We explore efficient estimation of statistical quantities, particularly rare event probabilities, for stochastic reaction networks. Consequently, we propose an importance sampling (IS) approach to improve the Monte Carlo (MC) estimator efficiency based on an approximate tau-leap scheme. The crucial step in the IS framework is choosing an appropriate change of probability measure to achieve substantial variance reduction. This task is typically challenging and often requires insights into the underlying problem. Therefore, we propose an automated approach to obtain a highly efficient path-dependent measure change based on an original connection in the stochastic reaction network context between finding optimal IS parameters within a class of probability measures and a stochastic optimal control formulation. Optimal IS parameters are obtained by solving a variance minimization problem. First, we derive an associated dynamic programming equation. Analytically solving this backward equation is challenging, hence we propose an approximate dynamic programming formulation to find near-optimal control parameters. To mitigate the curse of dimensionality, we propose a learning-based method to approximate the value function using a neural network, where the parameters are determined via a stochastic optimization algorithm. Our analysis and numerical experiments verify that the proposed learning-based IS approach substantially reduces MC estimator variance, resulting in a lower computational complexity in the rare event regime, compared with standard tau-leap MC estimators.

翻译：我们探索了随机反应网络中统计量（尤其是稀有事件概率）的高效估计方法。为此，提出了一种基于近似tau-leap方案的重要性采样（IS）方法，以提升蒙特卡洛（MC）估计器的效率。IS框架的关键步骤在于选择合适的概率测度变化，以实现显著的方差缩减。这一任务通常具有挑战性，需要深入理解问题本质。因此，我们提出了一种自动化方法，通过建立随机反应网络背景下寻找最优IS参数与随机最优控制公式之间的原始关联，在概率测度类中实现高效路径依赖的测度变化。最优IS参数通过求解方差最小化问题获得。首先，推导出相应的动态规划方程。由于解析求解该后向方程具有难度，我们提出近似动态规划公式以寻找近优控制参数。为缓解维度灾难，提出基于学习的方法，利用神经网络近似值函数，并通过随机优化算法确定参数。理论分析与数值实验表明，与标准tau-leap MC估计器相比，所提出的基于学习的IS方法在稀有事件场景下能显著降低MC估计器方差，从而降低计算复杂度。