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LE-PDE++: Mamba for accelerating PDEs Simulations

Aoming Liang, Zhaoyang Mu, Qi liu, Ruipeng Li, Mingming Ge, Dixia Fan

TL;DR

The Latent Evolution of PDEs method is designed to address the computational intensity of classical and deep learning-based PDE solvers by proposing a scalable and efficient alternative, and incorporates the Mamba model, an advanced machine learning model known for its predictive efficiency and robustness in handling complex dynamic systems with a progressive learning strategy.

Abstract

Partial Differential Equations are foundational in modeling science and natural systems such as fluid dynamics and weather forecasting. The Latent Evolution of PDEs method is designed to address the computational intensity of classical and deep learning-based PDE solvers by proposing a scalable and efficient alternative. To enhance the efficiency and accuracy of LE-PDE, we incorporate the Mamba model, an advanced machine learning model known for its predictive efficiency and robustness in handling complex dynamic systems with a progressive learning strategy. The LE-PDE was tested on several benchmark problems. The method demonstrated a marked reduction in computational time compared to traditional solvers and standalone deep learning models while maintaining high accuracy in predicting system behavior over time. Our method doubles the inference speed compared to the LE-PDE while retaining the same level of parameter efficiency, making it well-suited for scenarios requiring long-term predictions.

LE-PDE++: Mamba for accelerating PDEs Simulations

TL;DR

The Latent Evolution of PDEs method is designed to address the computational intensity of classical and deep learning-based PDE solvers by proposing a scalable and efficient alternative, and incorporates the Mamba model, an advanced machine learning model known for its predictive efficiency and robustness in handling complex dynamic systems with a progressive learning strategy.

Abstract

Partial Differential Equations are foundational in modeling science and natural systems such as fluid dynamics and weather forecasting. The Latent Evolution of PDEs method is designed to address the computational intensity of classical and deep learning-based PDE solvers by proposing a scalable and efficient alternative. To enhance the efficiency and accuracy of LE-PDE, we incorporate the Mamba model, an advanced machine learning model known for its predictive efficiency and robustness in handling complex dynamic systems with a progressive learning strategy. The LE-PDE was tested on several benchmark problems. The method demonstrated a marked reduction in computational time compared to traditional solvers and standalone deep learning models while maintaining high accuracy in predicting system behavior over time. Our method doubles the inference speed compared to the LE-PDE while retaining the same level of parameter efficiency, making it well-suited for scenarios requiring long-term predictions.

Paper Structure

This paper contains 13 sections, 11 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Our LE-PDE++ Framework
  5. Mamba for latent evolution
  6. LE-PDE++
  7. Progressive sampling policy as the learning objective
  8. Experiments
  9. Navier-Stokes Equation Datase (NS)
  10. Shallow Water Equation (SWE)
  11. Pollutant Transport Equation (PTE)
  12. Ablation and Comparison Results
  13. Conclusion

Figures (5)

  • Figure 1: The framework of LE-PDE++
  • Figure 2: Progressive sampling ratio with three different Setting
  • Figure 3: Performace on NS of LE-PDE++, the first row is the input state, the second row is the ground truth, the third row is the output of LE-PDE++, the final row is an absolute error.
  • Figure 4: Performace on SWE of LE-PDE++
  • Figure 5: Performace on PTE of LE-PDE++