# PG-Video-LLaVA: Pixel Grounding Large Video-Language Models

Shehan Munasinghe1∗, Rusiru Thushara1∗, Muhammad Maaz1, Hanoona Abdul
Rasheed1,
Salman Khan1,2, Mubarak Shah4, Fahad Khan1,3
1Mohamed bin Zayed University of AI, 2Australian National University
3Linköping University, 4University of Central Florida
Project: https://github.com/mbzuai-oryx/Video-LLaVA

###### Abstract

Extending image-based Large Multimodal Models (LMM) to videos is challenging
due to the inherent complexity of video data. The recent approaches extending
image-based LMM to videos either lack the grounding capabilities (e.g.,
VideoChat, Video-ChatGPT, Video-LLaMA) or do not utilize the audio-signals for
better video understanding (e.g., Video-ChatGPT). Addressing these gaps, we
propose PG-Video-LLaVA, the first LMM with pixel-level grounding capability,
integrating audio cues by transcribing them into text to enrich video-context
understanding. Our framework uses an off-the-shelf tracker and a novel
grounding module, enabling it to spatially and temporally localize objects in
videos following user instructions. We evaluate PG-Video-LLaVA using video-
based generative and question-answering benchmarks and introduce new
benchmarks specifically designed to measure prompt-based object grounding
performance in videos. Further, we propose the use of Vicuna over GPT-3.5, as
utilized in Video-ChatGPT, for video-based conversation benchmarking, ensuring
reproducibility of results which is a concern with the proprietary nature of
GPT-3.5. Our framework builds on SoTA image-based LLaVA model and extends its
advantages to the video domain, delivering promising gains on video-based
conversation and grounding tasks.

**footnotetext: Equal Contribution

##  1 Introduction

Recent efforts on Large Multimodal Models (LMMs), spearheaded by GPT-4V [25],
allow detailed conversations about images but generally do not scale well to
videos. The magnitude of video data scales far beyond other modalities due to
its massive volume on social and internet media. Furthermore, extending LMMs
to videos is challenging due to their complex dynamics with long temporal
context that needs to be understood accurately. Although recent approaches
towards video-LMMs such as VideoChat [15], Video-LLaMA [45], and Video-ChatGPT
[22] have demonstrated capabilities in video comprehension and dialogue, they
lack the crucial feature of visual grounding. Visual grounding in videos aims
to associate the LMM responses to specific objects within the video input.
Addressing this gap, we introduce PG-Video-LLaVA, the first video-LMM capable
of localizing objects appearing in LMM responses. This task leads to enhanced
intractability and demonstrates deep understanding of video content.

Figure 1: Spatio-temporal video grounding on example videos from Vid-STG [48]
(above) and HC-STVG [34] (below) datasets. PG-Video-LLaVA can generate textual
responses with referred objects grounded in the video content (_tennis racket_
and _man_ are localized in the top and bottom examples, respectively).

In PG-Video-LLaVA, we address the unique challenges posed by video data. The
model is designed to track objects within shorter video clips that maintain
consistent camera views, enabling accurate visual grounding across scenes and
motions. This tracking links spatio-temporal segments directly to
conversational elements, enhancing the model’s contextual understanding. A
salient feature of PG-Video-LLaVA is its modular design, allowing for easy
integration with existing grounding modules and the flexibility to adapt to
future enhancements in visual grounding technology. Moreover, PG-Video-LLaVA
enriches its capabilities by incorporating audio context. It achieves this by
leveraging video audio in a form understandable to LLM, which is particularly
useful in situations where the auditory information is essential to the
conversation. This inclusion broadens the model’s understanding, making it
more versatile in interpreting video content.

Furthermore, this work introduces an improved framework for benchmarking
video-based conversational models, pivoting from previous approaches [22] that
predominantly used the proprietary GPT3.5-Turbo model for evaluation. Given
that GPT3.5-Turbo is subject to changes at any time and lacks transparency due
to its closed-source nature, it presents challenges in terms of reliability
and reproducibility. To address this, we propose the use of Vicuna, an open-
source LLM for benchmarking. This shift not only enhances reproducibility but
also improves transparency in the evaluation process. We evaluate PG-Video-
LLaVA using on our improved benchmarks and show notable improvements over
existing video conversational models like Video-ChatGPT [22] and Video-LLaMA
[45] in ungrounded dialogues, achieving state-of-the-art (SoTA) performance.

The key contributions of this work are:

  * •

We propose PG-Video-LLaVA, the first video-based LMM with pixel-level
grounding capabilities, featuring a modular design for enhanced flexibility.

  * •

By incorporating audio context, PG-Video-LLaVA significantly enhances its
understanding of video content, making it more comprehensive and aptly suited
for scenarios where the audio signal is crucial for video understanding (e.g.,
dialogues and conversations, news videos, etc.).

  * •

We introduce improved quantitative benchmarks for video-based conversational
models. Our benchmarks utilize open-source Vicuna LLM to ensure better
reproducibility and transparency. We also propose benchmarks to evaluate the
grounding capabilities of video-based conversational models.

##  2 Related Works

Recent advancements in Large Multimodal Models (LMMs) [18, 50, 8] and Large
Language Models (LLMs) [7, 26, 36] have significantly transformed the
artificial intelligence landscape, particularly in natural language processing
and multimodal tasks. These breakthroughs have enhanced machine learning
models’ ability to understand and generate human-like text, while also
enabling more effective integration of various data types like images, sounds
and videos with textual information. This progress represents a major leap in
creating AI systems that can accurately interpret and interact with a diverse
range of content.

Large Language Models (LLMs): The natural language processing (NLP) field has
undergone a revolution with the advent of LLMs such as GPT [4], LLaMA [36],
OPT [46], and MOSS [27], particularly noted for their zero-shot learning
abilities and adaptability. The development of models like InstructGPT [28]
and ChatGPT [26] has further propelled advancements in conversational AI and
complex query handling, chiefly through instruction tuning. Within the LLaMA
framework, the emergence of open-source models such as Alpaca [35] and Vicuna
[7] exemplifies how instruction tuning can significantly boost model
performance. This shift towards open-source initiatives in language modeling,
highlighted by models like Alpaca and Vicuna, indicates a growing trend
towards more accessible and collaborative approaches in the field. In this
work, we build on the open-source Vicuna LLM and extend it with multimodal
capabilities. We also propose an open-source benchmark for video conversation
and reasoning tasks using Vicuna LLM that is reproducible for fair
evaluations.

Large Multimodal Models (LMMs): The field of AI has witnessed significant
advancements with the development of vision-language models like CLIP [30],
renowned for their impressive zero-shot capabilities using extensive image-
text pairs during training. These models have proven effective in a variety of
applications, from image detection and segmentation [17, 3] to more complex
tasks such as 3D modeling and video analysis [33, 23, 37, 31]. The
introduction of BLIP-2 marked a pivotal transition, pioneering the integration
of image features encoded by a visual encoder with text embeddings, setting
the stage for the evolution into Large Multimodal Models (LMMs). This
advancement influenced subsequent models like LLaVA [19], InstructBLIP [8],
and MiniGPT-4 [50], which further refined image-text feature alignment and
instruction tuning. VideoChat [15], Video-ChatGPT [22] and Video-LLaMA [45]
represents an extension of these LMMs, moving from image-based to video-based
applications, while models such as Otter [14], mPLUG-Owl [42], LLaMa-Adapter
[9], and InternGPT [21] continue to push the boundaries of multimodal
interaction. Despite these significant strides, challenges in achieving robust
visual grounding in LMMs highlight key areas for ongoing research and
development in this dynamic field. Further, effective integration of audio
signals within LMMs for comprehensive video understanding is an open research
question that this work aims to address.

