论文标题
非线性PDE的一些深部方案的近似误差分析
Approximation error analysis of some deep backward schemes for nonlinear PDEs
论文作者
论文摘要
最近提出的用于求解基于神经网络的高维非线性偏微分方程（PDE）的数值算法显示出其出色的性能。 我们回顾其中的一些并研究其收敛性。 这些方法依赖于向后的随机微分方程（BSDE）及其迭代时间离散化对PDE的概率表示。我们提出的算法，称为Deep Backward Multistep方案（MDBDP），是Turkedjiev Gobet的LSMDP方案的机器学习版本（Math。Comp。85，2016）。它通过向后诱导，通过神经网络通过向后触发，通过随机梯度下降执行的合适的二次损耗函数的顺序最小化来估算溶液及其梯度。 我们的主要理论贡献是提供MDBDP方案的近似误差分析以及在Beck，Becker，Cheridito，Jentzen，Neufeld（2019）中设计的半连接PDE的深层分裂（DS）方案。我们还补充了Hur {é}，Pham，Warin的DBDP方案的错误分析（Math。Comp。89，2020）。当PDE在MDBDP方案的溶液梯度和DBDP方案的半线性情况下，当PDE在溶液的梯度中是线性时，这会根据一类深Lipschitz连续组神经网络的神经元数量的收敛速率明显产生。我们通过一些数值测试与文献中的其他一些机器学习算法进行了比较，以说明结果。
Recently proposed numerical algorithms for solving high-dimensional nonlinear partial differential equations (PDEs) based on neural networks have shown their remarkable performance. We review some of them and study their convergence properties. The methods rely on probabilistic representation of PDEs by backward stochastic differential equations (BSDEs) and their iterated time discretization. Our proposed algorithm, called deep backward multistep scheme (MDBDP), is a machine learning version of the LSMDP scheme of Gobet, Turkedjiev (Math. Comp. 85, 2016). It estimates simultaneously by backward induction the solution and its gradient by neural networks through sequential minimizations of suitable quadratic loss functions that are performed by stochastic gradient descent. Our main theoretical contribution is to provide an approximation error analysis of the MDBDP scheme as well as the deep splitting (DS) scheme for semilinear PDEs designed in Beck, Becker, Cheridito, Jentzen, Neufeld (2019). We also supplement the error analysis of the DBDP scheme of Hur{é}, Pham, Warin (Math. Comp. 89, 2020). This yields notably convergence rate in terms of the number of neurons for a class of deep Lipschitz continuous GroupSort neural networks when the PDE is linear in the gradient of the solution for the MDBDP scheme, and in the semilinear case for the DBDP scheme. We illustrate our results with some numerical tests that are compared with some other machine learning algorithms in the literature.
