Solving high-dimensional partial differential equations using deep learning

Abstract

Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality." This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function. Numerical results on examples including the nonlinear Black-Scholes equation, the Hamilton-Jacobi-Bellman equation, and the Allen-Cahn equation suggest that the proposed algorithm is quite effective in high dimensions, in terms of both accuracy and cost. This opens up possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their interrelationships.

Keywords: Feynman–Kac; backward stochastic differential equations; deep learning; high dimension; partial differential equations.

Fig. 1.

Plot of ${\theta }_{u_0}$ as an approximation of $u\left(t=0,x=\left(100,\dots ,100\right)\right)$ against the number of iteration steps in the case of the $100$-dimensional nonlinear Black–Scholes equation with $40$ equidistant time steps ($N=40$) and learning rate $0.008$. The shaded area depicts the mean $\pm$ the SD of ${\theta }_{u_0}$ as an approximation of $u\left(t=0,x=\left(100,\dots ,100\right)\right)$ for five independent runs. The deep BSDE method achieves a relative error of size $0.46\%$ in a runtime of $\mathrm{1,607}$ s.

Fig. 2.

(Top) Relative error of the deep BSDE method for $u\left(t=0,x=\left(0,\dots ,0\right)\right)$ when $\lambda =1$ against the number of iteration steps in the case of the $100$-dimensional HJB Eq. 13 with $20$ equidistant time steps ($N=20$) and learning rate $0.01$. The shaded area depicts the mean $\pm$ the SD of the relative error for five different runs. The deep BSDE method achieves a relative error of size $0.17\%$ in a runtime of $330$ s. (Bottom) Optimal cost $u\left(t=0,x=\left(0,\dots ,0\right)\right)$ against different values of $\lambda$ in the case of the $100$-dimensional HJB Eq. 13, obtained by the deep BSDE method and classical Monte Carlo simulations of Eq. 14.

Fig. 3.

(Top) Relative error of the deep BSDE method for $u\left(t=0.3,x=\left(0,\dots ,0\right)\right)$ against the number of iteration steps in the case of the $100$-dimensional Allen–Cahn Eq. 15 with $20$ equidistant time steps ($N=20$) and learning rate $0.0005$. The shaded area depicts the mean $\pm$ the SD of the relative error for five different runs. The deep BSDE method achieves a relative error of size $0.30\%$ in a runtime of $647$ s. (Bottom) Time evolution of $u\left(t,x=\left(0,\dots ,0\right)\right)$ for $t\in \left[\mathrm{0,0.3}\right]$ in the case of the $100$-dimensional Allen–Cahn Eq. 15 computed by means of the deep BSDE method.

Fig. 4.

Illustration of the network architecture for solving semilinear parabolic PDEs with $H$ hidden layers for each subnetwork and $N$ time intervals. The whole network has $\left(H+1\right)\left(N-1\right)$ layers in total that involve free parameters to be optimized simultaneously. Each column for $t=t_1,t_2,\dots ,t_{N-1}$ corresponds to a subnetwork at time $t$. $h_n^1,\dots ,h_n^H$ are the intermediate neurons in the subnetwork at time $t=t_n$ for $n=\mathrm{1,2},\dots ,N-1$.