Autoencoder-assisted study of directed percolation with spatial long-range interactions

Abstract

Spatial L{\'{e}}vy-like flights are introduced as a way to absorbing phase transitions to produce non-local interactions. We utilize the autoencoder, an unsupervised learning method, to predict the critical points for $(1+1)$-d directed percolation with such spatial long-range interactions. After making a global coverage of the reaction-diffusion distance and taking a series of different values for the parameter \;${\beta}$\; in the distribution \;$P(r){\sim}1/r^{\beta}$\;, the critical points $P_c$ that can be continuously varied are obtained. And the dynamic decay of the particle density under the critical points was counted as a way to determine the critical exponent \;${\delta}$\; of the survival rate. We also investigate the active behavior of the system's particles under the critical point with increasing time steps, which allows us to determine the characteristic time $t_f$ of the finite-scale systems. And the dynamic exponents \;$z$\; are obtained using the scaling relation \;$t_f{\sim}L^{z}$\;. We find that the autoencoder can well identify this characteristic evolutionary behavior of particles. Finally, we discuss the compliance of the scaling form \;$1/{\delta}-({\beta}-2)/{\delta}z=2$\; in different \;${\beta}$\; intervals as well as a method to introduce a global scaling mechanism by generating a random walking step using the L{\'{e}}vy distribution.