A Deep-Genetic Algorithm (Deep-GA) Approach for High-Dimensional Nonlinear Parabolic Partial Differential Equations
This addresses computational bottlenecks in solving high-dimensional PDEs for applications like finance and control, but it is incremental as it builds on the existing deep-BSDE method.
The authors tackled the problem of solving high-dimensional nonlinear parabolic partial differential equations by proposing a deep-genetic algorithm (deep-GA) that embeds a genetic algorithm into the deep-BSDE method to optimize initial guess selection. The result showed comparable accuracy to deep-BSDE with significantly improved computational efficiency, as demonstrated on the Black-Scholes equation with default risk and the Hamilton-Jacobi-Bellman equation.
We propose a new method, called a deep-genetic algorithm (deep-GA), to accelerate the performance of the so-called deep-BSDE method, which is a deep learning algorithm to solve high dimensional partial differential equations through their corresponding backward stochastic differential equations (BSDEs). Recognizing the sensitivity of the solver to the initial guess selection, we embed a genetic algorithm (GA) into the solver to optimize the selection. We aim to achieve faster convergence for the nonlinear PDEs on a broader interval than deep-BSDE. Our proposed method is applied to two nonlinear parabolic PDEs, i.e., the Black-Scholes (BS) equation with default risk and the Hamilton-Jacobi-Bellman (HJB) equation. We compare the results of our method with those of the deep-BSDE and show that our method provides comparable accuracy with significantly improved computational efficiency.