A deep-genetic algorithm (deep-GA) approach for high-dimensional nonlinear parabolic partial differential equations. (English) Zbl 1538.91102
Summary: 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.
MSC:
| 91G60 | Numerical methods (including Monte Carlo methods) |
| 65M75 | Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs |
| 68W50 | Evolutionary algorithms, genetic algorithms (computational aspects) |
| 68T07 | Artificial neural networks and deep learning |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
Keywords:
high dimensionality; nonlinear equations; genetic algorithm; backward stochastic differential equationSoftware:
TensorFlowReferences:
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