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Abstract

High-dimensional Partial Differential Equations (PDEs) form the foundation of complex process modeling in various scientific and engineering applications, including finance, physics, and optimal control. However, classical numerical methods are adversely affected by the curse of dimensionality, making them inapplicable for large-scale problems. Recently, however, deep learning-based approaches have provided a new toolbox for these high-dimensional PDEs, including methods such as the Deep Backward Stochastic Differential Equation (Deep BSDE) method. Our approach draws on a more sophisticated deep learning backbone, using neural networks (in our case, a Residual Neural Network and a Long Short-Term Memory network (LSTM) integrated into the Deep BSDE setup. Despite their success in various applications, feed-forward neural networks have drawbacks, as they are not designed to handle issues like vanishing gradients or dependencies between timeframes. We apply our method to several benchmark problems, including the nonlinear Black–Scholes equation, the Hamilton--Jacobi--Bellman (HJB) equation, and the Allen–Cahn equation in high-dimensional settings (up to 100 dimensions). The ResNet/LSTM-based approach developed here achieves consistently better performance with lower relative errors than the baseline feedforward-based approach, faster convergence rates, and greater computational efficiency. Overall, the results highlight the promise of using advanced neural architectures in the Deep BSDE approach and provide a scalable and efficient method for high-dimensional PDEs. This study lays the foundation for data-driven computational approaches and opens new opportunities in scientific machine learning.

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