We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural networks, while the Hessian is approximated by automatic differentiation of the gradient at previous step. This methodology extends to the fully nonlinear case the approach recently proposed in \cite{HPW19} for semi-linear PDEs. Numerical tests illustrate the performance and accuracy of our method on several examples in high dimension with nonlinearity on the Hessian term including a linear quadratic control problem with control on the diffusion coefficient, Monge-Amp{\`e}re equation and Hamilton-Jacobi-Bellman equation in portfolio optimization.
翻译:我们提出了一个解决高维全非线性局部偏差方程式的数值方法。我们的算法估算同时,在后向时间引导溶液的同时,由多层神经网络将其梯度引入多层神经网络,而赫西安人则在前一步通过对梯度的自动区分进行近似。这个方法将最近在\cite{HPW19}中为半线性PDEs建议的办法扩大到完全非线性的情况。数字测试表明我们的方法在高维中几个例子中的性能和准确性,在赫西安术语上具有非线性,包括一个线性二次控制问题,在组合优化中控制扩散系数、Monge-Amp ⁇ ere等式和汉密尔顿-Jacobi-Bellman等式。