Hyperbolic balance laws with uncertain (random) parameters and inputs are ubiquitous in science and engineering. Quantification of uncertainty in predictions derived from such laws, and reduction of predictive uncertainty via data assimilation, remain an open challenge. That is due to nonlinearity of governing equations, whose solutions are highly non-Gaussian and often discontinuous. To ameliorate these issues in a computationally efficient way, we use the method of distributions, which here takes the form of a deterministic equation for spatio-temporal evolution of the cumulative distribution function (CDF) of the random system state, as a means of forward uncertainty propagation. Uncertainty reduction is achieved by recasting the standard loss function, i.e. discrepancy between observations and model predictions, in distributional terms. This step exploits the equivalence between minimization of the square error discrepancy and the Kullback–Leibler divergence. The loss function is regularized by adding a Lagrangian constraint enforcing fulfilment of the CDF equation. Minimization is performed sequentially, progressively updating the parameters of the CDF equation as more measurements are assimilated.

中文翻译：

---

学习动态统计流形

具有不确定（随机）参数和输入的双曲线平衡定律在科学和工程中无处不在。量化源自这些定律的预测中的不确定性，以及通过数据同化减少预测不确定性，仍然是一个开放的挑战。这是由于控制方程的非线性所致，其解高度非高斯且通常不连续。为了以计算上有效的方式改善这些问题，我们使用分布方法，这里采用随机系统状态累积分布函数 (CDF) 时空演化的确定性方程的形式，作为向前不确定性传播。通过在分布方面重新定义标准损失函数，即观测值和模型预测之间的差异，可以减少不确定性。这一步利用了平方误差差异的最小化和 Kullback-Leibler 散度之间的等价性。通过添加强制满足 CDF 方程的拉格朗日约束来正则化损失函数。最小化是按顺序执行的，随着更多测量被同化，逐渐更新 CDF 方程的参数。