MPFBench: A Large Scale Dataset for SciML of Multi-Phase-Flows: Droplet and Bubble Dynamics

Response:

To address the first weakness noted, all models in this study are trained and tested on the same set of 1,000 samples from the 2D bubble rise dataset to ensure consistency in evaluation. As you suggested, in future work, we plan to expand our analysis to include three levels of difficulty within the 2D dataset (easy, moderate, and hard). This will allow us to better understand model performance across varying levels of complexity and demonstrate the ability of SciML models to learn and generalize transient two-phase flow phenomena.

As you pointed out, incorporating 3D physics would provide valuable insights into real-world problems. However, the 3D dataset poses additional challenges, as it is computationally expensive and requires data-parallel training to accommodate sequences of timesteps. In future work, we plan to explore approaches such as data-parallel training to evaluate models under different scenarios for the complex phenomena occurring in 3D.

Additionally, we are sharing this dataset with the SciML community to encourage further research and the development of models capable of addressing higher-dimensional, transient two-phase flow problems. We will ensure that all typos are corrected in the camera-ready version and will carefully address the points raised in this feedback.

Questions:

1- How do you train the foundation models (scOT and Poseidon)? Are they retrained or fine-tuned based on pre-trained checkpoints?

scOT is trained from scratch, while Poseidon is fine-tuned based on a pre-trained checkpoint.

Response:

Thank you for your feedback. We have revised the manuscript to better emphasize the dataset's real-world applications, such as optimizing oxygen transfer in wastewater treatment and improving oil spill cleanup strategies, highlighting its relevance for industries reliant on multiphase flow modeling. To improve clarity for non-physics readers, we have expanded the description of the physics methods in the appendix, including why the Lattice Boltzmann Method and Allen-Cahn equations were chosen. We have also simplified technical explanations to make them more accessible to a general ML audience. Lastly, the anonymous repository link has been tested and is working properly. We hope these updates address your concerns.

Questions:

1- In appendix A, can you briefly describe the Allen Cahn equations a bit more for the readers? Specifically on big picture descriptions on how close is this to direct numerical simulation of Navier Stokes?

Allen-Cahn Equation and DNS of Navier-Stokes

The Allen-Cahn equation models phase separation and interface dynamics using an order parameter $(\phi)$. It simplifies interface tracking by representing interfaces as diffuse regions, avoiding explicit reconstruction. While it is not a standalone DNS of Navier-Stokes, it can be coupled with Navier-Stokes to simulate two-phase flows.

• Strengths: Efficient interface handling, natural inclusion of surface tension, and robust merging/splitting dynamics.
• Limitations: Artificial interface thickness $( \epsilon )$ can smooth out small-scale phenomena, limiting accuracy in high-Reynolds-number flows.

It is an efficient approximation for complex two-phase flows but less detailed than pure DNS.

2- In appendix A, what's the benefit of Lattice Boltzman methods vs conventional interface-capturing Finite Volume Solvers? Are there any cost-accuracy tradeoffs with your simulation approach? This could be useful for readers to know as well.

Lattice Boltzmann Method (LBM) offers simplicity, efficient interface tracking, and scalability due to its local operations. Unlike Finite Volume Solvers (FVS), LBM avoids complex equation coupling and explicit interface tracking.

• Benefits: Easy parallelization, natural multiphysics integration, and reduced complexity.
• Tradeoffs: LBM is less accurate for sharp interface or high-Reynolds flows, while FVS achieves higher accuracy but is costlier for complex flows.

LBM is ideal for scalable, efficient multiphase simulations, with some tradeoffs in interface resolution and accuracy.

3- Since Section 4, line 365. How were hyper parameters chosen?

The hyperparameters were carefully chosen through an extensive search on the dataset. We explored a wide range of values for parameters such as learning rate, number of layers, and hidden dimensions to identify the optimal configuration for our benchmarks. Details of the selected hyperparameters will be provided in the appendix for better reproducibility of our results.

4- Section 4 and 5 -- How many train/val/test splits?

The dataset was split into 70% for training, 10% for validation, and 20% for testing. We conducted experiments using a single train/val/test split.

5- Can you spend a paragraph or 2 explaining the broader applications of this dataset and importance of sequence to field and sequence predictions benchmarks in the intro?

We have revised the introduction to emphasize the broader applications of this dataset and expanded on the significance of sequence-to-field and sequence-to-sequence prediction benchmarks. Sequence-to-field predictions are essential for modeling steady-state behaviors, where the goal is to map an input sequence (e.g., parameters describing initial or boundary conditions) directly to the corresponding field outputs. In contrast, sequence-to-sequence predictions are designed to capture the transient dynamics of bubble rise by mapping an input sequence (e.g., time-series data describing earlier states of the system) to another sequence representing the system’s future states. This technique leverages concepts rooted in natural language processing, where seq2seq models transform one sequence into another through an encoder-decoder architecture. Originally developed for tasks like machine translation, seq2seq methods are well-suited for time-evolving physical systems as they can learn complex temporal dependencies. For example, in our dataset, seq2seq predictions enable accurate modeling of time-varying behaviors such as bubble deformation, oscillations, and breakup, which are critical for understanding and simulating multiphase flow phenomena.

