Scalable learning of macroscopic stochastic dynamics

TL;DR

The paper tackles the challenge of learning macroscopic stochastic dynamics for spatially extended systems without resorting to prohibitively expensive large-scale microscopic simulations. It proposes a framework that combines partial-evolution data collection within local patches, a closure-variable autoencoder to identify an autonomous macroscopic state, a stochastic differential equation model for the macroscopic dynamics, and a hierarchical upsampling procedure to generate large-system data from small-system trajectories. A theoretical result ensures equivalence to full-system dynamics under local interactions, and the method is demonstrated across stochastic SPDEs, Ising-type spin systems, and a NbMoTa alloy, showing accurate reproduction of equilibrium behavior and critical phenomena from small-system data. The approach offers a scalable pathway to understanding macroscopic material behavior, enabling exploration of phase transitions and diffusion in large systems with reduced computational cost, and it provides code for reproducibility.

Abstract

Macroscopic dynamical descriptions of complex physical systems are crucial for understanding and controlling material behavior. With the growing availability of data and compute, machine learning has become a promising alternative to first-principles methods to build accurate macroscopic models from microscopic trajectory simulations. However, for spatially extended systems, direct simulations of sufficiently large microscopic systems that inform macroscopic behavior is prohibitive. In this work, we propose a framework that learns the macroscopic dynamics of large stochastic microscopic systems using only small-system simulations. Our framework employs a partial evolution scheme to generate training data pairs by evolving large-system snapshots within local patches. We subsequently identify the closure variables associated with the macroscopic observables and learn the macroscopic dynamics using a custom loss. Furthermore, we introduce a hierarchical upsampling scheme that enables efficient generation of large-system snapshots from small-system trajectory distributions. We empirically demonstrate the accuracy and robustness of our framework through a variety of stochastic spatially extended systems, including those described by stochastic partial differential equations, idealised lattice spin systems, and a more realistic NbMoTa alloy system.

Figures (10)

• Figure 1: Schematic illustration of our framework. The hierarchical upsampling scheme generates the large-system distribution $\mathcal{D}$ from the small-system trajectory distribution $\mathcal{D}_s$ through multiple iterations, each consisting of an Upsample and a LocalRelax step. An example of one iteration for the Ising model is shown. For the partial evolution scheme, for every $\mathbf{x}_t \sim \mathcal{D}$, a patch $\mathcal{I}$ is first uniformly sampled, then the microscopic dynamics is evolved locally within the patch $\mathcal{I}$ for a short time to yield $\mathbf{x}_{t+\delta t, \mathcal{I}}$. For the closure modeling, an autoencoder is trained to discover the closure variables to the macroscopic observables, and the macroscopic dynamics are identified with the designed loss $\mathcal{L}_p$.
• Figure 2: Results on the stochastic Predator-Prey system. (a) The MMD is plotted as a function of time. (b) The average MMD over the entire simulation time is reported as a function of the hyperparameter $\lambda$.
• Figure 3: Test error as a function of $n_s$. The test error is the mean relative error of the mean macroscopic observables between ground-truth and predicted trajectories. (a) Results on the Curie-Weiss model. (b) Results on the Ising model.
• Figure 4: Results on the Curie-Weiss model with $n_s=16^2$. Mean and standard deviation of the magnetization are estimated from $20$ trajectories per method.
• Figure 5: Results on the Ising model with $n_s=16^2$. Mean and standard deviation are estimated from $20$ trajectories per method. (a) Magnetization statistics. (b) Domain wall density statistics.

Theorems & Definitions (2)

• Theorem 1
• proof : Proof of \ref{['thm:1']}