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

Abstract Determining the universality class of reaction–diffusion processes with long-range interactions in non-equilibrium phase transitions is both challenging and intriguing. Identifying critical points is fundamental for studying the phase transition characteristics of these universality classes. Unlike Monte Carlo simulations of statistical system observables, machine learning methods can extract evolutionary information from clusters of such systems, enabling a faster approach to phase transition regions. We developed a method that uses the one-dimensional encoding output of a stacked autoencoder (SAE) to determine the critical point in systems undergoing (1+1)-dimensional directed percolation with spatial long-range interactions. We validate this method by examining the power-law behavior of particle density at the critical point, which strongly supports our approach. As the system adheres to the scaling relation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mrow> <mml:mo>∼</mml:mo> </mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>z</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> at the critical point, we conducted extensive simulations at this critical probability to determine the dynamic exponent z . In addition, the SAE is also capable of identifying the characteristic time of critical states. Finally, we tested two other heavy-tailed distributions that generate random step lengths: the Lévy distribution and the Cauchy distribution, which introduce different global spreading mechanisms. This method remains effective for determining the critical points in these systems. Our findings highlight the promising applications of SAE techniques for processes involving long-range interactions.