Response:

While the dataset may appear relatively simple in its setup, it has significant real-world applicability in industries where multiphase flow phenomena are critical. By capturing bubble rise dynamics across varying surface tensions, density ratios, and bubble sizes, it provides valuable insights into processes such as aeration in wastewater treatment, where bubble behavior directly influences oxygen transfer efficiency, and in chemical reactors, where rising bubbles play a vital role in enhancing mass and heat transfer. Additionally, the dataset has practical applications in environmental remediation, such as predicting bubble behavior in oil spill cleanup, aiding in the design of effective mitigation strategies. Although it does not currently include interactions between bubbles, thermal exchanges, or interactions with walls, the high-fidelity data it provides is a foundational resource for developing predictive models and optimizing processes in these and other multiphase flow applications. This focus on fundamental bubble dynamics allows researchers to validate and generalize their models for a wide range of industrial applications.

The input to the models consists of the solution sequence for the input fields (velocity, pressure, and interface indicator) and does not include scalar quantities such as density ratio, viscosity ratio, Bond number, or Reynolds number. This distinction will be clarified in the text. The LBM solver was run for 400,000 timesteps, saving solutions every 4,000 timesteps to produce 100 uniformly distributed time snapshots. Details of the model hyperparameters and the mathematical formulas for the evaluation metrics (MSE and $L_2$ relative error) have been added to the appendix.

Questions:

1- In line 268, by interface indicator, do the authors refer to the signed-distance function?

No, the interface indicator does not refer to the signed-distance function. Instead, it refers to the volume fraction, which serves as an indicator of the phase distribution within the domain. Specifically:

• $c = 0.5$ indicates phase 1 (e.g., liquid),
• $c = -0.5$ indicates phase 2 (e.g., gas),
• $c = 0$ represents the interface between the two phases.

This volume fraction approach is commonly used in multiphase flow simulations to distinguish between phases and to capture the interface without explicitly defining the geometry. It provides a smooth transition across the interface, enabling numerical methods to handle phase boundaries effectively.

2- Looking at the figures in the manuscript, most flow fields seem symmetric with respect to the centered vertical axis, meaning that one could store only one half of the field in 2D and one quarter in 3D to describe the full flow field. Did you consider doing this to reduce the dataset size?

While storing only half or a quarter of the fields could reduce dataset size, it introduces significant drawbacks. It adds complexity to data handling, requiring additional preprocessing and postprocessing to reconstruct full fields during training and inference, which complicates machine learning workflows. Furthermore, modern data compression techniques, such as the .npz format, already minimize dataset size effectively without compromising completeness or fidelity. Given these considerations, we opted to retain the full flow fields to ensure the dataset remains straightforward, accurate, and versatile for a wide range of applications.

Response:

We have addressed the feedback by explicitly including a discussion of the dataset's license (CC-BY-NC) in the main paper, emphasizing its purpose for non-commercial research and educational use within the SciML community to benchmark and challenge models on this complex dataset. Additionally, detailed instructions on how to download, unzip, and access the dataset have been added to the appendix, ensuring that the steps provided on the Hugging Face website are now fully documented within the paper for ease of use.

Questions:

1- What is the license of the dataset (please justify your choice)

This dataset is intended for the SciML community to challenge state-of-the-art neural operators and foundation models and is licensed under CC-BY-NC for non-commercial use. The choice of this license ensures that the dataset is freely accessible for research and educational purposes while restricting its use for commercial applications without explicit permission, aligning with the open-access goals of the SciML community.

2- How do you plan to host the data (is it hugging face?)

The dataset will be hosted on Hugging Face, a widely used platform for machine learning datasets and models. Hugging Face provides:

• Easy accessibility for researchers and practitioners.
• A robust framework for organizing and managing large datasets.
• Built-in version control to track updates and changes.
• Seamless integration with popular machine learning tools and frameworks.

3- What are you plans to ensure it's maintances over years and potentially decades.

To ensure the dataset remains useful and relevant for years and decades, we have the following plans:

1. Long-term Hosting on Hugging Face:

   • Hugging Face provides stable hosting and long-term accessibility to datasets.
   • Regular updates to the repository will keep the dataset compatible with evolving machine learning frameworks.

2. Community Involvement:

   • Engage the SciML community to provide feedback and contribute to dataset maintenance and improvements.
   • Promote collaborative enhancements through GitHub tools linked to the dataset.

3. Dataset Expansion:

   • Commit to periodic updates, including new 3D simulations and potentially three-phase flows (e.g., bubble rise and droplet fall in stratified flow).
   • Expand the dataset to include more complex scenarios and physics to meet future research demands.

4. Backup and Redundancy:

   • Maintain multiple backups of the dataset in secure storage locations to prevent data loss.