AROMA: Preserving Spatial Structure for Latent PDE Modeling with Local Neural Fields

Summary

This paper presents a new pipeline for modeling PDE systems, especially for learning local neural fields. It designs a new encoder-decoder for absorbing any type of input, which avoids the constraint on meshes and cloud points. It can handle diverse geometries. A diffusion-based transformer architecture is used to model the latent dynamics. The numerical experiments have shown the effectiveness of the proposed method.

Strengths

- This paper proposes an interesting encoder-decoder for handling arbitrary input geometries. - This paper has done multiple experiments on different PDEs to compare the model performance. - The paper is well-written and well-organized. - This research topic is of interest to the general scientific machine-learning community.

Weaknesses

- The motivation of using a VAE-type encoder-decoder is not well justified. The authors might add more details about the benefits of using a VAE setup for an encoder-decoder instead of a deterministic auto-encoder. Also, my concern is that using Gaussian distribution might constrain the representation capability of the latent features. The real-world dynamics or complex turbulence dynamics sometimes present heavy-tailed characteristics. - This paper considers a diffusion-based transformer for learning latent dynamics. The training efficiency and computational memory might be an issue. Moreover, I think the clarification of its benefits is not sufficient. This module can be replaced by many other methods for modeling latent dynamics, such as NeuralODEs [1] or neural spectral methods [2]. It would be good to have an ablation study on the latent dynamics part. **Refs:** [1] Krishnapriyan, Aditi S., et al. "Learning continuous models for continuous physics." Communications Physics 6.1 (2023): 319. [2] Wu, Haixu, et al. "Solving high-dimensional pdes with latent spectral models." arXiv preprint arXiv:2301.12664 (2023).

Questions

- In lines 90-91, for the second stage of training, do you fix the encoder-decoder or pretrain encoder-decoder and then fine-tune the entire network? - The authors claim that the encoder-decoder is principled. I didn’t see a principle or theoretical guarantee behind that. - In Table 1, why does DNOT work worse than FNO on the 1D Burgers case? Also, as a standard baseline, FNO works pretty good among all baseline models and the proposed AROMA. - I am also curious why FNO is not considered in the experiments of temporal extrapolation. FNO can also be modified in an auto-regressive way.

Rating

6

Confidence

4

Soundness

2

Presentation

3

Contribution

2

Limitations

Please see my concerns in **Weaknesses** and **Questions**.

Summary

The paper proposes a framework using autoencoder and diffusion transformer for predicting the forward dynamics of time-dependent PDEs. Leveraging cross-attention and neural fields, the framework is able to handle different types of meshes and geometries. The authors demonstrate the effectiveness of their proposed framework on several 1D and 2D PDE problems with different geometries.

Strengths

1. The proposed framework adopted many existing techniques in a natural and reasonable way, such as using local neural fields and cross attention to pass information between nodes in different meshes, diffusion-based predictor/refiner. The motivation and benefits of various components in the framework are illustrated clearly. 2. The empirical performance of the proposed model is strong across several benchmarks. 3. The authors explain their method well and overall the paper is easy to follow.

Weaknesses

1. Many of the techniques used in the paper are taken from existing models (authors have properly acknowledged them), so the technical advancement for the paper might not be that much. Nevertheless, I think the authors have done a decent job tweaking them to work well, evidenced by the experiments, and explained why these techniques can be helpful to PDE modeling. 2. One issue in the experiment is that the authors have employed a diffusion predictor to predict the dynamics (which can be seen as a predictor-corrector scheme) whereas other baselines are deterministic single-step predictor. With that said, the proposed framework uses more NFE (proportional to the number of diffusion steps) than other baselines. The authors did provide an ablation on Burgers' which shows that AROMA without diffusion still outperforms other model, but it is unclear for other more chaotic and higher-dimensional cases.