Visual-Language Grounding: Grounded Large Language Models (LLMs) have made
notable progress in enhancing visual and language comprehension. A diverse
array of models including Kosmos-2 [29], Ferret [43], All-Seeing Model [38],
LISA [13], BuboGPT [49], Shikra [5], and GLaMM [32] have employed various
methodologies to master complex grounding tasks. These models demonstrate
proficiency in tasks like referring expression comprehension and image
segmentation, showcasing the advanced image understanding capabilities of
LLMs. Methodologically, Kosmos-2, Shikra, and All-Seeing focus predominantly
on creating language-based context for visual grounding. In contrast, BuboGPT
merges visual elements with language, and LISA leverages vision-language
embeddings for producing segmentation masks. Furthermore, GLaMM is adept at
generating natural language responses linked with object segmentation masks,
facilitating detailed visual-textual interactions. However, challenges remain,
such as LISA’s constrained performance in multi-object scenarios and the
limitations of BuboGPT and GLaMM to image-based applications, not extending to
video processing. To this end, we introduce PG-Video-LLaVA, a video
conversational model with pixel-level grounding capability. Further, PG-Video-
LLaVA incorporates audio transcripts alongside visual and textual data, aiming
to provide a more detailed understanding of video content.

##  3 PG-Video-LLaVA

Figure 2: Architecture of PG-Video-LLaVA:  PG-Video-LLaVA integrates a CLIP-
based visual encoder with a multimodal language model for video understanding.
The CLIP visual encoder extracts spatio-temporal features from videos by
averaging frame-level features across temporal and spatial dimensions. These
features are then projected into the LLM’s input space using a learnable
Multi-Layer Perceptron (MLP). The system features a grounding module for
spatially locating textual descriptions within video frames, a class-agnostic
object tracker, and an entity-matching module. Audio processing incorporates
voice activity detection, phoneme modeling, and Whisper-based audio
transcription, resulting in a multimodal pipeline that facilitates robust
video-question answering. The architecture is trained on a hybrid dataset of
video instructions, enabling the handling of diverse conversational contexts
with high accuracy.

###  3.1 Overview

In this paper, we introduce PG-Video-LLaVA, a novel Large Multimodal Model
(LMM) designed to align video and audio representations with a Large Language
Model (LLM). This integration equips PG-Video-LLaVA with the capability to
proficiently manage both video and audio data in conversational contexts.
Additionally, our method integrates a specialized plug-and-play module for
effective video grounding (see Figure 2).

In constructing PG-Video-LLaVA, our approach integrates sophisticated
mechanisms for aligning video and audio signals with language processing
capabilities, thereby facilitating a comprehensive multimodal analysis.
Central to our model is an advanced CLIP-based video encoder, which has been
specifically adapted to process both spatial and temporal dimensions of video
data. This adaptation enables a deeper understanding of video content, setting
PG-Video-LLaVA apart from conventional image-centric models.

For training, PG-Video-LLaVA utilizes a Video-ChatGPT [22] dataset comprising
100K video instructions derived from ActivityNet-200 [11], supplemented with
our 3K diverse, human-annotated video instructions. This diverse dataset
ensures that the model is well-equipped to handle a broad spectrum of video
contexts with high accuracy. In addition to visual processing, PG-Video-LLaVA
incorporates state-of-the-art audio analysis by leveraging advanced audio
transcription techniques, similar to those employed in WhisperX [2]. This
integration allows the model to process and understand audio inputs
effectively, enhancing its overall multimodal interpretation capabilities.

While PG-Video-LLaVA’s foundation is based on the LLaVA-1.5 [18] framework, it
is extended for videos to incorporate spatio-temporal representations, audio
understanding and visual grounding capabilities. Its unique combination of
enhanced video encoding, extensive training dataset, integrated audio
processing and grounding capability marks it as a forward step in the field of
LMMs.

###  3.2 Architecture

Our architecture utilizes the CLIP ViT-L/14@336 as the visual encoder, which,
unlike its original image-focused design, has been adapted for video
processing in PG-Video-LLaVA. This adaptation is crucial for the model to
capture spatio-temporal representations in videos effectively. In our model,
video samples are represented as $V_{i}\in\mathbb{R}^{T\times H\times W\times
C}$, where $T$ denotes the frame count. The encoder processes each of the $T$
frames independently, treating them as a series of images. This leads to the
generation of frame-level embeddings $x_{i}\in\mathbb{R}^{T\times h\times
w\times D}$, where $h=H/p$ and $w=W/p$, with $p$ being the patch size (14 for
ViT-L/14) and $N=h\times w$ indicating the total token count.

To construct a comprehensive video-level representation, we apply average
pooling across the temporal dimension of these frame-level embeddings,
resulting in a temporal representation $t_{i}\in\mathbb{R}^{N\times D}$. This
temporal pooling technique effectively amalgamates information across multiple
frames. Similarly, for spatial information, we achieve spatial representation
$z_{i}\in\mathbb{R}^{T\times D}$ through average pooling along the spatial
dimension. The final video-level features $v_{i}$ are a combination of these
temporal and spatial features, as shown in the equation:

$v_{i}=[t_{i}\quad z_{i}]\in\mathbb{R}^{(T+N)\times D}$ (1)

In our architectural design, the spatio-temporal feature extraction is
inspired by Video-ChatGPT [22], with an additional enhancement of employing a
higher resolution of 336×336 pixels to encode frame-level features.

Within the architecture of PG-Video-LLaVA, we have implemented a learnable
Multi-Layer Perceptron (MLP), designated as $g$, to serve as our cross-modal
connector. This MLP is intricately designed to project video-level features
into the embedding space of the language decoder. This is inspired from
LLaVA-1.5 [18], aiming to optimize the model’s multi-modal capabilities beyond
what could be achieved with a simple linear projection. The process yields
language embedding tokens $Q_{v}$, calculated as follows:

$Q_{v}=g(v_{i})\in\mathbb{R}^{(T+N)\times K}$ (2)

Text queries, denoted as $Q_{t}\in\mathbb{R}^{L\times K}$ where $L$ is the
length of the query, are tokenized to be dimensionally compatible with these
video embeddings. The combination of $Q_{v}$ and $Q_{t}$ is then fed into the
language decoder, facilitating the seamless integration of video and textual
data within the model (see Figure 2).

####  3.2.1 Audio Modality Integration

In PG-Video-LLaVA, we have integrated an audio processing pipeline that
significantly enhances the video-question answering capabilities by
incorporating audio cues from the input, drawing inspiration from the
architecture of WhisperX[2]. The process begins with the deployment of a Voice
Activity Detection (VAD) model. This model is crucial for pinpointing speech-
containing temporal segments within the audio track. Following the VAD’s
identification of speech segments, these segments undergo processing—cutting,
merging, and padding—to align with the input specifications of the Whisper
model [24]. Simultaneously, a phoneme segmentation model operates in parallel,
producing phone-level segmentations essential for the subsequent alignment of
raw transcriptions with the audio.

The VAD model serves a dual purpose: it not only identifies speech segments
but also aids in filtering out non-speech audio components. To enhance the
compatibility of transcriptions generated by Whisper with our model, we
integrate Whisper-AT[10]. This advanced version of the Whisper model
specializes in audio tagging. It annotates the audio stream with labels from
an extensive set of 527 audio event classes, allowing for precise temporal
resolution.

The audio transcripts are then subjected to a multi-stage filtering process.
Initially, a VAD-based filter is applied, followed by a phoneme-based forced
alignment using the Whisper model, ensuring temporally accurate text
transcriptions. Utilizing Whisper’s language identification feature, we
eliminate non-English speech segments at this stage. For each identified
sentence segment, we apply Whisper-AT [10] for audio tagging, focusing on the
top three predicted audio classes. Segments that do not predominantly feature
‘speech’, or where ‘music’ probabilities significantly exceed ‘speech’, are
excluded from further processing.

Finally, the integration of the audio transcript with the video component is
executed through a carefully designed prompt template. This template is
pivotal in guiding the system to understand user instructions, assimilate the
video frames, and incorporate the transcriptions generated by the automatic
speech recognition model. This structured approach ensures that PG-Video-LLaVA
efficiently leverages all available modalities—visual and auditory—thereby
enabling users to achieve task completion and query resolution based on a
comprehensive analysis of both visual and auditory content (refer to Figure 2
for details).

