taiwan University ♢Stanford University

###### Abstract

Synthetic data is widely used in speech recognition due to the availability of
text-to-speech models, which facilitate adapting models to previously unseen
text domains. However, existing methods suffer in performance when they fine-
tune an automatic speech recognition (ASR) model on synthetic data as they
suffer from the distributional shift commonly referred to as the synthetic-to-
real gap. In this paper, we find that task arithmetic is effective at
mitigating this gap. Our proposed method, SYN2REAL task vector, shows an
average improvement of 10.03% improvement in word error rate over baselines on
the SLURP dataset. Additionally, we show that an average of SYN2REAL task
vectors, when we have real speeches from multiple different domains, can
further adapt the original ASR model to perform better on the target text
domain.

Task Arithmetic can Mitigate Synthetic-to-Real Gap in
Automatic Speech Recognition

Hsuan Su♡ Hua Farn♡ Fan-Yun Sun♢ Shang-Tse Chen♡ Hung-yi Lee♡ ♡National Taiwan
University ♢Stanford University

##  1 Introduction

Existing automatic speech recognition (ASR) models have been found to lack
generalizability towards domains unseen during training Bartelds et al.
(2023); Radford et al. (2022); Sundar et al. (2023). Existing works, when
adapting an ASR model to a previously unseen domain, often rely on synthetic
speech data Su et al. (2024); Bataev et al. (2023); Joshi and Singh (2022);
Zheng et al. (2021); Yuen et al. (2023); Yang et al. (2023) due to its ease of
generation and availability. However, this approach often leads to performance
degradation due to acoustic mismatches such as intonations, background noise,
speaker accents, and environmental sound differences between synthetic and
real speech Su et al. (2024). This distributional shift is often referred to
as the synthetic-to-real gap. This paper tackles this problem, particularly
when adapting an ASR model from a source domain with text and real speech data
to a new target domain with only text data.

Our idea is that paired synthetic speech and real speech data within a single
domain can guide the adaptation of models trained on synthetic data to perform
better on real-world data in a new domain. Inspired by the new paradigm of
editing pre-trained neural networks by manipulating their weightsSung et al.
(2023); Tam et al. (2024), we propose to bridge the synthetic-to-real gap
using task vectors Huang et al. (2024); Bhardwaj et al. (2024). A task vector
is a representation that encodes the difference between two tasks, allowing
models to perform arithmetic operations to transition from one task to
another. In this paper, we show that we can apply simple arithmetic operations
to bridge the synthetic-to-real gap.

Figure 1: Overview of the SYN2REAL Task Vector Approach. The pre-trained model
is fine-tuned on source domain synthetic and real speech data, separately. The
difference between their parameters forms the SYN2REAL task vector. The
SYN2REAL task vector is then added to a model fine-tuned on target synthetic
data to overcome the synthetic-to-real gap.

Taking inspiration from Ilharco et al. (2023), we propose SYN2REAL task vector
for synthetic-to-real adaptation in ASR. Figure 1 provides an overview of the
SYN2REAL task vector approach. The top row illustrates the process of fine-
tuning models on synthetic and real speech data separately and then deriving
the SYN2REAL task vector from the differences in their parameters. The bottom
row demonstrates the application of this vector to a model fine-tuned on
synthetic target domain data, resulting in an adapted model with improved
performance by incorporating the acoustic characteristics of real speech.

Empirically, we conduct comprehensive experiments and ablation studies to
demonstrate the effectiveness of our approach. Applying the SYN2REAL task
vector results in an average improvement of 10.03% in word error rate (WER)
for unseen target domains compared to the model before applying our method.
Cosine similarity analysis of task vectors generated by different TTS systems
confirms that SYN2REAL vectors effectively capture domain-specific acoustic
information.

##  2 Related Works

#### ASR Text-only Domain Adaptation

In the context of automatic speech recognition (ASR), "text-only" domain
adaptation typically refers to scenarios where the target domain only provides
text data for training or fine-tuning the models. Previous work has explored
internal language models adaptation that finetune language models in ene-to-
end ASR models with CTC loss to improve the generalizability Chen et al.
(2023); Sato et al. (2022); Vuong et al. (2023).

The other direction is to adapt ASR models with synthetic speech. Zheng et al.
(2021) develop a method that provides synthetic audio for out-of-vocabulary
(OOV) words to boost recognition accuracy. Yang et al. (2023) works on
personalize ASR with synthetic speech. Bataev et al. (2023) focuses on
developing a mel-spectrogram generator to improve ASR models.

#### Task Arithmetic

The concept of task vector is introduced in Ilharco et al. (2023). Task
vectors are created by subtracting the weights of a fine-tuned model from
those of its corresponding pre-trained model. Different task vectors derived
from the same pre-trained models can then be adjusted and combined through
these simple arithmetic operations such as addition and subtraction to achieve
multi-task learning Zhang et al. (2023) and task forgetting Daheim et al.
(2023).

Recently, task vectors have shown promise in natural language processing (NLP)
Huang et al. (2024); Daheim et al. (2023); Bhardwaj et al. (2024); Zhang et
al. (2023). Daheim et al. 2023 used a task vector from a negatively fine-tuned
model to mitigate hallucinations. Zhang et al. (2023) proposed combining
parameter-efficient fine-tuning (PEFT) modules Hu et al. (2022); Liu et al.
(2022) arithmetically. Huang et al. (2024) obtained the Chat Vector by
subtracting the chat version of Llama 2 Touvron et al. (2023) from its pre-
trained version, enhancing dialogue capabilities and safety. Bhardwaj et al.
(2024) introduced RESTA, adding a safety vector to re-align safety for models
fine-tuned on downstream tasks. The application of task vectors is relatively
underexplored in ASR.Ramesh et al. (2024) applied task arithmetic to ASR
models and introduced a "task analogy" formulation, improving performance on
low-resource tasks using models trained on high-resource tasks. Unlike Ramesh
et al. (2024), we focus on using task vector to mitigate the distributional
shift between real and synthetic data.

##  3 Methodology

In this work, we aim to adapt ASR models from source domains with real speech
and text data to a new domain with only text data. More specifically, we adapt
ASR models using data synthesized from off-the-shelf text-to-speech (TTS)
models. However, ASR models often perform poorly due to acoustic differences
between synthetic data generated by TTS and real speech data. To overcome this
limitation, we introduce the SYN2REAL task vector, a novel approach that
bridges the gap between the acoustic characteristics of synthetic and real
speech data.

###  3.1 Problem Formulation

We assume a problem setting in which we have two domains: a source domain
$D_{s}$ and a target domain $D_{t}$. The source domain $D_{s}$ consists of
paired text and speech samples, denoted as $T_{s}$ and $S_{s}$, respectively.
The target domain $D_{t}$ contains only text data, denoted as $T_{t}$. This
problem setting is common as it is easy to generate synthetic text data,
whereas collecting paired real speech data is labor-intensive.

###  3.2 SYN2REAL Task Vector

To adapt ASR models to a previously unseen domain, we employ a common
methodology Joshi and Singh (2022) that utilizes synthetic data generated from
the target text $T_{t}$ for model adaptation.

Figure 2: Framework for SYN2REAL task vector in Domain Adaptation for ASR. The
framework illustrates the process of creating the SYN2REAL task vector by
subtracting the parameter differences between a model fine-tuned on synthetic
speech (Source Synthetic) and a model fine-tuned on real speech (Source Real)
from pretrained ASR (PASR). This task vector is then applied to the target
synthetic domain (Target Synthetic) to improve ASR performance by bridging the
gap between synthetic and real speech data.

Previous work in task arithmetic has demonstrated that vectors can encode
distinct capabilities, such as language or domain-specific features. We
hypothesize that the differences in acoustic properties between real and
synthetic speech are also learnable and can be isolated through parameter
arithmetic. Specifically, we assume that we have models fine-tuned on real and
synthetic data from the source domain, denoted as $\theta_{real}^{S}$ and
$\theta_{syn}^{S}$ respectively. The acoustic disparity between real and
synthetic speech is quantified by subtracting the parameter sets of these
models:

$\tau=\theta_{real}^{S}-\theta_{syn}^{S}$ (1)

Once the SYN2REAL vector $\tau$ is computed, we apply it to the model
parameters fine-tuned on the synthetic data in the target
domain$\theta_{syn}^{T}$, thereby enhancing its adaptation to the target
domain:

$\theta_{syn\\_new}=\theta_{syn}^{T}+\lambda\tau$ (2)

Where $\lambda$ is the scaling factor of SYN2REAL task vector.

