arXiv 1902.015992019-02math.PRcs.NAcs.NEmath.NAmath.OCstat.ML

高维非线性偏微分方程的深向后格式

Deep backward schemes for high-dimensional nonlinear PDEs

Côme Huré, Huyên Pham, Xavier Warin

备注：: 34 pages

摘要：

我们提出了新的机器学习方案来解决高维非线性偏微分方程（PDE）。依靠经典的向后随机微分方程（BDEs）表示，我们的算法同时估计的解决方案和其梯度的深度神经网络。这些近似是在每个时间步从最小化的损失函数递归定义的逆向归纳法。该方法被扩展到变分不等式产生的最优停止问题。我们分析了深度学习方案的收敛性，并根据神经网络的通用近似性提供了误差估计。数值结果表明，我们的算法给出了非常好的结果，直到50维（当然以上），对偏微分方程和变分不等式问题。对于偏微分方程的分辨率，我们的结果是非常相似的最近的方法在\cite{weinan2017deep}时，后者收敛到正确的解决方案或不发散。数值试验表明，所提出的方法不会像在\cite{weinan2017deep}中设计的算法那样陷入局部极小，并且不会出现发散。唯一的限制似乎是由于所考虑的深度神经网络无法表示在高维中具有过于复杂的复结构的解决方案。

英文摘要：

We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate simultaneously the solution and its gradient by deep neural networks. These approximations are performed at each time step from the minimization of loss functions defined recursively by backward induction. The methodology is extended to variational inequalities arising in optimal stopping problems. We analyze the convergence of the deep learning schemes and provide error estimates in terms of the universal approximation of neural networks. Numerical results show that our algorithms give very good results till dimension 50 (and certainly above), for both PDEs and variational inequalities problems. For the PDEs resolution, our results are very similar to those obtained by the recent method in \cite{weinan2017deep} when the latter converges to the right solution or does not diverge. Numerical tests indicate that the proposed methods are not stuck in poor local minimaas it can be the case with the algorithm designed in \cite{weinan2017deep}, and no divergence is experienced. The only limitation seems to be due to the inability of the considered deep neural networks to represent a solution with a too complex structure in high dimension.

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