论文标题
校准的自适应概率溶解器
Calibrated Adaptive Probabilistic ODE Solvers
论文作者
论文摘要
普通微分方程的概率求解器为初始值问题的解决方案分配了后验度量。该分布的关节协方差提供了（全局）近似误差的估计。此误差估计的收缩率是求解器的步长函数的函数，将其标识为符合良好的最坏情况误差，但其对特定步长大小的显式数值并不是对显式误差的良好估计。在解决这个问题时，我们介绍，讨论和评估了几种校准不确定性估计值的概率动机方法。数值实验表明，这些校准方法与自适应的阶梯尺寸选择有效相互作用，从而导致描述性且有效地计算的后代。我们通过对经典，广泛使用的dormand-rince 4/5 runge-kutta方法进行基准测试来证明该方法的效率。
Probabilistic solvers for ordinary differential equations assign a posterior measure to the solution of an initial value problem. The joint covariance of this distribution provides an estimate of the (global) approximation error. The contraction rate of this error estimate as a function of the solver's step size identifies it as a well-calibrated worst-case error, but its explicit numerical value for a certain step size is not automatically a good estimate of the explicit error. Addressing this issue, we introduce, discuss, and assess several probabilistically motivated ways to calibrate the uncertainty estimate. Numerical experiments demonstrate that these calibration methods interact efficiently with adaptive step-size selection, resulting in descriptive, and efficiently computable posteriors. We demonstrate the efficiency of the methodology by benchmarking against the classic, widely used Dormand-Prince 4/5 Runge-Kutta method.
