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A deep-genetic algorithm (deep-GA) approach for high-dimensional nonlinear parabolic partial differential equations.

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Title: A deep-genetic algorithm (deep-GA) approach for high-dimensional nonlinear parabolic partial differential equations.
Authors: Putri, Endah R.M.1 (AUTHOR) endahrmp@matematika.its.ac.id, Shahab, Muhammad L.1,2 (AUTHOR), Iqbal, Mohammad1 (AUTHOR), Mukhlash, Imam1 (AUTHOR), Hakam, Amirul1 (AUTHOR), Mardianto, Lutfi3 (AUTHOR), Susanto, Hadi2 (AUTHOR)
Source: Computers & Mathematics with Applications. Jan2024, Vol. 154, p120-127. 8p.
Subjects: Parabolic differential equations, Stochastic differential equations, Hamilton-Jacobi-Bellman equation, Machine learning, Partial differential equations, Genetic algorithms
Abstract: 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. • We introduce a deep-GA method. • The method provides a better accuracy and efficiency than the deep-BSDE solver. • Efficiency and accuracy is obtained by embedding a genetic algorithm (GA). • We solve the Black-Scholes with default risk and Hamilton-Jacobi-Bellman equations. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
Description
Abstract: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. • We introduce a deep-GA method. • The method provides a better accuracy and efficiency than the deep-BSDE solver. • Efficiency and accuracy is obtained by embedding a genetic algorithm (GA). • We solve the Black-Scholes with default risk and Hamilton-Jacobi-Bellman equations. [ABSTRACT FROM AUTHOR]
ISSN:08981221
DOI:10.1016/j.camwa.2023.11.022