Model | Evaluation Metrics
---|---
Correctness | Detail Orientation | Contextual Understanding | Temporal Understanding | Consistency
LLaMA Adapter [9] | 2.34 | 2.43 | 2.65 | 2.20 | 3.02
Video Chat [15] | 2.49 | 2.82 | 2.92 | 2.27 | 3.11
Video-LLaMA [45] | 2.29 | 2.61 | 2.68 | 2.17 | 2.87
Video-ChatGPT [22] | 2.51 | 2.53 | 2.85 | 2.32 | 3.10
PG-Video-LLaVA (7B) | 2.73 | 2.89 | 3.13 | 2.44 | 3.40
PG-Video-LLaVA (13B) | 2.86 | 2.95 | 3.23 | 2.53 | 3.49

Table 1: Performance benchmarking of video-based conversational models.
Comparative performance evaluation of PG-Video-LLaVA against various models
using the benchmarking framework from Video-ChatGPT [22]. The metrics include
correctness, detail orientation, contextual understanding, temporal
understanding, and consistency. The updated assessment pipeline incorporates
Vicuna1.5 [7] for enhanced reproducibility, replacing GPT-3.5. Results
indicate that PG-Video-LLaVA achieves favourable performance across all
metrics, particularly in contextual and temporal understanding, as compared to
foundational models and recent advancements in the field.

####  3.2.2 Grounding Module

In PG-Video-LLaVA, our spatial grounding approach starts with processing
video-question pairs to generate textual descriptions. These descriptions are
then used for grounding within the video frames. Key noun phrases are
extracted from the generated text using Vicuna, targeting the most critical
content aspects. Simultaneously, an image tagging model, RAM [47], tags visual
elements in each frame, creating a detailed map of the video content.

The video is segmented into smaller parts using PySceneDetect [1], based on
changes in scene composition. This segmentation facilitates a more focused
grounding process. In each segment, our grounding ensemble, composed of
GroundingDINO [20], DEVA [6], and SAM [12], utilizes the image tags to create
segmentation masks and tracking IDs for the identified visual elements.

The visual cues from these segmentation masks are then matched with the
textual noun phrases using CLIP [30]. This matching process links text to the
corresponding visual elements in the video, enhancing our understanding of the
content.

For quantitative analysis, we use the tagging output of the RAM model to
identify the most relevant noun phrase or referring expression in PG-Video-
LLaVA’s responses. These phrases are input into our grounding modules, which
then generate segmentation masks and tracking IDs. We measure the spatial
grounding accuracy of our model by calculating the Intersection over Union
(IoU) between these segmentation masks and ground truth bounding boxes.

This systematic approach enables PG-Video-LLaVA to effectively ground textual
descriptions within video content, thereby improving the performance and
interpretability of video-question answering systems.

Figure 3: Qualitative results comparison of Video-ChatGPT vs PG-Video-LLaVA
(Ours) Qualitative analysis of video descriptions generated by Video-ChatGPT,
PG-Video-LLaVA (7B), and PG-Video-LLaVA (13B) models. The evolution in model
performance is evident, with enhancements in the accuracy of information,
richness of descriptive detail, and alignment with the video’s context and
sequence of events as we move from the baseline Video-ChatGPT to the more
advanced PG-Video-LLaVA (13B) model.

##  4 Experiments

###  4.1 Implementation Details

We build our stronger baseline on top of LLaVA-1.5 which utilizes CLIP
ViT-L/14@336 as the image encoder and Vicuna 1.5 as the LLM. We only tune the
MLP projection layers during training, while keeping the rest of the
architecture frozen. We finetune the model for 3 epochs using a learning rate
of 2$e^{-5}$ and an overall batch size of 32. The training of our 7B and 13B
models took around 6 hours and 15 hours respectively on 4 A100 80GB GPUs.

For audio transcript extraction, base Whisper model is used. Our grounding
module is based on GroundingDINO-T variant and CLIP ViT-B/32. For the image-
tagging model we use RAM Swin-Large variant (with input size 384). DEVA
Tracker is applied under online-setting in our experiments.

Vicuna-13b-v1.5 model is used in performing video-based conversational
benchmarking, zero-shot question answering evaluation, and extracting the key
noun or referring expression from the model output in the quantitative
evaluation of the spatial grounding task. Further, Vicuna-13b-v1.5 was used to
implement the entity matching as in [49].

###  4.2 Stronger Baseline

This section provides an overview of the quantitative evaluations conducted to
determine the effects of the strengthened baseline on PG-Video-LLaVA. To
evaluate the impact of the enhanced baseline on PG-Video-LLaVA, we apply the
benchmarking framework from Video-ChatGPT[22]. This framework measures
performance on several axes critical for video-based conversational agents,
including correctness of information, detail orientation, contextual
understanding, temporal understanding, and consistency.

In order to facilitate a reliable and reproducible evaluation, we have updated
our assessment pipeline by replacing GPT-3.5 with Vicuna-13b-v1.5. This
adjustment addresses the limitations in reproducibility inherent to the
closed-source nature of GPT-3.5. Subsequently, we have re-assessed both PG-
Video-LLaVA and other recent models to ensure a fair and consistent
comparison. The results shown in Table 1 demonstrate that PG-Video-LLaVA
outperforms the foundational Video-ChatGPT model and exhibits superior
performance when compared to other recent contributions in the domain.

Following the quantitative assessment, the qualitative results in Figure 3
indicate the enhanced baseline’s impact on PG-Video-LLaVA’s performance. The
PG-Video-LLaVA (13B) model exhibits improved accuracy in the information
presented, a deeper level of descriptive detail, and a stronger alignment with
the context and temporal progression of the videos. This advancement is
particularly noticeable in the precise depiction of the child’s engagement
with their surroundings and the giraffe’s behaviour, indicating a refined
interpretation of both the activities and their settings. These qualitative
insights are consistent with the quantitative results, highlighting the
augmented baseline’s role in advancing PG-Video-LLaVA’s capacity in video
understanding.

Figure 4: Qualitative Results for Video Grounding: Visual representation of
the grounding capability of advanced video-conversational capabilities of PG-
Video-LLaVA. The highlighted regions in each video frame indicate the model’s
ability to identify and spatially locate key subjects mentioned in the textual
description, such as the giraffe, the statue, and the gymnast on a balance
beam.

Model |  VidSTG [48] |  HC-STVG [34]
---|---|---
Grounding DINO [20] | 25.3 | 19.5
Video-LLaMA [45] | 28.6 | 26.1
Video-ChatGPT [22] | 32.8 | 20.8
PG-Video-LLaVA (7B) | 34.2 | 28.3
PG-Video-LLaVA (13B) | 35.1 | 27.3

Table 2: Performance of PG-Video-LLaVA and other models on spatial grounding
task: Evaluated using the VidSTG and HC-STVG benchmarks, the results
demonstrate PG-Video-LLaVA’s favorable spatial grounding capabilities, as
evidenced by its ability to generate accurate descriptive responses and
effectively locate referring expressions within video frames. The table shows
the model’s progress, particularly in the 13B version, showcasing its
performance among other SoTA video-conversational models.

Model |  MSVD-QA [39] |  MSRVTT-QA [40] |  TGIF-QA [16] |  Activity Net-QA [44]
---|---|---|---|---
| Accuracy | Score | Accuracy | Score | Accuracy | Score | Accuracy | Score
FrozenBiLM [41] | 32.2 | – | 16.8 | – | 41.0 | – | 24.7 | –
LLaMA Adapter [9] | 53.7 | 3.3 | 45.6 | 3.2 | 54.3 | 3.3 | 37.3 | 3.2
Video LLaMA [45] | 48.6 | 3.2 | 32.8 | 2.8 | 51.4 | 3.4 | 27.1 | 2.9
Video-ChatGPT [22] | 62.6 | 3.6 | 50.0 | 3.3 | 66.5 | 3.7 | 40.8 | 3.3
PG-Video-LLaVA | 64.1 | 3.7 | 51.6 | 3.3 | 66.8 | 3.8 | 39.9 | 3.3

Table 3: Zeroshot video-based question-answering: Comparison of PG-Video-LLaVA
with other video generative models. The latest available models are used for
all the approaches and the benchmarks are calculated using open-source Vicuna
LLM. PG-Video-LLaVA performs better than the previously proposed video-based
conversational methods. Figure 5: Qualitative Results for Including Audio
Modality: The figure illustrates the integrated audio processing pipeline that
augments video-question answering with audio cues. It provides side-by-side
comparisons showing how audio cues offer additional context, leading to a more
accurate interpretation of the video content, as seen in the examples above.

