The accurate numerical solution of partial differential equations is a central task in numerical analysis allowing to model a wide range of natural phenomena by employing specialized solvers depending on the scenario of application. Here, we develop a variational approach for solving partial differential equations governing the evolution of high dimensional probability distributions. Our approach naturally works on the unbounded continuous domain and encodes the full probability density function through its variational parameters, which are adapted dynamically during the evolution to optimally reflect the dynamics of the density. For the considered benchmark cases we observe excellent agreement with numerical solutions as well as analytical solutions in regimes inaccessible to traditional computational approaches.
翻译:部分偏差方程式的准确数字解决办法是数字分析的一项核心任务,它允许根据应用的假设情况使用专门的解决方案来模拟广泛的自然现象。在这里,我们制定了一种可变办法,以解决关于高维概率分布演变的局部差异方程式。我们的方法自然地在无限制连续域上工作,并通过其变异参数编码全部概率的密度功能,这些参数在演进期间动态调整,以最佳地反映密度的动态。对于经过考虑的基准案例,我们观察到与数字解决方案以及传统计算方法所无法利用的制度中的分析解决方案达成了极好的一致。