This adjusted model, $\theta_{syn\\_new}$, is expected to perform more
robustly in the target domain as it incorporates the acoustic characteristics
of real speech, making it better suited for practical ASR tasks where real
speech is present.

####  SYN2REAL Ensemble

In the previous discussion, we assume no access to domain labels such as
’email’ or ’music’ from the source domain to emulate real-world situations
better. All real speech data falls in the source domain. In scenarios where we
have access to data of multiple domains, another approach is to create
SYN2REAL task vectors for each domain separately and then combine these
vectors. This method involves fine-tuning separate ASR models on each
individual source domain to obtain domain-specific SYN2REAL task vectors,
which are then averaged to form a comprehensive SYN2REAL task vector.

For each domain $i$ in source domain $S$. The task vector can be defined as:

$\tau_{i}=\theta_{real}^{S_{i}}-\theta_{syn}^{S_{i}}$ (3)

Where $\theta_{real}^{S_{i}}$ and $\theta_{syn}^{S_{i}}$ represent the model
parameters fine-tuned on real and synthetic data for domain $i$ respectively.
Once the vector $\tau_{i}$ is computed, we apply it to the model parameters
fine-tuned on synthetic target domain data $\theta_{syn}^{T}$, thereby
enhancing its adaptation to the target domain:

$\theta_{syn\\_new}=\theta_{syn}^{T}+\frac{\lambda}{|S|}\sum_{i=0}^{|S|}\tau_{i}$
(4)

Where $|S|$ is the number of source domains, and $\lambda$ is the scaling
factor for the task vector.

##  4 Experimental Setups

We design our experiments to answer the following questions: Q1: What is the
efficacy of SYN2REAL task vector?, Q2: How does SYN2REAL task vector perform
across different model sizes? Q3: Is SYN2REAL task vector effective on ASR
models other than Whisper?, Q4: Can we form SYN2REAL task vectors from other
TTS models?, Q5: What is the impact of the scaling factor $\lambda?$ , Q6: Do
SYN2REAL task vectors obtained with the same TTS have similar direction?

To answer the questions above and to mimic real-world use cases, we first form
a source domain ASR model by compiling synthetic and real speech data from
various domains. We then adapt this source domain ASR model to the target
domain using data synthesized by TTS models. SYN2REAL task vector is
constructed by subtracting the weights of an ASR model fine-tuned on synthetic
data from the weights of the same ASR model fine-tuned on real data, both
using the same pre-trained model as the starting point. Our goal is to improve
the performance of an ASR model on the target domain without using any real
speech from the target domain.

WER Target Domains Average Methods Alarm Audio Calendar Cooking Datetime Email
General IOT Lists Music News Play QA Recommendation Social Takeaway Transport
Weather Target Synthetic ASR (Baseline) 16.13 14.69 22.88 14.26 47.16 16.23
27.16 13.67 15.49 23.51 21.31 21.61 24.04 17.54 29.57 21.25 18.91 15.45 21.16
\+ SYN2REAL 15.65 13.68 22.64 14.36 40.29 16.15 16.87 12.49 15.22 17.03 21.25
20.77 23.88 15.19 21.87 18.03 16.90 20.38 19.04 Relative WER (%)$\uparrow$
2.95% 6.87 % 1.03 % -0.70 % 14.58 % 0.50 % 37.89 % 8.58 % 1.74 % 27.57 % 0.28
% 3.88 % 0.64 % 13.42 % 26.04 % 15.14 % 10.65 % -31.91% 10.03%

Table 1: Word Error Rate (WER) Performance Across Various Target Domains.
Comparison of the baseline Whisper model and the model enhanced with the
SYN2REAL task vector generated by BARK. The SYN2REAL task vector shows an
average WER reduction of 10.03% across various target domains. Target
Synthetic ASR refers to the baseline that is finetuned on 17 domains
(excluding the target domain) real+synthetic data followed by synthetic data
from the target domains in the SLURP dataset.

###  4.1 Dataset

SLURP Bastianelli et al. (2020) is a spoken language understanding dataset
containing 16521 utterances of human commands towards a virtual agent, based
on 200 pre-defined prompts such as “How would you ask for the time.” The
utterances are categorized into 18 domains (e.g., email, cooking, etc.). In
each of our experiments, we select one of these domains as the target domain
and combine the remaining 17 domains to form the source domain.

###  4.2 Text-to-Speech (TTS) Models

In our experiments, we prepare synthetic speech using two off-the-shelf TTS
models for text from the target domains.

#### BARK

BARK111https://github.com/suno-ai/bark is a transformer-based Vaswani et al.
(2023) autoregressive model, it is pretrained with similar architecture as
AudioLM Borsos et al. (2023) and Vall-E Wang et al. (2023). The input of BARK
contains prompts, transcription, and users. In our experiments, we do not
specify the speaker for BARK.

#### Speech T5

Speech T5 Ao et al. (2022) is a unified model framework that employs encoder-
decoder pre-training for self-supervised speech/text representation learning.
In our experiments, we randomly sample 5 speakers from 7931 speakers.

###  4.3 ASR Models

#### Whisper

Whisper Radford et al. (2022) is an encoder-decoder Transformer-based Vaswani
et al. (2023) model that supervised finetuned on 680,000 hours of labeled
audio data. All experiments are conducted using the Whisper small model,
except for the ablation study, where we experiment with models of different
sizes, including the base and tiny models, to validate our method.

#### Wav2Vec2-Conformer

Wav2Vec2 Baevski et al. (2020) is a framework for self-supervised learning of
speech representations that masks latent representations of the raw waveform
and solves a contrastive task over quantized speech representations.
Wav2Vec2-Conformer Wang et al. (2022) (referred to as Wav2vec2 in the
experiments) follows the same architecture as Wav2Vec2, but replaces the
Attention-block with a Conformer-block Wang et al. (2020) is the conformer
Gulati et al. (2020). We use the large
checkpoint222facebook/wav2vec2-conformer-rope-large-960h-ft with 618M
parameters with rotary position embeddings, pretrained and fine-tuned on 960
hours of Librispeech Panayotov et al. (2015) on 16kHz sampled speech audio to
conduct experiments

##  5 Results & Discussion

Here, we discuss our results in relation to the questions we set out to
answer.

###  5.1 What is the efficacy of SYN2REAL task vector?

To answer Q1, we apply our method by comparing the word error rate (WER)
across various target domains. We select one of these domains as the target
domain and combine the remaining 17 domains to form the source domain Table 1
presents the WER results for both the baseline ASR model fine-tuned on
synthetic speech data and the model enhanced with the SYN2REAL task vector.

The baseline model, fine-tuned solely on synthetic data, exhibits varying WERs
across different target domains, with an average WER of 20.15. This
performance highlights the challenge of adapting ASR models to real-world data
when trained on synthetic speech, primarily due to acoustic mismatches.

By applying the SYN2REAL task vector, we observe a significant reduction in
WER across most target domains. The SYN2REAL-enhanced model achieves an
average WER of 19.04, representing an average relative WER reduction of
10.03%. This improvement demonstrates the effectiveness of the SYN2REAL task
vector in bridging the gap between synthetic and real speech data, thus
enhancing the model’s adaptability to diverse real-world scenarios.

The SYN2REAL task vector shows particularly notable improvements in domains
such as ’Music’ (27.57% reduction), ’Takeaway’ (15.14% reduction), and
’Social’ (26.04% reduction). These results suggest that the task vector
effectively captures domain-specific acoustic variations, enabling the ASR
model to generalize better to unseen target domains. In the following
experiments we select the four domains includes two highest improved domains
(’Music’ & ’Social’), and the two lowest improved domains (’Weather’ &
’Cooking’) to conduct the experiments.

