论文标题
与神经网络的部分微分方程的变异蒙特卡洛方法
Variational Monte Carlo Approach to Partial Differential Equations with Neural Networks
论文作者
论文摘要
偏微分方程的准确数值解是数值分析中的一项核心任务，可以通过使用专用求解器来对广泛的自然现象进行建模，具体取决于应用程序的情况。在这里，我们开发了一种解决偏微分方程的变分方法，该方程是管理高维概率分布的演变的。我们的方法自然可以在无限的连续域上起作用，并通过其变异参数编码完整的概率密度函数，这些函数在演变过程中动态调整以最佳反映密度的动力学。对于经过考虑的基准案例，我们观察到与传统计算方法无法访问的政权中的数值解决方案以及分析解决方案的一致性。
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.