Questions

1. For systems that feature a decaying spectrum (e.g. fluid flow with relatively high viscosity), the proposed method is a good fit, which is not surprising. Will the method still perform well on slightly more chaotic systems like KS-equation or 2D Komogorov flow and how does the spectrum of reconstructed snapshots from decoder look like in these cases? 2. How many refinement steps does the model use? 3. There are some closed related works that not have been discussed. For example, the similar idea of learnable latent tokens have been explored in "Solving high-dimensional pdes with latent spectral models." Using pretrained autoencoder to dervie a mesh-reduced space and then learn to forecast dynamics in the latent space is also studied in "Predicting Physics in Mesh-reduced Space with Temporal Attention" and "Latent Neural PDE Solver: a reduced-order modelling framework for partial differential equations". 4. Typos: * line 228: 3D Shallow-Water Equation (Navier-Stokes1 × 10−3) * Table 4&5: "opochs"

Rating

7

Confidence

4

Soundness

3

Presentation

3

Contribution

2

Limitations

The authors have discussed the limitation in the conclusion part.

Summary

The paper introduces a novel framework designed to enhance the modeling of partial differential equations (PDEs) using local neural fields. It proposes a flexible encoder-decoder architecture that achieves smooth latent representations of spatial physical fields from various data types and employs a diffusion-based formulation to achieve greater stability and enable longer rollouts compared to conventional MSE training. The authors show the superior performance of their framework on 1D and 2D PDEs.

Strengths

The authors claim that the proposed method can handle a variety of data types including regular-grid inputs and point clouds, eliminating the need for patching and allowing efficient processing of diverse geometries. The diffusion-based formulation enhances stability and enables longer rollouts compared to traditional methods. AROMA demonstrates superior performance in simulating 1D and 2D equations, capturing complex dynamical behaviors effectively. The framework leverages attention blocks and neural fields, resulting in a model that is easy to train and achieves state-of-the-art results without requiring prior feature engineering.

Weaknesses

Although the proposed method reduces computational cost compared to existing transformer architectures, it may still be computationally expensive for very large datasets or extremely high-resolution simulations. The experiments are performed on benchmark datasets, and the performance on larger and real-world examples remains to be demonstrated. It is also unclear if the approach always produces stable results.

Questions

It is unclear what the authors mean by patching for data input types. The visual results for the cylinder and aerofoil cases are not provided. Is there any specific reason these were not included? What would be the training cost of extending this to 3D PDEs.

Rating

5

Confidence

5

Soundness

3

Presentation

3

Contribution

3

Limitations

While AROMA demonstrates effective performance on small-size datasets, its scalability to larger datasets and more complex real-world scenarios is not fully established. Scaling up neural field-based methods to handle larger volumes of data without compromising performance remains a challenge.

Summary

An innovative approach for improving the modeling of partial differential equations (PDEs) using local neural fields is presented in the paper "AROMA: Preserving Spatial Structure for Latent PDE Modeling with Local Neural Fields" (Attentive Reduced Order Model with Attention). Through the provision of a versatile encoder-decoder architecture that can handle a variety of data sources, including regular grids and point clouds, AROMA seeks to alleviate the shortcomings of current neural operator models.

Strengths

na

Weaknesses

na

Questions

1. What are the main drawbacks that AROMA seeks to solve with respect to current neural operator models for PDEs? 3. What function does the AROMA architecture's conditional transformer serve? 5. How can AROMA guarantee the accuracy and stability of its forecasting models? 5. How is the model's performance improved by the encoder and decoder's two-stage training process? 6. In the experiments, what kinds of datasets were employed, and how did AROMA fare in comparison to other models?

Rating

9

Confidence

5

Soundness

4

Presentation

4

Contribution

4

Limitations

na

Response

Thank you for your thoughtful remarks and detailed review. We greatly appreciate the time and effort you put into providing valuable insights regarding the different blocks of our architecture. We will certainly explore ways to better represent real-world and highly turbulent dynamics in our future work.

Reply to authors' rebuttal

Thanks for the response and update. The majority of the questions and concerns I raised have been addressed and I have adjust my rating accordingly.