However, it is important to note that some domains, such as ’Cooking’ and
’Weather,’ exhibit marginal improvements or slight degradation in WER. These
variations indicate that while the SYN2REAL task vector generally enhances
performance, further fine-tuning and domain-specific adjustments may be
necessary to optimize results across all target domains.

Overall, the results demonstrate that the SYN2REAL task vector is a promising
approach for improving ASR domain adaptation. By addressing the acoustic
mismatches between synthetic and real speech data, our method significantly
enhances the performance of ASR models in real-world applications.

###  5.2 How does SYN2REAL task vector perform across different model sizes?

To answer Q2, we analyze the effect of model size on the performance of ASR
adaptation using the SYN2REAL task vector. Table 2 presents the relative word
error rate (WER) improvements across different model sizes (Tiny, Base, Small)
and various target domains.

Relative WER $\uparrow$ Cooking Music Social Weather Average Tiny 41.11%
-13.47% 2.60% 30.42% 19.48% Base 1.49% 37.80% 5.00% 6.82% 14.70% Small -0.70%
27.56% 26.04% -31.91% 12.43%

Table 2: Relative WER Improvement Across Different Model Sizes after applying
SYN2REAL task vector. This table shows the relative WER improvement compared
to the Target Synthetic ASR for Whisper models of various sizes (Tiny, Base,
and Small).

The results indicate that the Base model achieves the highest average relative
WER improvement of 14.70% across all target domains. This model size shows
substantial gains, particularly in the ’Music’ (37.80%) and ’Social’ (5.00%)
domains, demonstrating its robustness in adapting to diverse acoustic
characteristics using the SYN2REAL task vector.

The Tiny model, while achieving a higher average improvement of 19.48%, shows
considerable performance gains in the ’Cooking’ (41.11%) and ’Weather’
(30.42%) domains. However, it experiences a performance degradation in the
’Music’ domain (-13.47%). This suggests that while the Tiny model can benefit
significantly from the SYN2REAL task vector in certain domains, its overall
adaptability might be limited compared to larger models due to its reduced
model size.

Interestingly, the Small model exhibits an average relative WER improvement of
12.43%, with significant performance enhancement in the ’Social’ (26.04%) and
’Music’ (27.56%) domains. However, it shows a notable degradation in the
’Weather’ domain (-31.91%), indicating potential overfitting or sensitivity to
specific acoustic variations.

These results highlight the importance of model size in ASR adaptation using
the SYN2REAL task vector. The Base model consistently provides balanced
performance across most domains, suggesting it strikes a good balance between
model size and performance. In contrast, the Tiny and Small models show
varying degrees of effectiveness.

Overall, the analysis demonstrates that while the SYN2REAL task vector
significantly improves ASR performance across different model sizes, the
extent of improvement is influenced by the model’s capacity.

###  5.3 Is SYN2REAL task vector effective on ASR models other than Whisper?

To validate the effectiveness of the SYN2REAL task vector on other ASR models,
we conduct additional experiments using the Wav2vec2-Conformer large model.

Wav2Vec2-Conformer Cooking Music Social Weather Average Target Synthetic ASR
(Baseline) 21.26 17.41 25.84 16.74 20.31 \+ SYN2REAL 18.88 14.33 21.48 13.36
17.01 Relative WER $\uparrow$ 11.21% 17.66% 16.87% 20.22% 16.25%

Table 3: WER of SYN2REAL task vector on Wav2Vec2-Conformer. This table shows
the WER and relative WER improvement across different target domains on
Wav2Vec2-Conformer model before and after applying SYN2REAL task vector.

Table 3 presents the WER results across various target domains, including
’Cooking’, ’Music’, ’Social’, and ’Weather’, comparing the baseline model
finetuned on synthetic speech with the model enhanced by the SYN2REAL task
vector. The Table 3 shows a significant reduction in WER when the SYN2REAL
task vector is applied. The average WER drops from 20.31 to 17.01,
representing an overall relative improvement of 16.25%.

The most notable improvement is observed in the ’social’ domain, with a
relative WER reduction of 16.87%. The ’Music’ domain also shows a substantial
improvement of 17.66%, indicating that the task vector successfully captures
and mitigates the acoustic variability associated with music-related speech.

In the ’Cooking’ and ’Weather’ domains, the WER reductions are 11.21% and
20.22%, respectively. While the improvement in the ’Cooking’ domain is more
modest, it still indicates that the SYN2REAL task vector enhances the model’s
adaptability to domain-specific acoustic characteristics.

Overall, the application of the SYN2REAL task vector significantly enhances
the performance of the Wav2vec2-Conformer large model across all tested
domains. These results validate the effectiveness of the SYN2REAL approach in
bridging the gap between synthetic and real speech data, ultimately improving
the robustness and versatility of ASR systems in diverse real-world scenarios.

Whisper Small +Speech T5 Cooking Music Social Weather Average Target Synthetic
ASR (Baseline) 16.94 16.04 53.34 16.27 25.65 \+ SYN2REAL 16.00 15.75 52.95
15.97 25.17 Relative WER $\uparrow$ 5.57% 1.77% 0.73% 1.82% 1.86%

Table 4: WER on Whisper small with SYN2REAL task vector derived from Speech T5
TTS models. This table shows the WER and relative WER improvement accross
different target domains on Whisper small with synthetic data from Speech T5.

###  5.4 Can we form SYN2REAL task vector from other TTS models?

To answer Q4, we conducted experiments using the Whisper Small model with
synthetic data generated by the Speech T5 model. Table 4 presents the WER
results across various target domains, including ’Cooking’, ’Music’, ’Social’,
and ’Weather’, comparing the baseline model finetuned on synthetic speech with
the model enhanced by the SYN2REAL task vector.

The results indicate that applying the SYN2REAL task vector leads to a
reduction in WER across all tested domains. The average WER drops from 25.65
to 25.17, representing an overall relative improvement of 1.86%.

The ’Cooking’ domain shows the highest relative WER reduction of 5.57%,
suggesting that the SYN2REAL task vector effectively adapts the model to this
specific domain. The ’Music’ and ’Weather’ domains also exhibit relative
improvements of 1.77% and 1.25%, respectively, indicating that the SYN2REAL
task vector helps mitigate the acoustic variations in these domains.

However, the improvement in the ’social’ domain is relatively modest, with a
relative WER reduction of only 0.73 %. This could be attributed to the high
baseline WER in this domain, suggesting that the synthetic data from Speech T5
might have limitations.

Overall, the application of the SYN2REAL task vector to the Whisper Small
model with Speech T5 synthetic data demonstrates consistent performance
enhancements, albeit with varying degrees of improvement across different
domains. These results validate the flexibility and effectiveness of our
approach in improving ASR models trained with synthetic data from different
TTS models.

Figure 3: WER vs. Scaling Factor across Different ASR Models & Different TTS
Models  The plot shows the average WER on ’Cooking’, ’Music’, ’Social’, and
’Weather’ target domains as a function of the scaling factor $\lambda$ for
various ASR models (Whisper and W2V2-conformer) and the TTS models (BARK and
Speech T5) to make SYN2REAL task vectors. We denote it as ’{ASR+TTS}’, such as
’Whisper Tiny+BARK’ in the figure. The scaling factor adjusts the magnitude of
the SYN2REAL task vector applied to each model.

###  5.5 What is the impact of the scaling factor $\lambda$?

This section investigates the effect of scaling the SYN2REAL task vector on
the WER of different ASR models. Figure 3 illustrates the WER as a function of
the scaling factor $\lambda$ for various ASR models and synthetic data,
including Whisper Tiny with BARK, Whisper Base with BARK, Whisper Small with
BARK, Whisper Small with Speech T5, and W2V2-Conformer with BARK.

The scaling factor $\lambda$ adjusts the magnitude of the SYN2REAL task vector
applied to the ASR models. We evaluated a range of scaling factors from 0.1 to
1.0 to determine the optimal balance that minimizes WER.

The results show that different models respond variably to changes in the
scaling factor. For Whisper Tiny+BARK, the curve is steeper, indicating that
smaller models may be more sensitive to larger adjustments from the SYN2REAL
task vector. In contrast, Whisper Base+BARK maintains relatively stable WER
values across different scaling factors, suggesting a more robust performance.

