AK–MCS
测试版 / 未完成beta / unfinished已知仍有错误；交互、措辞与数值表达正在复核。Known errors remain; interaction, wording, and numerical communication are under review.
一种将主动学习、Kriging 代理模型与 Monte Carlo Simulation 结合的可靠度方法。这里不从公式开始：先看见一个结构怎样成为一个点，一群点怎样长出一条边界，以及算法为什么只把真实计算花在最值得问的位置。A reliability method combining active learning, a Kriging surrogate, and Monte Carlo simulation. Rather than beginning with equations, see how a structure becomes a point, how a cloud of points produces a boundary, and why the algorithm spends high-fidelity evaluations only where they are most informative.
不确定性不止两个变量Uncertainty is not limited to two variables
先从两个能被画出来的维度开始：材料强度Begin with two dimensions that can be drawn: material strength 与初始几何缺陷and initial geometric imperfection 。金色取值点与结构响应保持同步。The golden sample point remains synchronized with the structural response.
真实可靠度问题通常还包括弹性模量、节点刚度、预紧力、摩擦系数等。画面只保留二维投影，但随机向量可以是任意维度。Real reliability problems commonly include elastic modulus, connection stiffness, preload, friction coefficient, and other variables. The image retains only a two-dimensional projection, while the random vector may have arbitrary dimension.
一次 realization，是所有变量同时取到一组具体值One realization is one simultaneous set of values for all variables
先看两个主变量的取值沿轨迹落入结构；其他高维变量随后一并汇入。结构与这一整组参数绑定，然后一起收缩成概率空间中的一个点。First, values of the two primary variables travel into the structure; the remaining higher-dimensional variables then join them. The structure binds to the complete parameter set and contracts into one point in probability space.
不是“结构消失成一个点”，而是同一个 realization 换了一种数学表示。The structure does not “disappear into a point”; the same realization is represented mathematically in another form.
然后，把同一个出生过程重复很多次Then repeat the same sampling process many times
前几个样本慢慢出生，之后逐渐加速。每一个点都代表一组新的随机变量取值，也就是一个新的可能结构。The first samples appear slowly, then accelerate. Each point represents a new set of random-variable values and therefore a new possible structure.
到这一幕结束时，我们只知道这些结构“可能出现在哪里”；它们是安全还是失效，仍然一个都不知道。At the end of this scene, we know only where these structures may occur; none has yet been classified as safe or failed.
灰色点只有可能性，还没有答案Gray points represent possibilities, not answers
一个候选样本进入真实模型，结构受荷、变形，模型返回一个极限状态函数值。只有此时，它才获得安全或失效的颜色。A candidate sample enters the high-fidelity model. The structure is loaded and deforms, and the model returns a limit-state function value. Only then is the point colored safe or failed.
真正慢的，不是生成样本；是真实求解The expensive step is not sampling, but high-fidelity evaluation
两千个样本尚且需要排队。稀有失效若需要百万级候选，而一次非线性有限元分析耗时十分钟，暴力 MCS 会把时间推到近二十年。Even two thousand samples require a queue. If rare failure demands one million candidates and each nonlinear finite-element analysis takes ten minutes, brute-force MCS approaches twenty years of computation.
只有少数已知答案，Kriging 怎样猜一个未知点？With only a few known responses, how does Kriging predict an unknown point?
先把一个没有真实答案的查询点放进空间。附近的训练点与它更相似，影响更强；远处训练点的影响更弱。Kriging 汇总这些已知信息，给出一个预测均值，并同时告诉我们这次预测有多不确定。Place a query point with no high-fidelity response in the space. Nearby training points are more similar and exert greater influence; distant points exert less. Kriging combines the known information to predict a mean and quantify predictive uncertainty.
把刚才的“问一个点”，重复到整个空间Repeat that one-point query across the entire space
同一个 Kriging 查询被逐点重复：每个位置都得到一个预测均值和一个不确定性。扫描线走过后，颜色留下均值，紫色雾感留下不确定性；直到整张连续预测场完整长出来。The same Kriging query is repeated point by point. Every location receives a predictive mean and uncertainty. As the scan passes, color records the mean and purple haze the uncertainty until a continuous prediction field appears.
边界不是画上去的，它从预测场中浮现The boundary is not drawn manually; it emerges from the prediction field
沿任意横截线，预测均值由正变负时会穿过零。把所有零点连起来，就得到 Kriging 预测的极限状态边界。Along any cross-section, the predictive mean crosses zero as it changes sign. Connecting all zero crossings produces the Kriging-predicted limit-state boundary.
选择最难判定正负号的点Select the point whose sign is hardest to determine
这里不是比较候选点到边界的几何距离。学习函数比较的是预测均值相对零点的标准化距离：零点离当前预测中心有多少个标准差？这个数越小，正负号越含糊，越值得调用真实模型。This is not the geometric distance from a candidate to the boundary. The learning function compares the standardized distance of the predictive mean from zero: how many predictive standard deviations separate zero from the current mean? The smaller the value, the more ambiguous the sign and the more informative a high-fidelity evaluation becomes.
A、B、C 的数值会在动画中逐一比较；最终被选中的 C 确实拥有最小的 U 值。The animation compares values at A, B, and C; the selected point C indeed has the smallest U value.
只问一次，整个模型都会改变One new query changes the entire model
选中的点进入真实模型，带着真实答案返回并加入训练集。局部不确定性收缩，旧边界退场，新边界重新生长。The selected point enters the high-fidelity model, returns with its evaluated response, and joins the training set. Local uncertainty contracts, the old boundary recedes, and a new boundary grows.
最后，概率从一群点的比例中出现Finally, probability emerges from a proportion of points
全部 MCS 候选由最终代理模型分类。安全点到左边，失效点到右边；失效样本占总样本的比例，就是失效概率估计。The final surrogate classifies all MCS candidates. Safe points move left and failed points right; the failed fraction of the total sample estimates failure probability.
许多可能Many possibilities
少量提问Few questions
一条边界One boundary
主动 Kriging–蒙特卡洛模拟Active Kriging–Monte Carlo Simulation：MCS 提供可能性，Kriging 推断未知，Active Learning 把真实计算只花在最值得问的位置。: MCS supplies possibilities, Kriging infers unknown responses, and active learning spends high-fidelity evaluations only where they are most informative.