Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed implementation schemes behind solving {high-dimensional} ODEs with a probabilistic numerical algorithm. This has not been possible before due to matrix-matrix operations in each solver step, but is crucial for scientifically relevant problems -- most importantly, the solution of discretised {partial} differential equations. In a nutshell, efficient high-dimensional probabilistic ODE solutions build either on independence assumptions or on Kronecker structure in the prior model. We evaluate the resulting efficiency on a range of problems, including the probabilistic numerical simulation of a differential equation with millions of dimensions.
翻译:普通差异方程式(ODEs)的概率求解器已成为对动态系统进行不确定性量化和推断的有效框架。 在这项工作中,我们用概率数字算法来解释解决{高维}代码的数学假设和详细实施方案。由于每个求解器步骤中的矩阵矩阵矩阵操作,这在以前是不可能的,但对于科学相关问题至关重要 -- -- 最重要的是,解决离散的{部分}差异方程式。在一个综合的、高效的高维概率的解决方案中,要么以独立假设为基础,要么以前一个模型中的克伦克尔结构为基础。我们评估由此产生的一系列问题的效率,包括数以百万维的差异方程式的概率数字模拟。