Notably, Whisper Small+BARK and Whisper Small + Speech T5 exhibit a U-shaped
trend, where moderate scaling factors (around $\lambda$ = 0.3 to 0.5) yield
the lowest WER. This indicates that an optimal scaling factor exists for these
models, which balances the incorporation of real speech characteristics
without overwhelming the model with excessive parameter adjustments. The
Wav2vec2-Conformer model consistently shows lower WER values across all
scaling factors, with the best performance at $\lambda$ = 0.5.

Overall, the analysis suggests that the optimal scaling factor $\lambda$
varies depending on the ASR model’s architecture and size. While smaller
models like Whisper Tiny+BARK may benefit from lower scaling factors, larger
and more robust models like W2V2-Conformer+BARK can effectively leverage
higher scaling factors. These findings highlight the importance of tuning the
scaling factor to achieve the best domain adaptation performance for different
ASR models.

###  5.6 Do SYN2REAL task vectors obtained with the same TTS have similar
directions?

Figure 4: Cosine Similarity between task vectors derived from Different TTS
Models. This heatmap shows the cosine similarity between task vectors
generated by BARK (B_) and Speech T5 (S_) models. Higher similarity values
between vectors from similar domains indicate effective acoustic-specific
information transfer by the SYN2REAL method.

To further validate the SYN2REAL approach, we conducted a cosine similarity
analysis between SYN2REAL task vectors derived by different text-to-speech
(TTS) models: BARK (denoted as B_) and Speech T5 (denoted as S_). Figure 4
presents the cosine similarity heatmap between these SYN2REAL task vectors.

The heat map reveals that SYN2REAL task vectors from similar TTS exhibit
higher cosine similarity, indicating that the SYN2REAL task vector effectively
captures the distributional shifts between different acoustic domains.

Moreover, the negative similarities between certain SYN2REAL task vectors,
such as ’B_recommendation’ and ’S_music’ (-0.67), highlight the distinct
acoustic features between these TTS synthetic data, further emphasizing the
effectiveness of the SYN2REAL approach in distinguishing and adapting to
different acoustic environments.

The overall trend observed in the heatmap supports the hypothesis that the
SYN2REAL task vectors not only bridge the gap between synthetic and real data
but also maintain consistency within similar acoustic environments. This
consistency is crucial for enhancing ASR performance across diverse target
domains, as it ensures that the task vectors can generalize well to new,
unseen domains.

##  6 SYN2REAL Task Vector given Domain Labels

In this section, we explore an alternative approach to generating SYN2REAL
task vectors, assuming we have access to domain labels for the data in the
source domains. This approach, which we refer to as SYN2REAL Ensemble,
involves generating separate SYN2REAL task vectors for each source domain and
then combining them to enhance the adaptation of the ASR model to the target
domain.

SYN2REAL Ensemble Cooking Music Social Weather Average Target Synthetic ASR
(Baseline) 14.26 23.51 29.57 15.45 20.70 \+ SYN2REAL Ensemble 14.46 16.98
21.13 15.11 16.92 Relative WER $\uparrow$ -1.40% 27.78% 28.54% 2.20% 18.25%

Table 5: WER Performance on Whisper Small Model with SYN2REAL Ensemble task
vectors. This table compares the word error rate (WER) of the baseline ASR
model fine-tuned on synthetic speech data with the WER of the model enhanced
with the SYN2REAL Ensemble task vectors across four target domains: ’Cooking’,
’Music’, ’Social’, and ’Weather’.

###  6.1 Performance of SYN2REAL Ensemble task vector

To evaluate the effectiveness of the SYN2REAL Ensemble task vector, we
conducted experiments using the Whisper Small model with synthetic speech
generated by the BARK TTS model. The experiments were carried out on four
target domains: ’cooking,’ ’music,’ ’social,’ and ’weather,’ using 17 source
domains to create the SYN2REAL Ensemble vectors. The results are presented in
Table 5. The results indicate that the SYN2REAL Ensemble approach provides
significant improvements in WER for several target domains compared to the
baseline method. The average WER is reduced from 20.70 to 16.92, representing
an overall relative improvement of 18.25%. This highlights the effectiveness
of using domain-specific task vectors to capture detailed acoustic
characteristics, leading to enhanced model adaptation.

Comparing the SYN2REAL Ensemble approach to the original SYN2REAL method, we
find that SYN2REAL Ensemble generally outperforms the original approach. The
detailed domain-specific information captured by the distinct task vectors
enhances model adaptation in most domains. However, in real-world scenarios,
we often do not have access to labels finer-grained domain labels.

###  6.2 Impact of the number of domains on the performance of SYN2REAL
Ensemble task vector

Figure 5: WER vs. Number of Source Domains for SYN2REAL Ensemble task vector.
This plot shows the word error rate (WER) of the Whisper small model and the
number of source domains used to generate the SYN2REAL Ensemble task vector
with BARK model. The x-axis represents the number of source domains, and the
y-axis represents the WER on average of the four domains (’Cooking’, ’Music’,
’Social’, ’Weather’).

Figure 5 shows the average WER across four target domains (’cooking,’ ’music,’
’social,’ and ’weather’) when we use different numbers of source domain data
to generate SYN2REAL Ensemble task vectors.

The results indicate that increasing the number of source domains generally
improves ASR performance. The WER decreases from 17.8 to 16.8 as the number of
source domains increases from 1 to 17. Notably, significant improvements are
observed within the incorporation of the first 5 source domains. This trend
suggests that incorporating more source domains helps capture diverse acoustic
characteristics, leading to better domain adaptation.

##  7 Conclusion

This paper introduces SYN2REAL task vector to address mismatches between
synthetic and real speech data. Future work will refine this approach and
extend its application to other types of tasks and data such as visual data,
contributing to more reliable speech and vision recognition systems.
Experiments showed significant WER reductions, averaging 10.03% across 18
domains. We also test various models, including Whisper Small, Whisper Base,,
Whisper Tiny and Wav2vec2-Conformer, with SYN2REAL showing robust performance.

##  8 Limitations

#### Domain-Specific Performance Variations

While the SYN2REAL task vector shows significant improvements in many target
domains, certain domains, such as ’Cooking’ and ’Weather,’ exhibit marginal
improvements or slight degradation in word error rate (WER). This suggests
that the task vector’s effectiveness may vary based on the specific
characteristics of different domains, indicating a need for further domain-
specific fine-tuning and adjustments.

#### Scaling Factor Sensitivity

The performance of the SYN2REAL-enhanced models is sensitive to the scaling
factor $\lambda$. Finding the optimal scaling factor requires careful tuning,
and the best value can vary between different ASR models and target domains.
This adds a layer of complexity to the implementation and may limit the
approach’s generalizability without additional adaptive scaling strategies.

#### Synthetic Data Quality

The approach relies heavily on the quality of synthetic speech data generated
by TTS systems. Variations in the quality and acoustic properties of synthetic
data across different TTS systems can impact the effectiveness of the SYN2REAL
task vector. Ensuring consistent quality in synthetic data is crucial for
achieving robust domain adaptation.

#### Model-Specific Dependencies

The observed improvements are model-dependent, with larger models like
Wav2Vec2-Conformer showing more substantial gains compared to smaller models
like Whisper Tiny. This indicates that the SYN2REAL task vector’s
effectiveness might be influenced by the underlying model architecture and
size, potentially limiting its applicability to a wider range of ASR models
without further optimization.

##  9 Acknowledgements

We specifically thank Ting-Yao Hu for all the insightful discussions and
constructive suggestions for this work.

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galactic nuclei.

Alhough we have focused on galactic nuclei, the results of our work can be
applied to other systems. On planetary system scales, massive (but not gap-
clearing) planetary cores interacting within a gaseous protoplanetary disk
would experience GDF that might lead to binary formation. Even once the gas in
the disk has disappeared, a disk of smaller bodies would exert dynamical
friction analogous to GDF on larger planetesimals. (In this way, our work is a
development on Goldreich et al. 2002; our $C_{g}$ is related to their drag
term $D$, and our result that $\tilde{\mathcal{R}}\propto C_{g}$ down to at
least $C_{g}\sim 10^{-5}$ suggests that the trend in their $L^{2}$ channel
continues for drag strengths several orders of magnitude lower than they
studied.)

