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Enhancing Computational Efficiency in Multiscale Systems Using Deep Learning of Coordinates and Flow Maps

Asif Hamid, Danish Rafiq, Shahkar Ahmad Nahvi, Mohammad Abid Bazaz

TL;DR

The paper tackles the computational burden of simulating time-dependent multiscale PDEs by learning a latent, low-dimensional coordinate system and a hierarchy of latent-space flow maps. It introduces latent HiTS (L-HiTS), which combines deep autoencoders to discover coordinates with a multiscale hierarchy of ResNet-based flow maps to advance reduced states across multiple time scales $\Delta t_m=2^{m-d}\Delta t$ while keeping accuracy. The method is demonstrated on two canonical PDEs—the FitzHugh–Nagumo model and the Kuramoto–Sivashinsky equation—showing state-of-the-art predictive performance with substantial speedups relative to the baseline multiscale HiTS, and revealing how latent dimension and model choice influence accuracy and efficiency. This approach enables efficient online, many-query predictions for large-scale multiscale systems, with open-source code and clear pathways for extensions to partial observations and adaptive time stepping.

Abstract

Complex systems often show macroscopic coherent behavior due to the interactions of microscopic agents like molecules, cells, or individuals in a population with their environment. However, simulating such systems poses several computational challenges during simulation as the underlying dynamics vary and span wide spatiotemporal scales of interest. To capture the fast-evolving features, finer time steps are required while ensuring that the simulation time is long enough to capture the slow-scale behavior, making the analyses computationally unmanageable. This paper showcases how deep learning techniques can be used to develop a precise time-stepping approach for multiscale systems using the joint discovery of coordinates and flow maps. While the former allows us to represent the multiscale dynamics on a representative basis, the latter enables the iterative time-stepping estimation of the reduced variables. The resulting framework achieves state-of-the-art predictive accuracy while incurring lesser computational costs. We demonstrate this ability of the proposed scheme on the large-scale Fitzhugh Nagumo neuron model and the 1D Kuramoto-Sivashinsky equation in the chaotic regime.

Enhancing Computational Efficiency in Multiscale Systems Using Deep Learning of Coordinates and Flow Maps

TL;DR

The paper tackles the computational burden of simulating time-dependent multiscale PDEs by learning a latent, low-dimensional coordinate system and a hierarchy of latent-space flow maps. It introduces latent HiTS (L-HiTS), which combines deep autoencoders to discover coordinates with a multiscale hierarchy of ResNet-based flow maps to advance reduced states across multiple time scales while keeping accuracy. The method is demonstrated on two canonical PDEs—the FitzHugh–Nagumo model and the Kuramoto–Sivashinsky equation—showing state-of-the-art predictive performance with substantial speedups relative to the baseline multiscale HiTS, and revealing how latent dimension and model choice influence accuracy and efficiency. This approach enables efficient online, many-query predictions for large-scale multiscale systems, with open-source code and clear pathways for extensions to partial observations and adaptive time stepping.

Abstract

Complex systems often show macroscopic coherent behavior due to the interactions of microscopic agents like molecules, cells, or individuals in a population with their environment. However, simulating such systems poses several computational challenges during simulation as the underlying dynamics vary and span wide spatiotemporal scales of interest. To capture the fast-evolving features, finer time steps are required while ensuring that the simulation time is long enough to capture the slow-scale behavior, making the analyses computationally unmanageable. This paper showcases how deep learning techniques can be used to develop a precise time-stepping approach for multiscale systems using the joint discovery of coordinates and flow maps. While the former allows us to represent the multiscale dynamics on a representative basis, the latter enables the iterative time-stepping estimation of the reduced variables. The resulting framework achieves state-of-the-art predictive accuracy while incurring lesser computational costs. We demonstrate this ability of the proposed scheme on the large-scale Fitzhugh Nagumo neuron model and the 1D Kuramoto-Sivashinsky equation in the chaotic regime.
Paper Structure (14 sections, 14 equations, 6 figures, 6 tables, 1 algorithm)

This paper contains 14 sections, 14 equations, 6 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Latent Hierarchical Time-Stepping (L-HiTS) scheme for multiscale PDEs
  4. Discovery of coordinates using deep autoencoders
  5. Discovery of multiscale flow maps using deep residual neural networks
  6. Combining deep AEs and multiscale HiTS scheme for multiscale predictions
  7. Simulation results
  8. The FitzHugh-Nagumo model
  9. The 1D Kuramoto-Sivashinsky equation
  10. Discussions and conclusion
  11. Effect of different latent dimensions
  12. Effect of individual and multiple models within the latest space
  13. Concluding remarks
  14. Hyperparameters for L-HiTS method

Figures (6)

  • Figure 1: Residual neural network architecture
  • Figure 2: Schematic diagram of an autoencoder. The high dimensional input state $\mathbf{x}(t)$ is restricted using an encoder function $\mathbf{f}_E$ to obtain latent vector $\mathbf{z}(t)$, which is then passed through the lifting mechanism of the decoder function $\mathbf{f}_D$ to obtain reconstructed state vector $\hat{\mathbf{x}}(t)$.
  • Figure 3: Block diagram of the proposed L-HiTS method
  • Figure 4: Results for the FHN model: (Plot A:) plot of the MSEs for autoencoder and PCA for testing dataset against varying latent dimensions (Plot B-C:) evolution of latent variables $\mathbf{z}_{1}(t)$ and $\mathbf{z}_{2}(t)$ obtained from AE and L-HiTS scheme, (Plot D:) true state $\mathbf{u}(\mathbf{x},t)$, (Plot E:) predicted state $\mathbf{\hat{u}}(\mathbf{x},t)$ from the L-HiTS scheme, (Plot F:) MSE between $\mathbf{\hat{u}}(\mathbf{x},t)$ and $\mathbf{\hat{u}}(\mathbf{x},t)$, (Plot G:) true state $\mathbf{v}(\mathbf{x},t)$, (Plot H:) predicted state $\mathbf{\hat{v}}(\mathbf{x},t)$ from the L-HiTS scheme, (Plot I:) MSE between $\mathbf{\hat{v}}(\mathbf{x},t)$ and $\mathbf{\hat{v}}(\mathbf{x},t)$.
  • Figure 5: Results for the 1D KS equation: (Plot A:) plot of MSEs for autoencoder and PCA for testing dataset against varying latent dimensions, (Plot B-I:) evolution of the latent variables $\mathbf{z}_1(t), \dots, \mathbf{z}_8(t)$ obtained from the AE model and L-HiTS scheme, (Plot J:) true solution of the state $\mathbf{u}(\mathbf{x},t)$, (Plot K:) predicted state $\mathbf{\hat{u}}(\mathbf{x},t)$ from the L-HiTS scheme, (Plot L:) MSE between $\mathbf{u}(\mathbf{x},t)$ and $\mathbf{\hat{u}}(\mathbf{x},t)$
  • ...and 1 more figures