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Learning Macroscopic Dynamics from Partial Microscopic Observations

Mengyi Chen, Qianxiao Li

TL;DR

The main idea of the approach is to map the training procedure on the macroscopic coordinates back to the microscopic coordinates, on which partial force computations can be used as stochastic estimation to update model parameters.

Abstract

Macroscopic observables of a system are of keen interest in real applications such as the design of novel materials. Current methods rely on microscopic trajectory simulations, where the forces on all microscopic coordinates need to be computed or measured. However, this can be computationally prohibitive for realistic systems. In this paper, we propose a method to learn macroscopic dynamics requiring only force computations on a subset of the microscopic coordinates. Our method relies on a sparsity assumption: the force on each microscopic coordinate relies only on a small number of other coordinates. The main idea of our approach is to map the training procedure on the macroscopic coordinates back to the microscopic coordinates, on which partial force computations can be used as stochastic estimation to update model parameters. We provide a theoretical justification of this under suitable conditions. We demonstrate the accuracy, force computation efficiency, and robustness of our method on learning macroscopic closure models from a variety of microscopic systems, including those modeled by partial differential equations or molecular dynamics simulations.

Learning Macroscopic Dynamics from Partial Microscopic Observations

TL;DR

The main idea of the approach is to map the training procedure on the macroscopic coordinates back to the microscopic coordinates, on which partial force computations can be used as stochastic estimation to update model parameters.

Abstract

Macroscopic observables of a system are of keen interest in real applications such as the design of novel materials. Current methods rely on microscopic trajectory simulations, where the forces on all microscopic coordinates need to be computed or measured. However, this can be computationally prohibitive for realistic systems. In this paper, we propose a method to learn macroscopic dynamics requiring only force computations on a subset of the microscopic coordinates. Our method relies on a sparsity assumption: the force on each microscopic coordinate relies only on a small number of other coordinates. The main idea of our approach is to map the training procedure on the macroscopic coordinates back to the microscopic coordinates, on which partial force computations can be used as stochastic estimation to update model parameters. We provide a theoretical justification of this under suitable conditions. We demonstrate the accuracy, force computation efficiency, and robustness of our method on learning macroscopic closure models from a variety of microscopic systems, including those modeled by partial differential equations or molecular dynamics simulations.

Paper Structure

This paper contains 31 sections, 5 theorems, 44 equations, 8 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Learning from Partial Observations
  4. Reduced Order Models
  5. Equation-free Framework
  6. Problem Setup
  7. Method
  8. Autoencoder for Closure Modeling
  9. Macroscopic Dynamics Identification
  10. Experiments
  11. Force Computation Efficiency
  12. Predator-Prey System
  13. Results
  14. Robustness to Different Latent Structures
  15. Lennard-Jones System
  16. ...and 16 more sections

Key Result

Theorem 1

Assume for any $\mathbf{x}\sim \mathcal{D}$, the eigenvalues of $\boldsymbol{\varphi}^{\prime}(\mathbf{x})\boldsymbol{\varphi}^{\prime}(\mathbf{x})^T$ are lower bounded by $b_1$ and upper bounded by $b_2$, $0<b_1\leq b_2$. Then: here C does not depend on $\boldsymbol{\theta}$ hence does not affect the optimization.

Figures (8)

  • Figure 1: Overview of our method. Left. Data generation workflow. For each configuration $\mathbf{x}$, forces on a subset of all the microscopic coordinates are calculated by the microscopic force calculator. Right. Macroscopic dynamics identification. The macroscopic dynamics is mapped to the microscopic space first, then compared with the forces on a subset of the microscopic coordinates.
  • Figure 2: Mean relative error on the test dataset of the Predator-Prey system. The black dashed line represents test error $=3\times 10^{-3}$.
  • Figure 3: Results on the Lennard-Jones system with 800 atoms and $N=4800$. Forces on 50 atoms are used to train $\mathcal{L}_{\mathbf{x}, p}$ for all the latent model structures. Each model is trained with ten repeats.
  • Figure 4: Number of force computations required to achieve $e_{\text{tol}}=3\times10^{-3}$ on Lennard-Jones system with different sizes. Forces on 50 atoms are used to train $\mathcal{L}_{\mathbf{x}, p}$ for systems of different sizes.
  • Figure 5: Visualization of the microscopic state of each system
  • ...and 3 more figures

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2: informal
  • Theorem
  • proof
  • Definition 1
  • Theorem 3
  • Theorem
  • proof