Our prescription for binary formation by SDF might also be relevant for
globular clusters, if some or all of the stars in these clusters are formed in
or dragged into thin gaseous disks.

Finally, we stress that our results are based on several assumptions and
approximations — the relative speed of the BHs is assumed to be much less than
the sound speed in the disks; the wakes of the BHs are assumed to not
interact; the BH masses are set to be equal and assumed to be much greater
than the masses of the stars in the nuclear star cluster (SDF); the spatial
excursions of the BH orbits are assumed to be not much larger than the Hill
radius; etc. All of these approximations are discussed in the text; they
likely add errors to our results, but in systems for which our assumptions are
valid these errors are small compared to the uncertainties in the typical
parameters of AGN disks and galactic nuclei. Furthermore, the approximate
agreement between our GDF capture rates, those from other recent works using
analytic GDF prescriptions (specifically DeLaurentiis et al., 2023; Qian et
al., 2024), and those from the hydrodynamic simulations of Whitehead et al.
(2023a) provide some assurance that our model captures the main features of
BBH formation by dynamical friction.

We thank Christopher Matzner, Norman Murray, Maya Fishbach, Aditya Vijaykumar,
and Jiaru Li for thoughtful comments and conversations. We gratefully
acknowledge Stanislav DeLaurentiis, Kecheng Qian, and Henry Whitehead for
sharing data from their respective papers. This work was supported in part by
the Natural Sciences and Engineering Research Council of Canada (NSERC),
funding reference number RGPIN-2020-03885.

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##  Appendix A Derivation of Stellar Dynamical Friction Equation

From equations (7.92) and (8.7) of Binney & Tremaine (2008), the dynamical
friction force acting on a body of mass $m$ moving with velocity vector
$\dot{\mathbf{r}}$ through a field of bodies with mass $m_{a}$ can be written

$m\ddot{\mathbf{r}}=-\frac{4\pi\mathbb{G}^{2}\rho_{s}m(m+m_{a})\ln\Lambda}{\sigma^{2}}G(X)\frac{\dot{\mathbf{r}}}{v},$
(A1)

where $\mathbb{G}$ is the gravitational constant;

$G(X)\equiv\frac{1}{2X^{2}}\text{erf}(X)-\frac{1}{X\sqrt{\pi}}\exp{(-X^{2})};$
(A2)

$X\equiv v/\sqrt{2}\sigma$, with $v=|\dot{\mathbf{r}}|$ and the one-
dimensional velocity dispersion of the field stars $\sigma$; $\rho_{s}$ the
local field density; and $\ln\Lambda$ the Coulomb logarithm, with $\Lambda$
defined in equation (7.84) of Binney & Tremaine (2008). If we have two bodies
with equal masses $m_{1}=m_{2}=m$, we can define some $\dot{\mathbf{s}}$ such
that their individual velocity vectors are given by
$\dot{\mathbf{r}}_{2}=\dot{\mathbf{r}}-\dot{\mathbf{s}}$ and
$\dot{\mathbf{r}}_{1}=\dot{\mathbf{r}}+\dot{\mathbf{s}}$ (where
$\dot{\mathbf{r}}$ is the center-of-mass velocity and $2\dot{\mathbf{s}}$ is
the relative velocity vector). Then the differential acceleration on these
bodies is

$\ddot{\mathbf{r}}_{2}-\ddot{\mathbf{r}}_{1}=-\frac{4\pi\mathbb{G}^{2}\rho_{s}(m+m_{a})\ln\Lambda}{\sigma^{2}}\mathbf{b},\quad\text{where}\quad\mathbf{b}\equiv
G(X_{2})\frac{\dot{\mathbf{r}}_{2}}{v_{2}}-G(X_{1})\frac{\dot{\mathbf{r}}_{1}}{v_{1}},$
(A3)

where we have abbreviated $X(v_{n})$ as $X_{n}$. For bodies on nearby,
initially circular orbits, as considered here, the Keplerian velocity
dominates the velocity vectors, and the separation of the bodies on their
initial orbits is small enough compared to their distance from the central
object that their Keplerian velocities are approximately equal; i.e.,
$\dot{\mathbf{r}}_{1}\simeq\dot{\mathbf{r}}_{2}$. Therefore we assume that
$|\dot{\mathbf{s}}|=|\dot{\mathbf{r}}_{1}-\dot{\mathbf{r}}_{2}|/2\ll|\dot{\mathbf{r}}|$
— this inequality may be violated for short periods of time during close
encounters between the two bodies, but this should not have a significant
effect on our results.

When $\dot{\mathbf{s}}=0$, the vector $\mathbf{b}=0$; we can Taylor expand
around $\dot{\mathbf{s}}=0$ to find the behavior of equation (A3) when
$|\dot{\mathbf{s}}|$ is small. This expansion gives

$\mathbf{b}\simeq\left.\frac{\partial\mathbf{b}}{\partial\dot{\mathbf{s}}}\right|_{\dot{\mathbf{s}}=0}\dot{\mathbf{s}}+\mathcal{O}(|\dot{\mathbf{s}}|^{2}).$
(A4)

The first-order term is a matrix
$(\partial\mathbf{b}/\partial\dot{\mathbf{s}})$ right-multiplied by a vector
$(\dot{\mathbf{s}})$; the $j$-th column of the matrix is given by

$\displaystyle\frac{\partial}{\partial\dot{s}_{j}}\left[G_{2}\frac{\dot{\mathbf{r}}_{2}}{v_{2}}-G_{1}\frac{\dot{\mathbf{r}}_{1}}{v_{1}}\right]=\frac{\partial
G_{2}}{\partial X_{2}}$ $\displaystyle\frac{\partial
X_{2}}{\partial\dot{s}_{j}}\frac{\dot{\mathbf{r}}_{2}}{v_{2}}+\frac{G_{2}}{v_{2}}\frac{\partial\dot{\mathbf{r}}_{2}}{\partial\dot{s}_{j}}-G_{2}\frac{\partial
v_{2}}{\partial\dot{s}_{j}}\frac{\dot{\mathbf{r}}_{2}}{v_{2}^{2}}$
$\displaystyle-\frac{\partial G_{1}}{\partial X_{1}}\frac{\partial
X_{1}}{\partial\dot{s}_{j}}\frac{\dot{\mathbf{r}}_{1}}{v_{1}}-\frac{G_{1}}{v_{1}}\frac{\partial\dot{\mathbf{r}}_{1}}{\partial\dot{s}_{j}}+G_{1}\frac{\partial
v_{1}}{\partial\dot{s}_{j}}\frac{\dot{\mathbf{r}}_{1}}{v_{1}^{2}},$ (A5)

where we have abbreviated $G(X_{n})$ as $G_{n}$. Note that $\partial
X_{n}/\partial\dot{s}_{j}=(\partial
v_{n}/\partial\dot{s}_{j})/\sqrt{2}\sigma$. Note also that

$\displaystyle\frac{\partial\dot{\mathbf{r}}_{1}}{\partial\dot{s}_{j}}=\hat{\mathbf{e}}_{j},\quad\quad$
$\displaystyle\quad\text{ }\quad\quad\frac{\partial
v_{1}}{\partial\dot{s}_{j}}=\frac{\dot{s}_{j}+\dot{r}_{1,j}}{v_{1}},$
$\displaystyle\frac{\partial\dot{\mathbf{r}}_{2}}{\partial\dot{s}_{j}}=-\hat{\mathbf{e}}_{j},\quad\quad$
$\displaystyle\text{and}\quad\quad\frac{\partial
v_{2}}{\partial\dot{s}_{j}}=\frac{\dot{s}_{j}-\dot{r}_{2,j}}{v_{2}}.$ (A6)

Plugging in these formulae, then evaluating the result at $\dot{\mathbf{s}}=0$
(equivalently, $\dot{\mathbf{r}}_{1}=\dot{\mathbf{r}}_{2}=\dot{\mathbf{r}}$),
equation (A) simplifies to