###  4.3 Spatial Grounding in Videos

To quantitatively assess PG-Video-LLaVA’s spatial grounding capability, we
conducted quantitative evaluations of PG-Video-LLaVA’s spatial grounding
capabilities using two benchmarks that are derived from the test set of the
VidSTG [48] and HC-STVG [34] datasets. Due to the novelty of integrating
spatial grounding within video-conversational models, we highlight the modular
nature of our grounding pipeline, which can be incorporated with other state-
of-the-art video conversation models. For the VidSTG dataset, we selectively
processed interrogative prompts to assess the grounding accuracy. The model
generates descriptive textual responses to these prompts, from which
Vicuna-13b-v1.5 extracts relevant referring expressions. These expressions are
then spatially grounded in the video frames using our grounding pipeline. For
the HC-STVG dataset, interrogative prompts are first mined from the text
captions using Vicuna and then used similarly to VidSTG prompts.

The results shown in Table 2 position PG-Video-LLaVA alongside alternative
methods using the same benchmarks, demonstrating our model’s enhanced ability
to accurately answer questions, thereby leading to improved spatial grounding
performance.

The qualitative results shown in Figure 4 emphasize the model’s refined
spatial grounding precision. The accurate overlay of masks on the subjects
within the videos confirms the model’s adeptness at correlating textual
descriptors with visual elements, a critical aspect of contextual
comprehension. This refined ability is crucial for applications that integrate
visual data with language, improving the model’s utility in environments that
demand rich, interactive visual and linguistic processing.

###  4.4 Zero-Shot Visual Question Answering

For PG-Video-LLaVA, zero-shot question-answering (QA) capabilities were
evaluated quantitatively using several established open-ended QA datasets:
MSRVTT-QA[40], MSVD-QA [39], TGIF-QA [16], and ActivityNet-QA [44]. These
datasets are benchmarks for assessing a model’s ability to generate accurate
answers without any dataset-specific fine-tuning. We adopted a zero-shot
evaluation methodology, utilizing Vicuna-13b-v1.5 to evaluate the model’s
understanding and predictive accuracy, with scores assigned on a scale from 1
to 5. Results are presented in Table 3.

In comparison to Video-ChatGPT, PG-Video-LLaVA demonstrates superior
performance, surpassing not only the predecessor but also other notable models
in the field, such as FrozenBiLM[41] and Video Chat[15]. The results from our
evaluations indicate that PG-Video-LLaVA has significantly enhanced its
ability to comprehend video content and generate contextually relevant
answers, thus establishing a new state-of-the-art in zero-shot VideoQA.

As shown in Figure 4, our method is able to visually ground the key objects in
the given video. Improvement in the model’s capability to describe the content
in the video is demonstrated in Figure 3. Further, it can be observed that
adding the audio modality helps make correct outputs, whereas the model
without audio modality fails to capture those details from visual content
alone (Figure 5).

##  5 Conclusion

In this work, we introduced PG-Video-LLaVA, a novel video-based conversational
model equipped with pixel-level grounding capabilities. PG-Video-LLaVA
enhances image-based conversational models by extracting spatio-temporal
features essential for comprehensive video understanding. It incorporates
filtered audio transcripts to enrich the interpretation of visual scenes where
audio cues are pivotal. Additionally, we developed a novel grounding module
capable of tracking and generating pixel-level grounding of objects within
videos. To promote reproducibility, we propose quantitative benchmarks for
video-based conversational models, utilizing the open-sourced LLM Vicuna
instead of GPT-3.5, as employed by previous approaches. These benchmarks are
specifically designed to evaluate grounding capabilities. In summary, this
work represents the first effort to integrate grounding capabilities into
video-based LMMs.

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# Suppression of turbulent dynamo in time irreversible turbulence

A.V. Kopyev 1, A.S. Il’yn 1,2, V.A. Sirota 1 and K.P. Zybin 1,2 111Electronic
addresses<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1
P.N.Lebedev Physical Institute of RAS, 119991, Leninskij pr.53, Moscow, Russia
2 National Research University Higher School of Economics, 101000,
Myasnitskaya 20, Moscow, Russia

###### Abstract

The conventional theory of magnetic field generation in a turbulent flow
considers time-reversible random flows. However, real turbulent flows are
known to be time irreversible: the presence of energy cascade is an intrinsic
property of turbulence. We generalize the ’standard’ model to account for the
irreversibility. We show that even small time asymmetry leads to significant
suppression of the dynamo effect at low magnetic Prandtl numbers, increases
the generation threshold and may even make generation impossible for any
magnetic Reynolds number. We calculate the magnetic energy increment as a
function of the parameters of the flow.

Introduction.  The theory of passive magnetic field transport in a turbulent
flow has numerous applications and, in particular, is likely to explain the
existence of magnetic field in many astrophysical objects Parker ;
Brandenburg-rev ; Sokoloff ; Kuznetso ; Kit . The basic idea of turbulent
dynamo is that the magnetic lines stretch as they are carried by turbulent
motion. However, a very wide range of magnetized astrophysical objects of
different scales implies very different parameters of the medium. A small-
scale turbulent dynamo, i.e., a dynamo that takes place at scales much less
than the integral scale of turbulence FGV ; Brandenburg12 , depends on two
basic parameters: the hydrodynamic Reynolds number Re and the magnetic
Reynolds number Rm. Their ratio Rm/Re=Pm is called magnetic Prandtl number: it
indicates whether the viscous scale $r_{\nu}$ of turbulence is larger (Pm$>1$)
or smaller than the resistive scale $r_{d}$. While both Rm and Re are large in
cosmic plasmas, their ratio varies crucially, and Pm is either very large or
very small in astrophysical objects. The Prandtl number determines the small-
scale dynamo mechanism Rincon .

The high-Pm turbulence is observed in interstellar and intergalactic media
Rincon ; Han . In these objects, the characteristic scale of magnetic field
generation is deep inside the viscous range of turbulence, so the stretching
of magnetic lines is exponential zeld ; the existence of the dynamo effect for
this case is confirmed by different theoretical approaches FGV ; Chertkov99 ;
Kazantsev  and by numerical simulations schek-large-Pm .

To the contrary, in stellar and planetary magnetism, in protostellar disks
etc., one observes low-Pm media (Pm $\sim 10^{-3}\div 10^{-7}$) Rincon ;
Rempel-rev . In this case, small-scale dynamo can occur only in the inertial
range of turbulence, at scales $\rho\gtrsim r_{d}\gg r_{\nu}$. The existence
of this effect is still questionable. Direct numerical simulations are very
difficult to perform in this range of parameters Iskakov-Schek07 ; Brand-
bottleneck ; some authors report the existence of generation, and even the
existence of a universal limit Rmc that guarantees the dynamo for any Re
Schek-Iskakov07 ; Warnacke , although their estimates for the threshold Rmc
differ essentially, and the values of Pm and especially Re that they achieve
(Pm$\gtrsim 10^{-3}$, Re$\lesssim 10^{5}$) are still far from the observed in
astrophysic objects. Simulations performed by means of the shell model Buchlin
; Verma-Kumar ; Frick-rev  and the implicit large-eddy simulations Rempel-14 ;
Hotta-16 ; Kapyla-18 ; Rempel-rev  allow to get small-scale dynamo at more
realistic parameters, however, the results may depend significantly on the
model assumptions Haugen ; Petrosy-rev . Observations of the Sun show the
presence of small-scale magnetic field, but it is unclear whether it is,
indeed, generated in the inertial range by turbulent motion Petrovay , or it
is merely a result of fragmentation of the large-scale field produced by some
other process; Stenflo Stenflo-1 ; Stenflo-2  argues that the latter is more
likely. Theoretical consideration predicts the possibility of the dynamo
effect in the low-Pm limit Vain . Kazantsev Kazantsev  has found the
dependence of the growth increment on the scaling properties of isotropic
incompressible flow with infinite Re (infinite integral scale of turbulence),
under the assumption of Gaussian and delta correlated velocity statistics. In
later papers, the model was extended for the case of finite Re NRS ; VK ; Rog
; Vinc ; Arponen ; Bold ; Mal ; Schober .

