In this paper, we propose a semigroup method for solving high-dimensional elliptic partial differential equations (PDEs) and the associated eigenvalue problems based on neural networks. For the PDE problems, we reformulate the original equations as variational problems with the help of semigroup operators and then solve the variational problems with neural network (NN) parameterization. The main advantages are that no mixed second-order derivative computation is needed during the stochastic gradient descent training and that the boundary conditions are taken into account automatically by the semigroup operator. Unlike popular methods like PINN \cite{raissi2019physics} and Deep Ritz \cite{weinan2018deep} where the Dirichlet boundary condition is enforced solely through penalty functions and thus changes the true solution, the proposed method is able to address the boundary conditions without penalty functions and it gives the correct true solution even when penalty functions are added, thanks to the semigroup operator. For eigenvalue problems, a primal-dual method is proposed, efficiently resolving the constraint with a simple scalar dual variable and resulting in a faster algorithm compared with the BSDE solver \cite{han2020solving} in certain problems such as the eigenvalue problem associated with the linear Schr\"odinger operator. Numerical results are provided to demonstrate the performance of the proposed methods.
翻译:在本文中, 我们提出一个基于神经网络解决高维椭圆部分偏差方程式( PDEs ) 及相关的天价问题的半组方法。 对于 PDE 问题, 我们使用半组操作员帮助将原始方程式重新配置为变异问题, 然后用神经网络参数化解决变异问题。 主要的好处是, 在随机梯度梯度下降训练期间, 不需要混合二级衍生衍生衍生物计算, 并且半组操作员会自动考虑边界条件 。 与 PINN \ cite {laisi2019 物理} 和 Deep Ritz {cite{weinan2018deep} 等流行方法不同, 在Dirichlet 边界条件仅通过惩罚功能强制实施并从而改变真正解决方案的情况下, 我们提出的方法能够解决边界条件的变异问题, 即使在使用半组操作员添加惩罚功能时, 也提供正确的真实的解决方案。 对于半组操作员来说, 提出一种原始方法, 以简单的 二次变数 20 和 直线性 算算算算算算法, 提供了一种快速的 解 的 。 快速 解算法,, 以简单的 解 解 20 解 解 的 。, 以 解算 快速 的 解 解 解 解 解 20 解 解 解 解 20 的 解 解 解 20 解 。