$\frac{\partial}{\partial\dot{s}_{j}}\left[G_{2}\frac{\dot{\mathbf{r}}_{2}}{v_{2}}-G_{1}\frac{\dot{\mathbf{r}}_{1}}{v_{1}}\right]=-2\left(\frac{\partial
G}{\partial X}\frac{\dot{r}_{j}\dot{\mathbf{r}}}{\sqrt{2}\sigma
v^{2}}-G\frac{\dot{r}_{j}\dot{\mathbf{r}}}{v^{3}}+G\frac{\hat{\mathbf{e}}_{j}}{v}\right).$
(A7)

In index notation, then, the first-order expansion is

$\mathbf{b}\simeq\frac{\partial}{\partial\dot{s}_{j}}\left[G_{2}\frac{\dot{r}_{2,i}}{v_{2}}-G_{1}\frac{\dot{{r}}_{1,i}}{v_{1}}\right]s_{j}=\left[\frac{\partial
G}{\partial X}\frac{\dot{r}_{i}\dot{r}_{j}\hat{\mathbf{e}}_{i}}{\sqrt{2}\sigma
v^{2}}-G\frac{\dot{r}_{i}\dot{r}_{j}\hat{\mathbf{e}}_{i}}{v^{3}}+G\frac{\hat{\mathbf{e}}_{j}}{v}\right](\dot{\mathbf{r}}_{2}-\dot{\mathbf{r}}_{1})_{j}.$
(A8)

We assume that all components of the velocity other than the local Keplerian
velocity $v_{K}\hat{\mathbf{e}}_{y}$ are negligible, so to lowest order,
$v_{j}\dot{\mathbf{r}}\simeq v_{K}^{2}\delta_{jy}\hat{\mathbf{e}}_{y}$ (and
therefore $v\simeq v_{K}$). Then the expansion becomes

$\mathbf{b}\simeq\left[\frac{\partial G_{K}}{\partial
X}\frac{\delta_{jy}\hat{\mathbf{e}}_{y}}{\sqrt{2}\sigma}-G_{K}\frac{\delta_{jy}\hat{\mathbf{e}}_{y}}{v_{K}}+G_{K}\frac{\hat{\mathbf{e}}_{j}}{v_{K}}\right](\dot{\mathbf{r}}_{2}-\dot{\mathbf{r}}_{1})_{j}.$
(A9)

Note $G_{K}=G(v_{K}/\sqrt{2}\sigma)$. The relevant derivative of $G(X)$ is

$\frac{\partial G}{\partial
X}=\left(\frac{2}{X^{2}\sqrt{\pi}}+\frac{2}{\sqrt{\pi}}\right)\exp(-X^{2})-\frac{1}{X^{3}}\text{erf}(X).$
(A10)

We can rewrite equation (A9) as

$\mathbf{b}\simeq\frac{1}{v_{K}}\left[\Gamma^{*}_{1}(\dot{\mathbf{r}}_{2}-\dot{\mathbf{r}}_{1})+(\Gamma^{*}_{2}-\Gamma^{*}_{1})(v_{2y}-v_{1y})\hat{\mathbf{e}}_{y}\right],$
(A11)

introducing the dimensionless constants $\Gamma^{*}_{1}$ and $\Gamma^{*}_{2}$,
which are defined as (using $v_{K}/\sigma=\sqrt{2}X_{K}$)

$\displaystyle\Gamma^{*}_{1}\equiv G_{K}$
$\displaystyle=\frac{1}{2X_{K}^{2}}\text{erf}(X_{K})-\frac{1}{X_{K}\sqrt{\pi}}\exp(-X_{K}^{2})$
(A12) $\displaystyle\Gamma^{*}_{2}\equiv\frac{1}{\sqrt{2}\sigma}\frac{\partial
G_{K}}{\partial X}$
$\displaystyle=-\frac{1}{X_{K}^{2}}\text{erf}(X_{K})+\left(\frac{2}{X_{K}\sqrt{\pi}}+\frac{2X_{K}}{\sqrt{\pi}}\right)\exp(-X_{K}^{2})$
(A13)

For a spherically symmetric and ergodic stellar system, following a density
profile $\rho(r)\propto r^{-\gamma}$, one can solve the Jeans equations to
obtain

$\sigma=\frac{v_{K}}{\sqrt{\gamma+1}}\quad\quad\text{and}\quad\quad
X_{K}=\sqrt{\frac{\gamma+1}{2}},$ (A14)

so

$\displaystyle\Gamma^{*}_{1}$
$\displaystyle=\frac{1}{\gamma+1}\text{erf}\left(\sqrt{\frac{\gamma+1}{2}}\right)-\sqrt{\frac{2}{\pi(\gamma+1)}}\exp\left(-\frac{\gamma+1}{2}\right)$
(A15) $\displaystyle\Gamma^{*}_{2}$
$\displaystyle=-\frac{2}{\gamma+1}\text{erf}\left(\sqrt{\frac{\gamma+1}{2}}\right)+\left(2\sqrt{\frac{2}{\pi(\gamma+1)}}+\sqrt{\frac{2(\gamma+1)}{\pi}}\right)\exp\left(-\frac{\gamma+1}{2}\right)$
(A16)

Note that $\Gamma^{*}_{1}$ and $\Gamma^{*}_{2}$ depend only on the logarithmic
density slope $\gamma$ of the star cluster. We now plug our expression for
$\mathbf{b}$ (eq. A11) back into equation (A3) to find

$\boldsymbol{\rho}^{\prime\prime}_{\text{SDF}}=-C^{*}_{s}\left(\Gamma^{*}_{1}\boldsymbol{\rho}^{\prime}+(\Gamma^{*}_{2}-\Gamma^{*}_{1})\eta^{\prime}\hat{\boldsymbol{\eta}}\right),$
(A17)

where we have also converted to the dimensionless spatial and temporal
coordinates $\boldsymbol{\rho}=\mathbf{x}/3^{1/3}R_{\text{H}}$ and
$t_{d}=\Omega t$ (with $R_{H}$ the mutual Hill radius and $\Omega$ the local
Keplerian frequency). The constant $C^{*}_{s}$ is given by

$C^{*}_{s}=\frac{4\pi\mathbb{G}^{2}\rho_{s}(m+m_{a})\ln\Lambda}{\Omega
v_{K}^{3}}(\gamma+1)=4\pi
r^{3}\rho_{s}\frac{m+m_{a}}{M_{\bullet}^{2}}\ln\Lambda(\gamma+1),$ (A18)

where $M_{\bullet}$ is the mass of the central SMBH and $r$ is the distance
from that to the bodies of interest. In the text, we define $C_{s}\equiv
C^{*}_{s}/(\gamma+1)$, $\Gamma_{1}\equiv(\gamma+1)\Gamma^{*}_{1}$, and
$\Gamma_{2}\equiv(\gamma+1)\Gamma^{*}_{2}$ in order to make $C_{s}$
independent of $\gamma$. In keeping $C_{s}$ constant throughout a given
interaction, we neglect any changes to the orbital radius $r$ over its
duration.


supplementary

#  Ab initio elasticity at finite temperature and stress in ferroelectrics

Mark A. Mathis1, Chris A. Marianetti1 1 Department of Applied Physics and
Applied Mathematics, Columbia University, New York, NY 10027

###### Abstract

Computing the temperature and stress dependence of the full elastic constant
tensor from first-principles in non-cubic materials remains a challenging
problem. Here we circumvent the aforementioned challenge via the generalized
quasiharmonic approximation in conjunction with the irreducible derivative
approach for computing strain dependent phonons using finite difference,
explicitly including dipole-quadrupole contributions. We showcase this
approach in ferroelectric PbTiO3 using density functional theory, computing
all independent elastic constants and piezoelectric strain coefficients at
finite temperature and stress. There is good agreement between the
quasiharmonic approximation and the experimental lattice parameters close to 0
K. However, the quasiharmonic approximation overestimates the temperature
dependence of the lattice parameters and elastic constant tensor,
demonstrating that a higher level of strain dependent anharmonic vibrational
theory is needed.