However, these theoretical works consider only time-reversible random flows.
Actually, the time asymmetry is associated with the third-order velocity
correlator, which is zero in the classical Kazantsev model. Meanwhile, the
real turbulent flows are known to be time irreversible: this follows from the
existence of energy cascade K41 ; Frisch . The account of small time
irreversibility in the high-Pm limit has shown the decrease of all magnetic
field statistical moments growth rate ApJ ; feedback ; stationar . So, there
is a reasonable question: how does it affect the magnetic field generation in
the low-Pm case?

In this Letter, we consider the influence of the third order velocity
correlator on the ability of a low-Pm conductive fluid to generate magnetic
field. We find the instability criterion and show that the magnetic generation
in a turbulent flow may be suppressed completely if the third correlator is
large enough. We show that in this case, even for infinite Rm, there is no
generation for Pm small enough but finite.

Definitions and the $V^{3}$ model.  The evolution of magnetic field in a
(random) velocity field ${\bf v}({\bf r},t)$ is described by the equation

$\frac{\partial\mathbf{B}(\mathbf{r},t)}{\partial
t}=\mathrm{rot}\bigl{[}\mathbf{v}(\mathbf{r},t)\times\mathbf{B}(\mathbf{r},t)\bigr{]}+\kappa\Delta\mathbf{B}(\mathbf{r},t)$
(1)

where $\kappa$ is the magnetic diffusivity. The magnetic field energy in the
problems under consideration is smaller than the kinetic energy of the flow at
all scales, and the feedback influence of the magnetic field on the velocity
dynamics is proportional to $B^{2}$, so one can neglect this feedback and
consider ${\bf v}({\bf r},t)$ as stationary random process with given
statistics.

We consider isotropic and homogenous stationary flow in incompressible fluid.
The Kazantsev equation for the second-order magnetic field correlator was
derived for Gaussian $\delta$-correlated in time velocity field,

$\langle
v_{i}(\mathbf{r},t)v_{j}(\mathbf{r}_{1},t_{1})\rangle=D_{ij}(\mathbf{r}-\mathbf{r}_{1})\,\delta_{\epsilon}(t-t_{1})$
(2)

Here $\delta_{\epsilon}$ is not a ’physical’ singularity but a regularized
$\delta$ function: a narrow peak with unitary square and width smaller than
all physical scales. To avoid ambiguities, for any real (short-correlated)
velocity field one can define $D_{ij}$ by the integral

$D_{ij}(\bm{\rho})=\int\langle
v_{i}(\mathbf{r},t)v_{j}(\mathbf{r}+\bm{\rho},t+\tau)\rangle\mathrm{d}\tau$
(3)

(It does not depend on ${\bf r}$ and $t$ because of space and time
homogeneity.) To take account of the third-order velocity correlator, we
introduce

$F_{ijk}(\bm{\rho}_{1},\bm{\rho}_{2})=\int\langle
v_{i}(\mathbf{r},t)v_{j}(\mathbf{r}+\bm{\rho}_{1},t_{1})v_{k}(\bm{r}+\bm{\rho}_{2},t_{2})\rangle\mathrm{d}t_{1}\mathrm{d}t_{2}$
(4)

Now we use the $V^{3}$ model JoSS ; ApJ ; Spectrum . It is a generalization of
the Kazantsev-Kraichnan model Kazantsev ; Kraichnan-passive  and implies two
assumptions: 1) the third order correlator is assumed to be singular in time:

$\displaystyle\langle
v_{i}(\mathbf{r},t)v_{j}(\mathbf{r}+\mathbf{\rho}_{1},t_{1})v_{k}(\mathbf{r}+\mathbf{\rho}_{2},t_{2})\rangle=F_{ijk}(\mathbf{\rho}_{1},\mathbf{\rho}_{2})\frac{\delta_{\epsilon}(t-t_{1})\delta_{\epsilon}(t-t_{2})+\delta_{\epsilon}(t-t_{1})\delta_{\epsilon}(t_{1}-t_{2})+\delta_{\epsilon}(t-t_{2})\delta_{\epsilon}(t_{1}-t_{2})}{3},$
(5)

and 2) the higher order correlators are set to zero. We will discuss the
validity of these assumptions a few strings later.

Under these assumptions, we derive the generalized Kazantsev equation (see
Supplementary for derivation):

$\begin{array}[]{rl}\frac{\partial}{\partial
t}G&=2\kappa\left(G^{\prime\prime}_{\rho\rho}+\frac{4G^{\prime}_{\rho}}{\rho}\right)+bG^{\prime\prime}_{\rho\rho}+\left(b^{\prime}+4\frac{b}{\rho}\right)G^{\prime}_{\rho}+\left(b^{\prime\prime}+4\frac{b^{\prime}}{\rho}\right)G\\\
&+2cG^{\prime\prime\prime}_{\rho\rho\rho}+3\left(c^{\prime}+\frac{4c}{\rho}\right)G^{\prime\prime}_{\rho\rho}+\left(3c^{\prime\prime}+\frac{12c^{\prime}}{\rho}+\frac{8c}{\rho^{2}}+\frac{8d}{\rho}\right)G^{\prime}_{\rho}+\left(c^{\prime\prime\prime}+\frac{6c^{\prime\prime}}{\rho}+\frac{4c^{\prime}}{\rho^{2}}-\frac{4c}{\rho^{3}}+\frac{4d^{\prime}}{\rho}+\frac{12d}{\rho^{2}}\right)G\end{array}$
(6)

Here

$G(\rho,t)=\langle B_{L}(\mathbf{r}+\bm{\rho},t)B_{L}(\mathbf{r},t)\rangle$
(7)

is the magnetic field correlator, and

$\begin{array}[]{l}b(\rho)={n}_{i}{n}_{j}\left(D_{ij}(0)-D_{ij}(\rho)\right),\\\
c(\rho)=\tfrac{1}{2}{n}_{i}\Pi_{jk}F_{ijk}(\rho,\rho),\\\
d(\rho)=(\tfrac{1}{4}\Pi_{ij}\Pi_{kl}-\tfrac{1}{2}n_{i}n_{k}\Pi_{jl})\frac{\partial}{\partial{\rho_{1}}_{l}}F_{ijk}(\bm{\rho}_{1},\bm{\rho})\bigl{|}_{\bm{\rho}_{1}=\bm{\rho}},\\\
\mathbf{n}=\bm{\rho}/\rho\ ,\ \ \Pi_{ij}=\delta_{ij}-n_{i}n_{j}\ ,\ \
B_{L}={\bf B}\cdot{\bf n}\end{array}$ (8)

represent the velocity correlators.

The applicability of the first assumption in long-term asymptotics for smooth
velocity field (Batchelor regime, $\rho<r_{\nu}$) was proved in fin-time . For
rough velocity field (inertial range, $\rho>r_{\nu}$) the correctness of this
assumption remains a hypothesis: we suppose that the effect of the finite
correlation time is weaker than that of the asymmetry.

The insufficiency of the finite-correlation time corrections for the Kazantsev
model was confirmed in DNS Mason . The second assumption produces non-physical
artefacts, as any non-Gaussian model with a finite number of cumulants does
Klyatskin ; Mill ; MY . Specifically, Eq. (6) contains the third order
derivative, while the Kazantsev equation ($c=d=0$) is the second order
differential equation. So, in addition to corrections to the two ’Kazantsev’
modes, Eq. (6) provides a new eigenmode that is non-physical. The account of
the whole set of the higher order correlators would destroy this ’parasite’
mode, but one can choose them small enough to have no significant effect on
the two ’physical’ modes.

For large Pm, it is enough to restrict the consideration by the viscous scales
range. Eq.(6) then reduces to the equation derived in ApJ . For small Pm,
careful analysis of the inertial range is needed.

