The Finite State Projection (FSP) method approximates the Chemical Master Equation (CME) by restricting the dynamics to a finite subset of the (typically infinite) state space, enabling direct numerical solution with computable error bounds. Adaptive variants update this subset in time, but multiscale systems with widely separated reaction rates remain challenging, as low-probability bottleneck states can carry essential probability flux and the dynamics alternate between fast transients and slowly evolving stiff regimes. We propose a flux-based adaptive FSP method that uses probability flux to drive both state-space pruning and time-step selection. The pruning rule protects low-probability states with large outgoing flux, preserving connectivity in bottleneck systems, while the time-step rule adapts to the instantaneous total flux to handle rate constants spanning several orders of magnitude. Numerical experiments on stiff, oscillatory, and bottleneck reaction networks show that the method maintains accuracy while using substantially smaller state spaces.

翻译：有限状态投影（FSP）方法通过将动力学限制在（通常无限的）状态空间的一个有限子集上来近似化学主方程（CME），从而能够直接进行数值求解并给出可计算的误差界。自适应变体随时间更新该子集，但具有广泛分离反应速率的多尺度系统仍然具有挑战性，因为低概率瓶颈状态可能携带关键的概率通量，且动力学在快速瞬态和缓慢演化的刚性状态之间交替。我们提出了一种基于通量的自适应FSP方法，该方法利用概率通量同时驱动状态空间剪枝和时间步长选择。剪枝规则保护具有较大出射通量的低概率状态，从而保持瓶颈系统中的连通性，而时间步长规则则根据瞬时总通量自适应调整，以处理跨越多个数量级的速率常数。针对刚性、振荡性和瓶颈反应网络的数值实验表明，该方法在保持精度的同时，使用了显著更小的状态空间。