论文标题
基于神经网络的高维委员会功能的半群方法
A semigroup method for high dimensional committor functions based on neural network
论文作者
论文摘要
本文提出了一种基于神经网络的新方法，用于计算满足Fokker-Planck方程的高维委员会函数。该新方法没有使用部分微分方程，而是基于差分运算符的半群来使用积分公式。然后，通过将委员会功能作为神经网络参数化来解决新公式的变分形式。这种新方法有两个主要好处。首先，随机梯度下降类型算法可以应用于委员会功能的训练，而无需计算任何混合二阶导数。此外，与以前的方法通过惩罚术语强制执行边界条件的方法不同，新方法会自动考虑边界条件。提供数值结果以证明所提出的方法的性能。
This paper proposes a new method based on neural networks for computing the high-dimensional committor functions that satisfy Fokker-Planck equations. Instead of working with partial differential equations, the new method works with an integral formulation based on the semigroup of the differential operator. The variational form of the new formulation is then solved by parameterizing the committor function as a neural network. There are two major benefits of this new approach. First, stochastic gradient descent type algorithms can be applied in the training of the committor function without the need of computing any mixed second-order derivatives. Moreover, unlike the previous methods that enforce the boundary conditions through penalty terms, the new method takes into account the boundary conditions automatically. Numerical results are provided to demonstrate the performance of the proposed method.
