Imam Mukhlash

Endah Rokhmati Merdika Putri, Muhammad Luthfi Shahab, Mohammad Iqbal et al.

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.

Muhammad Luthfi Shahab, Imam Mukhlash, Hadi Susanto

This work investigates the use of shallow physics-informed neural networks (PINNs) for solving forward and inverse problems of nonlinear partial differential equations (PDEs). By reformulating PINNs as nonlinear systems, the Levenberg-Marquardt (LM) algorithm is employed to efficiently optimize the network parameters. Analytical expressions for the neural network derivatives with respect to the input variables are derived, enabling accurate and efficient computation of the Jacobian matrix required by LM. The proposed approach is tested on several benchmark problems, including the Burgers, Schrödinger, Allen-Cahn, and three-dimensional Bratu equations. Numerical results demonstrate that LM significantly outperforms BFGS in terms of convergence speed, accuracy, and final loss values, even when using shallow network architectures with only two hidden layers. These findings indicate that, for a wide class of PDEs, shallow PINNs combined with efficient second-order optimization methods can provide accurate and computationally efficient solutions for both forward and inverse problems.