Overview
This research introduces Mesh-Native Convolution (MNC), a tokenization layer engineered to process data represented on unstructured meshes. The primary objective of MNC is to facilitate the adaptation of physics foundation models, initially trained on structured grid data, to handle unstructured mesh-based data. The layer aims to provide a locality-aware encoding mechanism for unstructured meshes that is analogous to the functionality of convolutional encoders used for structured grid data.
Physics foundation models are increasingly utilized as versatile emulators for spatiotemporal systems. Their effectiveness relies on tokenization layers that translate physical field data into a latent representation suitable for dynamic processing. While convolutional encoders are a common and efficient choice for structured data due to their locality awareness, they are not applicable to scientific computing problems involving unstructured meshes, such as simulations on domains with curved boundaries.
Research Context
The field of physics foundation models frequently employs tokenization layers to convert physical field data into a latent representation. For structured grid data, convolutional neural network (CNN) encoders are a typical selection, offering locality-aware processing and efficient training. However, a significant class of scientific computing problems involves data represented on unstructured meshes. Examples include simulations performed on domains characterized by curved boundaries. For these scenarios, CNNs are unsuitable.
Instead, unstructured mesh data traditionally necessitate the use of graph neural network (GNN) architectures. These architectures, while capable of handling unstructured data, are generally more expensive to train than CNNs. Furthermore, the information encoding methods used by current GNN forms differ from those of CNNs, which creates a challenge for portability and direct application of models trained in one data regime to the other.
Approach
The proposed approach centers on the development of Mesh-Native Convolution (MNC). MNC functions as a locality-aware tokenization layer specifically designed for unstructured mesh-based data. Its design intent is to encode information in a manner analogous to how classical CNN encoders process structured grid data. This analogy in encoding is intended to bridge the representational gap between structured and unstructured data formats.
The theoretical consistency of the MNC approach with classical CNN encoders is supported by theoretical results. The research demonstrates that this tokenization method allows models to be trained once on mesh data and subsequently deployed to arbitrary unstructured meshes, including irregular ones, with minimal retraining effort. A key aspect of the approach is its application to retrofitting grid-pretrained foundation models, allowing them to be adapted to unstructured meshes efficiently.
Findings
The integration of Mesh-Native Convolution (MNC) as a tokenization layer for unstructured mesh data yields several findings:
- MNC enables the development of models that, once trained on mesh data, can be deployed to arbitrary unstructured meshes, including irregular ones, with minimal retraining.
- The tokenization approach can efficiently retrofit grid-pretrained foundation models onto unstructured meshes.
- Theoretical results confirm the consistency of MNC with classical CNN encoders.
- The retrofitting approach demonstrated favourable performance.
- In many cases, this approach reached acceptable accuracy even when deployed in a zero-shot configuration.
- The retrofitting approach outperformed existing baselines after only a few epochs of fine-tuning.
- The method provides an efficient bridge between the structured grid data regime and the unstructured mesh data regime for physics foundation models.
Why This Matters
The introduction of Mesh-Native Convolution (MNC) addresses a fundamental challenge in applying physics foundation models across different data representations. By enabling efficient adaptation of models from structured grids to unstructured meshes, it enhances the utility of these models for a broader spectrum of scientific computing problems. This bridge between data regimes could streamline the development and deployment of emulators for complex spatiotemporal systems without extensive re-training from scratch for each data type.