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DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training (2403.03542v4)

Published 6 Mar 2024 in cs.LG, cs.NA, and math.NA

Abstract: Pre-training has been investigated to improve the efficiency and performance of training neural operators in data-scarce settings. However, it is largely in its infancy due to the inherent complexity and diversity, such as long trajectories, multiple scales and varying dimensions of partial differential equations (PDEs) data. In this paper, we present a new auto-regressive denoising pre-training strategy, which allows for more stable and efficient pre-training on PDE data and generalizes to various downstream tasks. Moreover, by designing a flexible and scalable model architecture based on Fourier attention, we can easily scale up the model for large-scale pre-training. We train our PDE foundation model with up to 0.5B parameters on 10+ PDE datasets with more than 100k trajectories. Extensive experiments show that we achieve SOTA on these benchmarks and validate the strong generalizability of our model to significantly enhance performance on diverse downstream PDE tasks like 3D data. Code is available at \url{https://github.com/thu-ml/DPOT}.

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Citations (15)
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Summary

  • The paper presents an auto-regressive denoising pre-training strategy that significantly enhances PDE solution operator generalization.
  • It leverages Fourier attention and temporal aggregation to efficiently handle diverse, large-scale PDE datasets.
  • DPOT outperforms existing models by reducing prediction errors by up to 52% while scaling to 0.5 billion parameters.

Insightful Overview of DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training

The paper "DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training" introduces a novel framework for pre-training neural operators on large-scale data derived from partial differential equations (PDEs). This approach targets the foundational task of learning solution operators for PDEs in scientific machine learning, addressing the complexities inherent in handling PDE datasets that exhibit significant heterogeneity in terms of dimensions, scales, and numerical ranges.

Key Contributions and Model Architecture

DPOT leverages an auto-regressive denoising pre-training strategy, which is designed to enhance the model's generalization ability across multiple PDE tasks. This innovative approach involves corrupting training data with Gaussian noise and predicting future timesteps from these noisy inputs, thereby improving robustness and transferability to various downstream PDE scenarios. The model employs a flexible transformer architecture based on Fourier attention, allowing it to efficiently scale up to 0.5 billion parameters—a current milestone in the field—when pre-training on comprehensive datasets containing over 100,000 trajectories from more than ten types of PDEs.

The architecture of DPOT is distinguished by several components:

  • Temporal Aggregation Layer: This layer is designed to extract PDE properties by aggregating information from adjacent timesteps, which helps infer the specifics of the PDE governing the data sample.
  • Fourier Attention Layer: By utilizing a kernel integral transformation in the frequency domain, this layer circumvents the quadratic scaling issues associated with traditional attention mechanisms, aiding in approximating PDE solutions efficiently.
  • Multi-head structure: Facilitating learning in distinct representation subspaces, it enhances parameter efficiency and scalability.

Numerical Results and Implications

In terms of empirical outcomes, DPOT delivers state-of-the-art (SOTA) performance across numerous benchmarks. The auto-regressive denoising approach not only outperforms existing architectures like MPP and FNO but also reduces errors significantly—up to 52% on some tasks. Its scalability is evidenced by the consistent improvement in performance noted as model sizes increase from 7 million to 0.5 billion parameters.

The fine-tuning results on diverse downstream tasks, including high-resolution turbulence flow prediction, 3D Navier-Stokes equations, and steady-state PDEs, underscore the versatility and adaptability of DPOT. It excels in transferring learned representations to higher-dimensional and data-scarce tasks, consolidating its utility as a foundational model in scientific machine learning.

Theoretical Insights

The theoretical underpinnings of DPOT are reinforced by a universal approximation theorem for its Fourier attention layers, which assures that they can approximate any continuous operator functions to a desired degree of accuracy. This positions DPOT as a robust framework capable of capturing the intricate dependencies in PDE data.

Future Directions

The research opens several avenues for further exploration. The integration of advanced noise-crafting techniques and adaptive learning rates could potentially enhance pre-training stability and efficiency. Moreover, extending the DPOT framework to support real-time simulations and integrating it with reinforcement learning environments could vastly expand its applicability.

In conclusion, the DPOT framework represents a significant step forward in the domain of large-scale PDE pre-training, providing a scalable solution capable of generalizing across diverse PDE tasks. Its contributions to robustness, flexibility, and computational efficiency mark it as a promising candidate for future research and application in scientific computing.

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