H. Susanto

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, Hadi Susanto

This research introduces an extended application of neural networks for solving nonlinear partial differential equations (PDEs). A neural network, combined with a pseudo-arclength continuation, is proposed to construct bifurcation diagrams from parameterized nonlinear PDEs. Additionally, a neural network approach is also presented for solving eigenvalue problems to analyze solution linear stability, focusing on identifying the largest eigenvalue. The effectiveness of the proposed neural network is examined through experiments on the Bratu equation and the Burgers equation. Results from a finite difference method are also presented as comparison. Varying numbers of grid points are employed in each case to assess the behavior and accuracy of both the neural network and the finite difference method. The experimental results demonstrate that the proposed neural network produces better solutions, generates more accurate bifurcation diagrams, has reasonable computational times, and proves effective for linear stability analysis.

Muhammad Luthfi Shahab, Fidya Almira Suheri, Rudy Kusdiantara et al.

This paper introduces a framework based on physics-informed neural networks (PINNs) for addressing key challenges in nonlinear lattices, including solution approximation, bifurcation diagram construction, and linear stability analysis. We first employ PINNs to approximate solutions of nonlinear systems arising from lattice models, using the Levenberg-Marquardt algorithm to optimize network weights for greater accuracy. To enhance computational efficiency in high-dimensional settings, we integrate a stochastic sampling strategy. We then extend the method by coupling PINNs with a continuation approach to compute snaking bifurcation diagrams, incorporating an auxiliary equation to effectively track successive solution branches. For linear stability analysis, we adapt PINNs to compute eigenvectors, introducing output constraints to enforce positivity, in line with Sturm-Liouville theory. Numerical experiments are conducted on the discrete Allen-Cahn equation with cubic and quintic nonlinearities in one to five spatial dimensions. The results demonstrate that the proposed approach achieves accuracy comparable to, or better than, traditional numerical methods, especially in high-dimensional regimes where computational resources are a limiting factor. These findings highlight the potential of neural networks as scalable and efficient tools for the study of complex nonlinear lattice systems.

V. R. Tjahjono, S. F. Feng, E. R. M. Putri et al.

Recent developments in applied mathematics increasingly employ machine learning (ML)-particularly supervised learning-to accelerate numerical computations, such as solving nonlinear partial differential equations. In this work, we extend such techniques to objects of a more theoretical nature: the classification and structural analysis of fractal sets. Focusing on the Mandelbrot and Julia sets as principal examples, we demonstrate that supervised learning methods-including Classification and Regression Trees (CART), K-Nearest Neighbors (KNN), Multilayer Perceptrons (MLP), and Recurrent Neural Networks using both Long Short-Term Memory (LSTM) and Bidirectional LSTM (BiLSTM), Random Forests (RF), and Convolutional Neural Networks (CNN)-can classify fractal points with significantly higher predictive accuracy and substantially lower computational cost than traditional numerical approaches, such as the thresholding technique. These improvements are consistent across a range of models and evaluation metrics. Notably, KNN and RF exhibit the best overall performance, and comparative analyses between models (e.g., KNN vs. LSTM) suggest the presence of novel regularity properties in these mathematical structures. Collectively, our findings indicate that ML not only enhances classification efficiency but also offers promising avenues for generating new insights, intuitions, and conjectures within pure mathematics.