Rough velocity field. In the inertial range of hydrodynamical turbulence,
$b(\rho)\propto\rho^{4/3}$; indeed, for Kolmogorov turbulence Vain ; VK ; Rog
one has

$\delta v(\rho)\propto\rho^{1/3},\,\,\tau_{c}(\rho)\propto\rho/\delta
v\propto\rho^{2/3}\,\Rightarrow\,\,b(\rho)\propto\delta
v^{2}\tau_{c}\propto\rho^{4/3}$

Following Kazantsev Kazantsev , we hereafter consider a more general rough
velocity field with the power-law structure function
$b(\rho)\propto\rho^{1+s}$. From dimensional analysis we then get

$\left\\{\begin{array}[]{rl}&\delta v(\rho)\propto\rho^{s}\\\
&\tau_{c}(\rho)\propto\rho^{1-s}\end{array}\right.\Rightarrow\left\\{\begin{array}[]{llll}b(\rho)&&\propto&\rho^{1+s}\\\
c(\rho)&&\propto&\zeta\,\rho^{2+s}\\\
d(\rho)&&\propto&-\beta\,\zeta\,\rho^{1+s},\end{array}\right.$ (9)

Thus, we define

$\zeta=\frac{c(\rho)}{\rho\,b(\rho)}\ ,\ \zeta\beta=-\frac{d(\rho)}{b(\rho)}$
(10)

The $\zeta$ and $\beta$ are dimensionless parameters that characterize the
third order correlator and that are constants inside the hydrodynamic inertial
range of scales; $\zeta$ must be small to ensure the applicability of the
$V^{3}$ model, while $\beta$ is, generally, not restricted. From the analysis
JoT  of numerical simulations JHTDB-1 ; JHTDB-3  it follows that $\beta\simeq
2$. We note that in (9),(10) we neglect possible intermittency: there is no
evidence of anomalous scaling for time-integrated functions.

We also introduce the convenient length scale

$r_{d}=\rho\left(2\kappa/b(s)\right)^{1/(1+s)}\ ,\rho\gg r_{\nu}$

For $\rho\gg r_{d}$, the molecular diffusion is negligibly small, and one can
omit the first bracket in (6). Then

$\begin{array}[]{rl}\frac{1}{2\kappa}\frac{\partial}{\partial
t}G=&\left(\frac{\rho}{r_{d}}\right)^{1+s}\left(G^{\prime\prime}+\frac{5+s}{\rho}G^{\prime}+\frac{(4+s)(1+s)}{\rho^{2}}G\right)+\\\
&+\zeta\,\left(\frac{\rho}{r_{d}}\right)^{1+s}\biggl{(}2\rho\,G^{\prime\prime\prime}+({18+3s})G^{\prime\prime}+\frac{3(5+s)(2+s)-8(\beta-1)}{\rho}G^{\prime}\left.+\frac{(4+s)\bigl{(}(4+s)(1+s)-4\beta\bigr{)}}{\rho^{2}}G\right),\
\rho\gg r_{d}\end{array}$ (11)

Generation threshold. In the case $\zeta=0$, (6) turns into the Kazantsev
equation.The magnetic field generation in this model is possible for $s>0$,
while for $s\leq 0$ the magnetic field decreases Kazantsev . Indeed, setting
$\frac{\partial}{\partial t}G=0$ in (11) we get two stationary modes, which
for $\rho\gg r_{d}$ obey the scaling law:

$G=\rho^{-\frac{s+4}{2}}\left(C_{+}\rho^{\eta_{K+}}+C_{-}\rho^{\eta_{K-}}\right)\
,\eta_{K\pm}=\pm\frac{\sqrt{3}}{2}\sqrt{-s(s+4)}$ (12)

The exponential in time solutions

$G=e^{\Gamma t}G(\rho)\ ,\partial G/\partial t=\Gamma G$ (13)

of the Kazantsev equation exist if and only if the stationary solution
oscillates. (This can be derived from the oscillation theorem, keeping in mind
that the Kazantsev equation reduces to a Schr$\ddot{{\rm o}}$dinger equation;
less formally, this implies that the solution (12) must match both with the
’viscous’ solution at $\rho<r_{d}$ and with the large-scale solution at
$\rho\to\infty$.)

We now use the same approach to find the magnetic field generation threshold
for $\zeta\neq 0$. Taking $\rho\gg r_{d}$ for the stationary mode of (11) we
get the power-law solutions $G\propto\rho^{-\frac{s+4}{2}+\eta_{\pm}}$ with
the characteristic equation for the powers $\eta_{\pm}$:

$\chi(\eta)=\left(\eta^{2}+\frac{3}{4}s(s+4)\right)\left(1+2\zeta\eta\right)-8\zeta\beta\eta=0$
(14)

Two of the solutions are close to those found for $\zeta=0$. The third
solution is $\sim\zeta^{-1}$, and it is a non-physical artefact. Again, the
existence of generation corresponds to oscillating stationary mode
eigenfunctions, i.e., to imaginary roots of (14). The minimum of the function
$\chi(\eta)$, to the second order in $\zeta$, is

$\chi_{min}=\frac{3}{4}s(s+4)-\zeta^{2}\left(4\beta-\frac{3}{4}s(s+4)\right)^{2}$

We see that the presence of $\zeta$ of any sign lowers the minimum. The
generation is still possible if $\chi_{min}>0$, i.e., if

$|\zeta|<\zeta_{cr}=\frac{\sqrt{\frac{3}{4}s(s+4)}}{|4\beta-\frac{3}{4}s(s+4)|}\simeq\frac{\sqrt{3s}}{4|\beta|}$
(15)

For $s=1/3$, $\beta=2$ we get $\zeta_{cr}\simeq 1/7$. For larger $\zeta$,
there is no dynamo.

We note that the role of $\beta$ is crucial: actually, one can see from (14)
that the terms $\zeta\beta$ are more effective to damp the oscillations than
the terms with $\zeta$ alone. This means that the influence of $d(\rho)$ in
(6) is more important than of $c(\rho)$.

Figure 1: The maximal growh rate $\Gamma$ as a function of $s$ for different
values $\zeta\beta$ in the minimal $V^{3}$ model; $\Gamma$ is normalized by
$DPm^{(1-s)/(1+s)}$ where $D$ is characteristic frequency of small-scale
eddies, $D=-\frac{3}{4}\left.\frac{\partial^{2}}{\partial\rho^{2}}\langle
v_{L}(\bm{\rho})v_{L}({\bf 0})\rangle\right|_{\rho=0}$.  Figure 2: The
critical value $(\zeta\beta)_{cr}(s)$ calculated in the frame of ’minimal’
$V^{3}$ model and $V^{3}$ model (with $\beta=2$).

Growth increment  Even if the generation is not completely suppressed, its
rate can be significantly slowed down. To investigate the dependence of the
generation rate on the time irreversibility, we focus on the terms in (6) that
contain $d$, and set $c=0$. We call this ’minimal model’. The estimate of
$\zeta_{cr}$ obtained in the ’minimal model’ coincides with (15) to the first
order in $s$.

We are looking for exponentially growing solutions (13). Now, the ’minimal
model’ equation is a second-order ordinary differential equation containing
the growth increment $\Gamma$; it has a discrete spectrum of solutions for
$\Gamma>0$ if $\zeta$ satisfies (15).

By means of (blobs, , Appendix A), we find numerically the maximal possible
$\Gamma$ for given $s$ and $\zeta$, assuming $\beta=2$. We normalize it by
$Pm^{\frac{1-s}{1+s}}$, so that it becomes independent of the regularization
as Pm$\to 0$ Schober . The results are presented in Fig.1. We see that they
differ significantly from the Kazantsev case: for $s=1/3$ the generation
suppression is essential even for $\zeta\beta\lesssim 0.2$. The vertical
asymptotes in the logarithmic coordinates correspond to the generation
threshold. The dependence of the critical value of $\zeta\beta$ on $s$ is
given in Fig.2. (For analytical expression, see Supplementaty.) It confirms a
good agreement between the $V^{3}$ and the ’minimal’ $V^{3}$ models (with and
without $c(\rho)$) up to $s=1/3$. The whole analysis is performed in the limit
of infinite integral scale, i.e., Rm$=\infty$; thus, from Fig.2 it follows
that, in the frame of our model, for $\zeta\beta\gtrsim 1/3$ generation is
impossible for any magnetic Reynolds number if Pm is small enough.