Ferroelectric materials have been widely studied due to technological
importance and interesting physics [1, 2, 3, 4, 5]. Many ferroelectric
materials are band insulators, which are typically well described by density
functional theory (DFT). For prototypical ferroelectric band insulators, the
Born-Oppenheimer potential generated from appropriate exchange-correlation
approximations within DFT produce temperature dependent structural phase
transitions consistent with experiment [6, 7, 8, 9, 10, 11, 12, 13, 14, 15],
indicating an accurate representation of the vibrational free energy. However,
evaluating piezoelectric properties at finite temperatures and stress requires
the computation of relevant strain curvatures of the DFT based vibrational
free energy at finite temperatures and stress (i.e. the elastic constants).
Computing these elastic constants requires encoding or sampling the
vibrational Hamiltonian as a function of strain, evaluating the vibrational
free energy in some approximation as a function of strain, and evaluating the
second strain derivatives of the free energy. Each of the aforementioned tasks
presents substantial theoretical and computational challenges.

The standard approximation for computing finite temperature elastic constants
is the quasiharmonic approximation (QHA) [16, 17, 18], yielding reasonable
agreement with experimental measurements for a variety of cubic systems [19,
20, 21, 22, 23, 24, 25]. The computational cost of executing the QHA within
DFT is appreciable [26], as evidenced by the sparsity of temperature dependent
elastic constant computations for non-cubic systems available in the
literature [27, 28, 29], and we are not aware of any published results at
finite temperature and stress. The aforementioned limitations can be mitigated
by using the recently developed generalized quasiharmonic approximation [30],
which leverages the irreducible derivative approach to computing phonons [31,
32]. Here we showcase the power of the generalized QHA by studying the
displacive ferroelectric PbTiO3 (spacegroup P4$mm$) using DFT, computing the
lattice parameters, full elastic constant tensor, and piezoelectric strain
coefficients at finite temperature and stress. PbTiO3 is an ideal candidate to
study within the generalized QHA, as the low symmetry ferroelectric phase
persists to roughly $T=760$ K [33, 34, 35].

DFT calculations were performed using the Vienna ab initio simulation package
(VASP) [36, 37, 38, 39] with the projector augmented wave (PAW) method [40,
41] unless otherwise stated. The generalized gradient approximation (GGA)
revised for solids (PBEsol) [42] and the strongly constrained and
appropriately normed (SCAN) [43] exchange-correlation functionals were used.
Convergence of the strain dependent phonons was achieved with a kinetic energy
cutoff of 1000 eV and a $\Gamma$-centered k-point mesh of
16$\times$16$\times$16 for the primitive unit cell with corresponding mesh
densities being used for supercells. Details of the PAW potentials, finite
difference calculations, and Fourier interpolation are provided in Sec. I of
the supplemental material (SM) [44]. The computationally relaxed crystal
structures using the PBEsol and SCAN functionals are compared with low
temperature experimental measurements in Table 1. Due to the significant
overestimation of the $c$ lattice parameter by the SCAN functional,
computations use the PBEsol functional unless otherwise stated.

Method | $a$ | $c$ |  $z$(Ti) |  $z$(O1,2) |  $z$(O3)
---|---|---|---|---|---
PBEsol | 3.872 | 4.214 | 0.539 | 0.623 | 0.118
SCAN | 3.865 | 4.341 | 0.545 | 0.638 | 0.139
PBEsol | 3.891 | 4.164 | 0.539 | 0.618 | 0.112
Mestric et al. | 3.891 | 4.168 | 0.542 | 0.629 | 0.124
Table 1:  Lattice parameters and direct atomic coordinates along the
$z$-direction. (Top) Results of DFT relaxation using the PBEsol and SCAN
functionals. (Bottom) $T=1$ K QHA results compared with measurements at $T=12$
K [45].

We begin by showcasing the computed phonons in the supercell
$\hat{S}_{BZ}=4\hat{1}$ (see Fig. 1 $(a)$), achieving good agreement to
previous computations [46]. The dipole-dipole contribution to Fourier
interpolated phonons is included [47, 48, 30] shown as the red lines, where
the dielectric tensor and Born effective charges were calculated from density
functional perturbation theory [49, 50] within VASP. Along the path from the
$\Gamma$ point to the R point, there are interpolated imaginary phonon
frequencies caused by a deficiency in the Fourier interpolation, which was
elucidated in previous work [51]. Supplementing the interpolation with the
dipole-quadrupole interactions has demonstrated the ability to remove these
spurious imaginary frequencies from the interpolation [51]. Our Fourier
interpolation of the phonons including the dipole-dipole and the dipole-
quadrupole contributions shown as the blue lines does not contain any spurious
imaginary frequencies, consistent with results in Ref. [51]. Computation of
the dipole-quadrupole contribution has been implemented analogously to the
dipole-dipole contribution, where dynamical quadrupoles were computed [52]
using density functional theory implemented in the (ABINIT) package [53, 54]
using the PBEsol optimized norm-conserving Vanderbilt pseudopotential
(ONCVPSP) [55]. Our results illustrate that both dipole-dipole and dipole-
quadrupole corrections to the Fourier interpolation can straightforwardly be
utilized in our irreducible derivative approaches, which are based on the
finite difference method.

We now present selected generalized Gruneisen parameters,

$\displaystyle\gamma_{i,\mathbf{q}\ell}=-\frac{1}{\omega_{\mathbf{q}\ell}}\frac{\partial\omega_{\mathbf{q}\ell}}{\partial\epsilon_{i}},$
(1)

which encapsulate the first order strain dependence of the phonons along a
given strain (see Fig. 1 $(b)$ and $(c)$). Computation of the full elastic
constant tensor within the QHA requires the phonons to be computed as a
function of all symmetrically distinct strains, whereas thermal expansion
computations only use strains associated with changes in volume. Thus, panel
$b$ shows the $A_{1}^{\circ}$ Gruneisen parameter associated with expansion
along the $z$-axis, where the symmetrized representations of the identity
strains are denoted as
$\epsilon_{A_{1}^{\square}}=\frac{1}{\sqrt{2}}(\epsilon_{xx}+\epsilon_{yy})$
and $\epsilon_{A_{1}^{\circ}}=\epsilon_{zz}$. Integration of the density of
states yields averaged Gruneisen parameters of
$\bar{\gamma}_{A_{1}^{\square}}$=1.78 and $\bar{\gamma}_{A_{1}^{\circ}}$=0.28.
Panel $c$ shows the $B_{1}$ Gruneisen parameter, where
$\epsilon_{B_{1}}=\frac{1}{\sqrt{2}}(\epsilon_{xx}-\epsilon_{yy})$. Symmetry
selection rules and first order perturbation theory require the Gruneisen
parameter to be zero along various directions in reciprocal space.

(a)

(b)

(c) Figure 1:  Phonons and specific Gruneisen parameters computed from DFT
(diamonds) within the $4\hat{1}$ and $2\hat{S}_{C}$ supercells, respectively,
and Fourier interpolated (lines). Left panels show values along high symmetry
directions, and right panels show the density of states. (a) Computed phonons
which are Fourier interpolated with dipole-dipole (red) or dipole-dipole and
dipole-quadrupole (blue) contributions. Gruneisen parameters computed with
strains transforming like the (b) $A_{1}^{\circ}$ and (c) $B_{1}$ irreducible
representations.

