Data-driven Effective Modeling of Stochastic Chemical Reaction Networks

Chemical reaction networks are widely used to model stochastic dynamics in biophysical processes. Their exact probabilistic description via the chemical master equation becomes intractable for high-dimensional systems due to the exponential growth of the state space. This difficulty is further exacerbated when metastable transitions and rare events dominate the dynamics. Here, we present an advanced neural-network approach that effectively tackles large reaction networks and rare-event dynamics within a unified framework. Building on autoregressive representations of the full joint probability distribution, we achieve efficient training through second-order optimization methods, including natural gradient descent and the time-dependent variational principle. We leverage enhanced-sampling strategies that enable accurate resolution of low-probability regions governing rare transitions. We demonstrate the method on a multistable genetic toggle switch and a mitogen-activated protein kinase (MAPK) cascade consisting of 16 species and 35 reactions with dense coupling. We further quantify rare events in spatially extended Schlögl models in both one- and two-dimensional lattices, reaching system sizes with extremely large state spaces up to $3\times10^{15}$. These results highlight the potential of neural modeling as a flexible and scalable framework for studying high-dimensional chemical reaction networks and rare events.

Exact stochastic simulation of continuous-time Markov chains (CTMCs) is essential when discreteness and noise drive system behavior, but the hard categorical event selection in Gillespie-type algorithms blocks gradient-based learning. We eliminate this constraint by decoupling forward simulation from backward differentiation, with hard categorical sampling generating exact trajectories and gradients propagating through a continuous massively-parallel Gumbel-Softmax straight-through surrogate. Our approach enables accurate optimization at parameter scales over four orders of magnitude beyond existing simulators. We validate for accuracy, scalability, and reliability on a reversible dimerization model (0.09% error), a genetic oscillator (1.2% error), a 203,796-parameter gene regulatory network achieving 98.4% MNIST accuracy (a prototypical deep-learning multilayer perceptron benchmark), and experimental patch-clamp recordings of ion channel gating (R^2 = 0.987) in the single-channel regime. Our GPU implementation delivers 1.9 billion steps per second, matching the scale of non-differentiable simulators. By making exact stochastic simulation massively parallel and autodiff-compatible, our results enable high-dimensional parameter inference and inverse design across systems biology, chemical kinetics, physics, and related CTMC-governed domains.

Stochastic reduced-order models are widely used to represent the effective dynamics of complex systems, but estimating their drift and diffusion coefficients from data remains challenging. Standard approaches often rely on short-time trajectory increments, state-space partitioning, or repeated simulation of candidate models, which become unreliable or computationally expensive for high-dimensional systems, coarse temporal sampling, or unevenly sampled data. We introduce a data-driven calibration method based on a novel relationship between the coefficients of a stochastic reduced model and the conditional score of the finite-time transition density, defined as the gradient of the logarithm of the transition density with respect to the initial state. The resulting identity expresses derivatives of lagged correlation functions as stationary expectations over observed lagged pairs involving this conditional score and the unknown model coefficients. This formulation allows the drift and diffusion structure to be constrained directly from finite-lag statistics, without differentiating trajectories, partitioning state space, or repeatedly integrating candidate reduced models during calibration, yielding a least-squares fitting problem over stationary lagged pairs. We validate the approach on three systems of increasing complexity: an analytically tractable Cox--Ingersoll--Ross diffusion, a two-dimensional nonequilibrium diffusion with affine multiplicative noise, and a periodic soft-spin stochastic Landau--Lifshitz chain. Across these tests, the inferred models preserve the invariant statistics while reproducing finite-lag dynamical correlations. The framework provides a scalable route for learning stochastic reduced-order models from data that reproduce prescribed statistical and dynamical properties.