Summarizing, the presence of even small irreversibility is shown to suppress
partially or completely the magnetic field generation. The mixed velocity and
gradients correlator $d(\rho)$ is shown to play a crucial role in this anti-
dynamo effect. The effect is strong enough to be significant, e.g., for small-
scale solar dynamo.

The authors are grateful to Professor A.V. Gurevich for his permanent
attention to their work. The authors are thankful to Professor L.L.
Kitchatinov for drawing our attention to the problem of Solar magnetism. We
are also much obliged to A.M. Kiselev for providing the computation programs
written by himself to collapse the tensor indexes. This work of A.V. Kopyev
was supported by the Foundation for the Advancement of Theoretical Physics and
Mathematics (BASIS).

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##  I SUPPLEMENTARY NOTE 1. GENERALIZED KAZANTSEV EQUATION

It is well-known that solenoidality and statistical homogeneity and isotropy
in space restrict two point correlator to one scalar function:

$\langle
B_{i}(\mathbf{r}+\bm{\rho},t)B_{j}(\mathbf{r},t)\rangle=G(\rho,t)n_{i}n_{j}+\frac{\rho}{2}\,G^{\prime}_{\rho}(\rho,t)\,(\delta_{ij}-n_{i}n_{j})$

Small-scale kinematical dynamo theory usually considers development of this
correlator in time. Kazantsev show that the closed equation on $G(\rho,t)$ can
be derived when the velocity field is stationary, Gaussian and delta-
correlated in time. In this case all statistical properties of the velocity
field are governed by the pair correlator (see (3) in the main text):

$D_{ij}(\mathbf{0})-D_{ij}(\bm{\rho})=b(\rho)\,\delta_{ij}+\frac{\rho}{2}\,b^{\prime}(\rho)\,(\delta_{ij}-n_{i}n_{j})$
$None$

To derive a dynamical equation on $G(\rho,t)$ in the frame of ’$V^{3}$ model’
two additional two-point correlators are needed:

$\displaystyle F_{ijk}(\bm{\rho},\bm{\rho})=\iint\langle
v_{i}(\mathbf{r},t)v_{j}(\mathbf{r}+\bm{\rho},t+\tau_{1})v_{k}(\mathbf{r}+\bm{\rho},t+\tau_{2})\rangle\mathrm{d}\tau_{1}\mathrm{d}\tau_{2},$
$\displaystyle\frac{\partial}{\partial{\rho_{1}}_{l}}F_{ijk}(\bm{\rho}_{1},\bm{\rho})\bigl{|}_{\bm{\rho}_{1}=\bm{\rho}}=\iint\langle
v_{i}(\mathbf{r},t)A_{jl}(\mathbf{r}+\bm{\rho},t+\tau_{1})v_{k}(\mathbf{r}+\bm{\rho},t+\tau_{2})\rangle\mathrm{d}\tau_{1}\mathrm{d}\tau_{2}$

where $A_{jl}=\partial v_{j}/{\partial r_{l}}$ is a velocity gradient tensor.
The first of the correlators can be restricted in our case by the only scalar
function LL :

$F_{ijk}(\bm{\rho},\bm{\rho})=c(\rho)n_{i}\delta_{jk}-\left(c(\rho)+\frac{\rho}{2}\,c^{\prime}(\rho)\right)\left(n_{j}\delta_{ik}+n_{k}\delta_{ij}\right)-\left(c(\rho)-\rho\,c^{\prime}(\rho)\right)n_{i}n_{j}n_{k}$
$None$

The second one requires one more scalar function JoT :

$\displaystyle\frac{\partial}{\partial{\rho_{1}}_{l}}F_{ijk}(\bm{\rho}_{1},\bm{\rho})\bigl{|}_{\bm{\rho}_{1}=\bm{\rho}}=$
$\displaystyle-\frac{c(\rho)}{2\rho}\delta_{il}\delta_{jk}-d(\rho)n_{i}n_{j}\delta_{kl}+\left(d(\rho)-\frac{c^{\prime}(\rho)}{2}\right)\delta_{ij}\delta_{kl}+\left(\frac{c(\rho)}{\rho}-d(\rho)+c^{\prime}(\rho)\right)\delta_{ik}\delta_{jl}+$
$None$
$\displaystyle+\left(\frac{c(\rho)}{\rho}+d(\rho)-c^{\prime}(\rho)\right)n_{i}n_{k}\delta_{jl}-\left(\frac{c(\rho)}{2\rho}+2d(\rho)-\frac{c^{\prime}(\rho)}{2}\right)n_{k}n_{l}\delta_{ij}+$
$\displaystyle+\left(\frac{c(\rho)}{2\rho}-\frac{c^{\prime}(\rho)}{2}\right)\left(n_{i}n_{l}\delta_{jk}+n_{j}n_{k}\delta_{il}\right)-\left(\frac{3c(\rho)}{2\rho}-\frac{3c^{\prime}(\rho)}{2}+\frac{\rho
c^{\prime\prime}(\rho)}{2}\right)n_{i}n_{j}n_{k}n_{l}+$
$\displaystyle+\left(-\frac{c(\rho)}{2\rho}+2d(\rho)+\frac{c^{\prime}(\rho)}{2}+\frac{\rho
c^{\prime\prime}(\rho)}{2}\right)n_{j}n_{l}\delta_{ik}$

Let us introduce the notation:

$\displaystyle B_{\alpha}=B_{\alpha}(\mathbf{r},t),\quad
B^{\prime}_{\alpha}=B_{\alpha}(\mathbf{r}^{\prime},t),\quad
v_{\alpha}=v_{\alpha}(\mathbf{r},t)$ $\displaystyle,\quad
v_{\alpha}^{1}=v_{\alpha}(\mathbf{r}_{1},t),\quad
v_{\alpha}^{2}=v_{\alpha}(\mathbf{r}_{2},t),\quad$
$\displaystyle\partial_{\alpha}=\frac{\partial}{\partial
r_{\alpha}},\quad\partial^{\prime}_{\alpha}=\frac{\partial}{\partial
r^{\prime}_{\alpha}},\quad\partial^{\rho}_{\alpha}=\frac{\partial}{\partial\rho_{\alpha}}$
$\displaystyle,\quad\partial^{1}_{\alpha}=\frac{\partial}{\partial{r_{1}}_{\alpha}},\quad\partial^{2}_{\alpha}=\frac{\partial}{\partial{r_{2}}_{\alpha}}$

Then from the induction equation (1) one can obtain:

$\frac{\partial}{\partial t}\langle
B_{\alpha}B^{\prime}_{\beta}\rangle=-\mathrm{e}_{\alpha
mn}\mathrm{e}_{npq}\partial_{m}^{\rho}\langle
v_{p}B_{q}B^{\prime}_{\beta}\rangle-\mathrm{e}_{\beta
mn}\mathrm{e}_{npq}\partial_{m}^{\rho}\langle
v_{p}B_{q}B^{\prime}_{\alpha}\rangle+2\kappa\,\partial^{\rho}_{m}\partial^{\rho}_{m}\langle
B_{\alpha}B^{\prime}_{\beta}\rangle$

where $\mathrm{e}_{npq}$ is Levi-Civita symbol. The latter equation is non-
closed. However the procedure of splitting Klyatskin  can make it the closed
one. A detailed performance of this procedure in the frame of $V^{3}$ model is
given in (ApJ, , Appendix B) for smooth velocity field. The case of rough
velocity field is quite analogous, so we just point out some key moments and
differences between two cases. The splitting procedure is based on Furutsu-
Novikov formula, which can be significantly simplified in the frame of $V^{3}$
model ($\delta/\delta v_{i}$ means a functional derivative):

