Tracking large chemical reaction networks and rare events by neural networks

Chemical reaction networks are widely used to model stochastic dynamics in chemical kinetics, systems biology and epidemiology. Solving the chemical master equation that governs these systems poses a significant challenge due to the large state space exponentially growing with system sizes. The development of autoregressive neural networks offers a flexible framework for this problem; however, its efficiency is limited especially for high-dimensional systems and in scenarios with rare events. Here, we push the frontier of neural-network approach by exploiting faster optimizations such as natural gradient descent and time-dependent variational principle, achieving a 5- to 22-fold speedup, and by leveraging enhanced-sampling strategies to capture rare events. We demonstrate reduced computational cost and higher accuracy over the previous neural-network method in challenging reaction networks, including the mitogen-activated protein kinase (MAPK) cascade network, the hitherto largest biological network handled by the previous approaches of solving the chemical master equation. We further apply the approach to spatially extended reaction-diffusion systems, the Schl"ogl model with rare events, on two-dimensional lattices, beyond the recent tensor-network approach that handles one-dimensional lattices. The present approach thus enables efficient modeling of chemical reaction networks in general.