Having computed the strain dependence of the phonons, we apply the generalized
quasiharmonic approximation to compute the $a$ and $c$ lattice parameters at
finite temperature and stress (see Fig. 2). We compare with experimental
measurements at various temperatures under unstressed conditions [33, 56, 57,
45, 58]. The $a$ and $c$ lattice parameters are in good agreement within the
experimental measurements, as there is at most 0.5% difference between various
measurements at any given temperature. Computation of the lattice parameters
at a given temperature $T$ and stress $\bm{\sigma}$ is achieved through the
Biot strain function,

$\displaystyle\tilde{\bm{\epsilon}}(T,\bm{\sigma}),\hskip
8.53581pt\textrm{where}\hskip
8.53581pt\tilde{\bm{\sigma}}(T,\tilde{\bm{\epsilon}}(T,\bm{\sigma}))=\bm{\sigma}.$
(2)

where definitions and notation are equivalent to Ref. [30] (see Eqs. 21-26).
The phonons are computed on a grid of strains and compared with a Taylor
series expansion of the phonons in strain, where convergence of the thermal
expansion is achieved when including first, second, and third strain
derivatives within the Taylor series (see Sec. I of the SM [44]). The crystal
structure predicted at $T=1$ K by the QHA differs from the values obtained
from DFT relaxations due to zero point motion, and are compared with
experimental measurements at $T=12$ K [45] (see Table 1). The predicted shift
in lattice parameters due to zero-point motion yields remarkable agreement
with the values obtained from experiment, however there are discrepancies in
the predicted direct atomic coordinates. The change in lattice parameters with
temperature is overestimated by the quasiharmonic approximation which is well
known for anharmonic materials [59, 60]. Under the application of
$\sigma_{A_{1}^{\square}}=-0.95$ GPa and $\sigma_{A_{1}^{\circ}}=-0.15$ GPa,
the $a$ and $c$ lattice parameter computations agree with experimental
measurements at room temperature.

(a)

(b) Figure 2:  The $a$ (panel $a$) and $c$ (panel $b$) lattice parameters
computed with the QHA (lines) under unstressed (blue) and stressed (red)
conditions compared with previous experimental measurements (diamonds) [33,
56, 57, 45, 58].

Having computed the lattice parameters, we now discuss the strain curvature of
the free energy at finite temperature and stress. There are three
experimentally relevant quantities related to the free energy curvature at
finite stress [61]: the free energy curvature $C_{ij}$, the elastic wave
propagation coefficient $S_{ij}$, and the stress-strain coefficient $B_{ij}$.
Additionally, elastic constants in ferroelectrics can be measured under
boundary conditions of constant electric field $E$ or constant electric
displacement field $D$. The relation between the two boundary conditions is
given by [62],

$\displaystyle C_{ij}^{D}(T,\bm{\sigma})=$ $\displaystyle
C_{ij}^{E}(T,\bm{\sigma})+\sum_{\alpha\beta}\big{(}e_{\alpha
i}(T,\bm{\sigma})\times$ $\displaystyle e_{\beta
j}(T,\bm{\sigma})((\hat{\epsilon}^{\infty,S}(T,\bm{\sigma}))^{-1})_{\alpha\beta}\big{)},$
(3)

where $\hat{\epsilon}^{\infty,S}$ is the relaxed-ion dielectric tensor at
fixed strain and $e_{\alpha i}$ is the relaxed-ion piezoelectric stress
coefficient [63, 64, 65, 62]. Our evaluation of $\hat{\epsilon}^{\infty,S}$
and $e_{\alpha j}$ at finite temperature $T$ and stress $\bm{\sigma}$ is
achieved by using the value at the strain corresponding to
$\tilde{\bm{\epsilon}}(T,\bm{\sigma})$.

We compute the elastic constant tensor at finite temperature and stress, and
compare with experimental values measured under unstressed conditions [66, 67,
68] (see Fig. 3). The elastic constants which were not measured at multiple
temperatures are shown in Sec. II of the SM [44]. There is good agreement
within the experimental values at room temperature, where the largest
disagreement is a 4% difference in the measured $C_{44}^{D}$ values. At room
temperature, the QHA with PBEsol yields good agreement for $C_{11}^{E}$ and
$C_{44}^{E}$, however $C_{33}^{D}$ and $C_{44}^{D}$ are overestimated and
$C_{66}$ is underestimated. The change in the predicted elastic constants with
temperature is greater than the change observed in experiment, likely due to
an overestimation of the thermal expansion with temperature (see Fig. 2) and
due to the neglect of explicit phonon interactions within the QHA [59, 60].
The change in the constant $D$-field elastic constants with temperature is
significantly greater than the change seen in experimental measurements, and
is due to the temperature dependence of $e_{ij}$ and
$\hat{\epsilon}^{\infty,S}_{ij}$. These relaxed-ion quantities depend on the
$\Gamma$-point dynamical matrix, and therefore the temperature dependence is
overestimated due to the aforementioned volume and anharmonic effects.

(a)

(b) Figure 3:  Selected elastic constants under constant $E$ fields or $D$
fields computed with the QHA (lines) under unstressed (solid) and stressed
(dashed) conditions compared with previous experimental measurements (markers)
[67, 68]. The axial and shear elastic constants are shown in panels $a$ and
$b$, respectively.

We proceed with the computation of the piezoelectric strain coefficients as a
function of temperature and stress, and compare with existing experimental
values measured under unstressed conditions [68, 67, 69] (see Fig. 4). The
piezoelectric strain coefficients $d_{ij}$ are constructed using the elastic
constant tensor and the piezoelectric stress coefficients [70, 62],

$\displaystyle d_{\alpha
i}(T,\bm{\sigma})=\sum_{j}((\hat{B}^{E}(T,\bm{\sigma}))^{-1})_{ij}e_{\alpha
j}(T,\bm{\sigma}),$ (4)

where $\hat{B}_{ij}^{E}$ denotes the stress-strain coefficient under a
constant electric field. The experimentally measured values of the
piezoelectric strain coefficients [68, 67, 69] show quantitative
inconsistency, as the values vary as much as 50% for $d_{33}$ and 20% for
$d_{31}$. Our quasiharmonic predictions of the $d_{33}$ and $d_{31}$
overestimate the experimental measurements, and the discrepancy can be
explained by differences in the $C_{A_{1}^{\square}A_{1}^{\circ}}^{E}$ and
$C_{33}^{E}$ elastic constants. We compute values of $d_{33}=$ 201.6 pC/N and
$d_{31}=-41.1$ pC/N using Eq. 4, in good agreement with our QHA predictions,
by using the experimental values from Ref. [67] and substituting the values of
$C_{A_{1}^{\square}A_{1}^{\circ}}^{E}$ and $C_{33}^{E}$ to the room
temperature computed values of 108.3 GPa and 56.2 GPa, respectively. The
temperature dependence of the $d_{31}$ piezoelectric coefficient agrees with
the only temperature dependent experimental measurement [69] from low
temperatures up to room temperature, where the QHA predictions increase
rapidly.

(a)

(b)

(c) Figure 4:  Piezoelectric strain coefficients $-d_{31}$, $d_{33}$, and
$d_{15}$ computed with QHA (lines) under unstressed (blue) and stressed (red)
conditions compared with existing experimental measurements (circles) [68, 67,
69] are shown in panels $a$, $b$, and $c$, respectively.

In summary, we have demonstrated the application of the generalized
quasiharmonic approximation to a non-cubic crystal, ferroelectric PbTiO3,
under conditions of finite temperature and stress. The irreducible derivative
approach to computing phonons from finite difference yields the strain
dependent phonons, where dipole-quadrupole effects are successfully
incorporated in the Fourier interpolation. The thermal expansion, elastic
constants, and piezoelectric strain coefficients are computed at finite
temperature and stress. The temperature dependence of the thermal expansion
and elastic constants at zero stress are over estimated by the quasiharmonic
approximation, illustrating the need to solve the vibrational Hamiltonian of
PbTiO3 using a strain-dependent theory which explicitly accounts for phonon
interactions. Our observed limitations of the QHA are not unexpected, as
discrepancies of the QHA are well known in various anharmonic materials [59,
60]. Advances in the computation of finite temperature vibrational properties
from DFT using more sophisticated approximations than the QHA have been
achieved [71, 72, 73, 74, 75, 76], however we are not aware of the application
of any of these theories to the computation of the elastic constant tensor at
finite temperature. These more advanced theories can be straightforwardly
applied as a function of strain to compute thermal expansion and elastic
constants at finite temperature and stress using the general formalism
outlined previously [30], which will be the subject of future work.

This work was supported by the grant DE-SC0016507 funded by the U.S.
Department of Energy, Office of Science. This research used resources of the
National Energy Research Scientific Computing Center, a DOE Office of Science
User Facility supported by the Office of Science of the U.S. Department of
Energy under Contract No. DE-AC02-05CH11231.

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