Accurate generation of stochastic dynamics based on multi-model Generative Adversarial Networks

TL;DR

This work demonstrates that Generative Adversarial Networks can learn stochastic dynamics on a lattice by using a conditional GAN trained on KMC data. A simple noise-regularization scheme, together with a multi-model ensemble, stabilizes training and yields quantitative agreement with analytic equilibrium distributions and first-passage times, despite intrinsic GAN oscillations near Nash equilibrium where losses approach $\log 2$. The multi-model averaging significantly improves predictive accuracy for both equilibrium and kinetic properties, outperforming single-model predictions. The study also shows transfer-learning potential and possible discrimination between stochastic processes via retraining dynamics, suggesting GANs as a powerful tool for tackling complex stochastic dynamics in physics.

Abstract

Generative Adversarial Networks (GANs) have shown immense potential in fields such as text and image generation. Only very recently attempts to exploit GANs to statistical-mechanics models have been reported. Here we quantitatively test this approach by applying it to a prototypical stochastic process on a lattice. By suitably adding noise to the original data we succeed in bringing both the Generator and the Discriminator loss functions close to their ideal value. Importantly, the discreteness of the model is retained despite the noise. As typical for adversarial approaches, oscillations around the convergence limit persist also at large epochs. This undermines model selection and the quality of the generated trajectories. We demonstrate that a simple multi-model procedure where stochastic trajectories are advanced at each step upon randomly selecting a Generator leads to a remarkable increase in accuracy. This is illustrated by quantitative analysis of both the predicted equilibrium probability distribution and of the escape-time distribution. Based on the reported findings, we believe that GANs are a promising tool to tackle complex statistical dynamics by machine learning techniques

Figures (10)

• Figure 1: Graphical representation of the random walk. Lattice points correspond to energy minima. In order to reach a new lattice position, the particle has to overcome the diffusion barrier $E_b$. Gray curve represent the potential profile, dashed black curve interpolates $E_i$.
• Figure 2: (a) $G$ and $D$ losses as a function of the number of epochs while minimizing equations \ref{['eq::loss_functions']} directly. Theoretical Nash equilibrium value of $\log 2$ is reported for reference. The behavior is clearly non-convergent. (b) Comparison of a generated trajectory (dark blue line) at the end of procedure in (a) and one from the dataset (transparent green line). (c) Lossplot obtained when Gaussian noise ($\sigma=0.25$) is added in the training procedure. (d) Example of a generated trajectory (dark blue line) obtained at the end of training procedure of (c) and one from the dataset (transparent green line).
• Figure 3: Equilibrium distribution obtained by random walk trajectories from different models. Models A, B, C and D in (a) refer to individual models picked at different epochs. (b) reports the equilibrium distribution obtained by Kinetic Monte Carlo simulations. (c) has been obtained by the multi-model approach. Black error bars represent confidence intervals (not reported in (a)). L2 distances between equilibrium distributions and the analytical one are also reported.
• Figure 4: First passage time distribution obtained by generated trajectories (dark blue line), by KMC simulations (dashed-dotted green line) and by a Model A (dashed gray line). Inset reports first passage times as predicted by models at different epochs and by the multi-model approach. Models A, B, C and D are the same of figure \ref{['fig::equilibrium_distributions']}. Black dashed line corresponds to the analytical mean value. Values for passages from the left (right) minimum to the right (left) one are reported separately.
• Figure 5: (a) Lossplot obtained by training a GAN on trajectories obtained by the random walk on continuous values. (b) Reports the same loss functions in the case of training on discrete dynamics.