$\displaystyle\langle v_{p}B_{q}B^{\prime}_{r}\rangle$
$\displaystyle=\sum_{N=1}^{\infty}\frac{1}{N!}\int\,d\mathbf{r}_{1}dt_{1}\dots
d\mathbf{r}_{N}dt_{N}\langle
v_{p}(\mathbf{r},t)v_{i_{1}}(\mathbf{r}_{1},t_{1})\dots
v_{i_{N}}(\mathbf{r}_{N},t_{N})\rangle_{c}\langle\frac{\delta^{N}\bigl{(}B_{q}(\mathbf{r},t)B_{r}(\mathbf{r}^{\prime},t)\bigr{)}}{\delta
v_{i_{1}}(\mathbf{r}_{1},t_{1})\dots\delta
v_{i_{N}}(\mathbf{r}_{N},t_{N})}\rangle$
$\displaystyle=\int\,d\mathbf{r}_{1}dt_{1}\langle
v_{p}(\mathbf{r},t)v_{i_{1}}(\mathbf{r}_{1},t_{1})\rangle\,\langle\frac{\delta\bigl{(}B_{q}(\mathbf{r},t)B_{r}(\mathbf{r}^{\prime},t)\bigr{)}}{\delta
v_{i_{1}}(\mathbf{r}_{1},t_{1})}\rangle$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{1}{2}\int\,d\mathbf{r}_{1}dt_{1}d\mathbf{r}_{2}dt_{2}\langle
v_{p}(\mathbf{r},t)v_{i_{1}}(\mathbf{r}_{1},t_{1})v_{i_{2}}(\mathbf{r}_{2},t_{2})\rangle\,\langle\frac{\delta^{2}\bigl{(}B_{q}(\mathbf{r},t)B_{r}(\mathbf{r}^{\prime},t)\bigr{)}}{\delta
v_{i_{1}}(\mathbf{r}_{1},t_{1})\delta v_{i_{2}}(\mathbf{r}_{2},t_{2})}\rangle$
$\displaystyle=\frac{1}{2}\int\,d\mathbf{r}_{1}D_{pj}(\mathbf{r}-\mathbf{r}_{1})\left(\langle\tfrac{\delta
B_{q}}{\delta v_{j}^{1}}B^{\prime}_{r}\rangle+\langle B_{q}\tfrac{\delta
B^{\prime}_{r}}{\delta v_{j}^{1}}\rangle\right)$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{1}{6}\int\,d\mathbf{r}_{1}d\mathbf{r}_{2}X_{pjk}(\mathbf{r}-\mathbf{r}_{1},\mathbf{r}-\mathbf{r}_{2})\left(\langle\tfrac{\delta^{2}B_{q}}{\delta
v_{j}^{1}\delta v_{k}^{2}}B^{\prime}_{r}\rangle+\langle\tfrac{\delta
B_{q}}{\delta v_{j}^{1}}\tfrac{\delta B^{\prime}_{r}}{\delta
v_{k}^{2}}\rangle+\langle\tfrac{\delta B_{q}}{\delta v_{k}^{2}}\tfrac{\delta
B^{\prime}_{r}}{\delta v_{j}^{1}}\rangle+\langle
B_{q}\tfrac{\delta^{2}B^{\prime}_{r}}{\delta v_{j}^{1}\delta
v_{k}^{2}}\rangle\right)$

Finding the functional derivatives (see ApJ ) one obtain the following
equation:

$\displaystyle\frac{\partial}{\partial t}\langle
B_{\alpha}B^{\prime}_{\beta}\rangle=2\kappa\,\partial^{\rho}_{m}\partial^{\rho}_{m}\langle
B_{\alpha}B^{\prime}_{\beta}\rangle-\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle-\bigl{(}\mathrm{e}_{\alpha mn}\mathrm{e}_{npq}\delta_{\beta
r}+\mathrm{e}_{\beta mn}\mathrm{e}_{npq}\delta_{\alpha
r}\bigr{)}\mathrm{e}_{j_{1}jk_{1}}\partial^{\rho}_{m}\partial^{\rho}_{i_{1}}$
$\displaystyle\left(-\mathrm{e}_{qi_{1}j_{1}}D_{pj}(\mathbf{0})\langle
B_{k_{1}}B^{\prime}_{r}\rangle+\mathrm{e}_{ri_{1}j_{1}}D_{pj}(\bm{\rho})\langle
B_{q}B^{\prime}_{k_{1}}\rangle\right)-$
$\displaystyle-\bigl{(}\mathrm{e}_{\alpha mn}\mathrm{e}_{npq}\delta_{\beta
r}+\mathrm{e}_{\beta mn}\mathrm{e}_{npq}\delta_{\alpha
r}\bigr{)}\mathrm{e}_{j_{1}jk_{1}}\mathrm{e}_{j_{2}kk_{2}}$
$\displaystyle\times$
$\displaystyle\times\partial^{\rho}_{m}\Bigl{(}2\mathrm{e}_{qi_{1}j_{1}}\mathrm{e}_{ri_{2}j_{2}}$
$\displaystyle\partial^{\rho}_{i_{2}}\bigl{(}\tfrac{\partial}{\partial{\rho_{1}}_{i_{1}}}F_{kpj}(\bm{\rho}_{1},\bm{\rho})\bigl{|}_{\bm{\rho}_{1}=\bm{\rho}}\langle
B_{k_{1}}B^{\prime}_{k_{2}}\rangle-
F_{kpj}(\bm{\rho})\partial^{\rho}_{i_{1}}\langle
B_{k_{1}}B^{\prime}_{k_{2}}\rangle\bigr{)}+$
$\displaystyle+\mathrm{e}_{ri_{1}j_{1}}\mathrm{e}_{k_{1}i_{2}j_{2}}$
$\displaystyle\partial^{\rho}_{i_{1}}\bigl{(}\tfrac{\partial}{\partial{\rho_{1}}_{i_{2}}}F_{pjk}(\bm{\rho}_{1},\bm{\rho})\bigl{|}_{\bm{\rho}_{1}=\bm{\rho}}\langle
B_{q}B^{\prime}_{k_{2}}\rangle-
F_{pjk}(\bm{\rho})\partial^{\rho}_{i_{2}}\langle
B_{q}B^{\prime}_{k_{2}}\rangle\bigr{)}\Bigr{)}.$

Substituting (S1), (S2) and (S3) into the latter expression and convoluting
enormous number of summands with the aid of computer algebra, one arrive at
the generalized Kazantsev equation (6).

Note that in the case of smooth velocity field considered in ApJ  functions
$c$ and $d$ are bounded by the symmetries of the flow:
$c(\rho)/d(\rho)=-\rho/6$ or $\beta=6$. In the case of the rough field
kinematical reasons are insufficient to obtain the relation between two
functions JoT . If one suppose that the correlation times of $\langle
v_{i}(\mathbf{r},t)v_{j}(\mathbf{r}+\bm{\rho},t+\tau_{1})v_{k}(\mathbf{r}+\bm{\rho},t+\tau_{2})\rangle$
and $\langle
v_{i}(\mathbf{r},t)A_{jl}(\mathbf{r}+\bm{\rho},t+\tau_{1})v_{k}(\mathbf{r}+\bm{\rho},t+\tau_{2})\rangle$
coincide, the numerical simulation data gives $\beta\simeq 2.$ JoT

##  II SUPPLEMENTARY NOTE 2. CRITICAL ANTIDYNAMO IRREVERSIBILITY IN V3 MODEL

It is convenient to introduce a notation

$\sigma=\frac{3s(s+4)}{4}.$

Consider (14). The minimum of cubic parabola must have a non-negative value to
have no generation. The following condition $\zeta>\zeta_{cr}(s,\beta)$ can be
obtained analytically. In minimal $V^{3}$ model ($\zeta\beta\gg\zeta$) the
cubic parabola degenerates into the quadratic one and the solution is obvious:

$(\zeta\beta)_{cr}=\frac{\sqrt{\sigma}}{4}$

In general $V^{3}$ model the analitycal solution can also be found:

$\zeta_{cr}=\frac{\sqrt{\sigma^{2}+10\beta\sigma-2\beta^{2}+2\sqrt{\beta}(\beta+2\sigma)^{3/2}}}{2(4\beta-\sigma)^{3/2}}$

These dependences are depicted in Figure 2 in the main text.

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