This work applies the first order adjoint sensitivity analysis methodology to a reduced-order model of BWR dynamics to compute the exact first order sensitivities of the model’s state-functions with respect to the model’s initial conditions and parameters. These sensitivities are computed using exact forward and adjoint sensitivity functions, which are shown to yield identical results, within machine accuracy, in all of the phase-space regions characterizing this BWR-model, which comprise: (i) the “stable” region, in which the state-functions converge asymptotically to time-independent values; (ii) the “limit-cycle” regions, in which the state-functions oscillate periodically in time, and (iii) the “chaotic” region, in which the state-functions oscillate among infinitely many unstable attractors aperiodically in time. The exact results using the adjoint sensitivity analysis methodology are contrasted with the unreliable results produced by “brute-force” methods using finite-differences, which are often used in the literature to compute approximate response sensitivities to parameters. The finite-difference methods are shown to produce reasonable approximations when used inside the stable region but completely fail when used in the oscillatory regions.

The sensitivity analysis results in the “stable” Region 1 indicate that, after an early oscillatory phase characterized by high-amplitude oscillations caused by the initial perturbation, the sensitivities of the state-variable reach the exact time-independent values predicted theoretically. In the “limit-cycle” Regions, the sensitivities of the state functions oscillate with increasing amplitudes among the respective unstable equilibrium points. In the chaotic region, the sensitivities oscillate with amplitudes that increase exponentially in time, reaching very large values (10²³) very rapidly after the onset of the initial perturbation, thus confirming the model’s extreme sensitivity to parameter perturbations in this region.

The major novelty resulting from this work is the first-ever demonstration that the First-Level Forward Sensitivity System (1st-LFSS) and the First-Level Adjoint Sensitivity System (1st-LASS) can be used reliably to compute the exact 1st-order sensitivities of state functions with respect to the model parameters not only in the stable region in phase-space, but also in the “limit-cycle” and “chaotic” regions, in contradistinction with the failure of methods based on finite-differences. Subsequent work will use the sensitivities presented in this work to compute the uncertainties induced in the reduced-order BWR-model’s state functions by uncertainties in this BWR-model’s parameters and initial conditions.

这项工作将一阶伴随敏感度分析方法应用于BWR动力学的降阶模型，以计算模型状态函数相对于模型的初始条件和参数的确切一阶敏感度。这些敏感度是使用精确的前向和伴随敏感度函数计算得出的，这些函数表明，在表征该BWR模型的所有相空间区域中，在机器精度范围内可以产生相同的结果，这些区域包括：（i）“稳定”区域，状态函数渐近收敛到与时间无关的值；（ii）“极限循环”区域，其中状态函数会定期发生周期性振荡；以及（iii）“混沌”区域，其中状态函数会在无限多个不稳定吸引子之间随时间周期性发生振荡。使用伴随灵敏度分析方法的精确结果与使用有限差分的“蛮力”方法产生的不可靠结果形成对比，该方法经常在文献中用于计算对参数的近似响应灵敏度。当在稳定区域内使用时，有限差分方法显示出合理的近似值，而在振荡区域中使用时，则完全失效。

在“稳定”区域1中的灵敏度分析结果表明，在以初始扰动引起的高振幅振荡为特征的早期振荡阶段之后，状态变量的灵敏度达到理论上预测的与时间无关的精确值。在“极限循环”区域中，状态函数的灵敏度随着各个不稳定平衡点之间的幅度增加而振荡。在混沌区域中，灵敏度随时间呈指数增加的幅度振荡，在初始扰动开始后非常迅速地达到非常大的值（10 ²³），从而确认了模型对该区域中参数扰动的极端敏感性。

这项工作产生的主要新颖性是有史以来第一次证明，可以可靠地使用第一级前向灵敏度系统（1st-LFSS）和第一级伴随灵敏度系统（1st-LASS）来计算精确的一阶状态函数对模型参数的敏感性不仅在相空间的稳定区域内，而且在“极限环”和“混沌”区域内，与基于有限差分的方法的失败相反。随后的工作将使用本工作中介绍的敏感性，通过该BWR模型的参数和初始条件的不确定性来计算降阶BWR模型的状态函数中引起的不确定性。