Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations (1804.07010v1)

Abstract: Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for solving forward and inverse problems associated with partial differential equations, we circumvent the tyranny of numerical discretization by devising an algorithm that is scalable to high-dimensions. In particular, we approximate the unknown solution by a deep neural network which essentially enables us to benefit from the merits of automatic differentiation. To train the aforementioned neural network we leverage the well-known connection between high-dimensional partial differential equations and forward-backward stochastic differential equations. In fact, independent realizations of a standard Brownian motion will act as training data. We test the effectiveness of our approach for a couple of benchmark problems spanning a number of scientific domains including Black-Scholes-Barenblatt and Hamilton-Jacobi-BeLLMan equations, both in 100-dimensions.

• The paper introduces a deep learning framework that approximates high-dimensional PDE solutions using forward-backward stochastic neural networks.
• It leverages automatic differentiation to compute precise derivatives, eliminating the limitations of traditional grid-based numerical methods.
• The approach efficiently addresses complex test cases like the 100-dimensional Black-Scholes and Hamilton-Jacobi-Bellman equations, demonstrating scalability and high accuracy.

Forward-Backward Stochastic Neural Networks: An Examination

The paper in question presents a novel approach to solving high-dimensional partial differential equations (PDEs) using deep learning techniques, specifically through the introduction of forward-backward stochastic neural networks (FBSNNs). The authors propose a methodology grounded in the connection between high-dimensional PDEs and forward-backward stochastic differential equations (FBSDEs), aiming to circumvent the traditional issues of numerical discretization related to dimensionality.

Technical Contributions

The central contribution lies in approximating the unknown solution of a PDE using a deep neural network. By leveraging automatic differentiation, the authors aim to efficiently compute derivatives without resorting to numerical or symbolic differentiation, thus maintaining computational precision at machine level. This approach is contrasted with classical numerical methods such as finite elements, finite differences, or spectral methods, which inherently suffer from the curse of dimensionality due to their dependence on spatio-temporal grids.

Several notable equations are put forth:

• The general form of FBSDEs is described, relating them to quasi-linear PDEs, and emphasizing the deterministic function of time and space in the solution.
• The loss function employed in the neural network training process is derived from discretizing the FBSDE using the Euler-Maruyama scheme. The derivation and setup are meticulous, showcasing that the parameters of the neural network can be learned through the minimization of this loss function.

Results and Numerical Analysis

The methodology is validated against several high-dimensional test problems, notably including the 100-dimensional Black-Scholes-Barenblatt and Hamilton-Jacobi-BeLLMan equations. It demonstrates effectiveness in solving these equations, presenting reliable approximations of the solution over the entire space-time domain, unlike prior works that only approximate solutions at initial points.

Strong numerical results illustrate the validity of this approach:

• The framework successfully delivers solutions upon a single training round, where conventional methods would require iterative retraining for different time instances.
• Evaluations showcase the model's ability to produce solutions with accuracy comparable to specialized state-of-the-art algorithms but with enhanced efficiency in computation across different spatial inputs.

Implications for Future Research

The applicability of FBSNNs spans various domains, including stochastic control, theoretical economics, and mathematical finance, areas where FBSDEs are prevalent. The framework's scalability and adaptability open avenues for extending its implementation to second-order backward stochastic differential equations and potentially reduce computational overhead in stochastic control problems.

Future advancements might explore optimizing neural architecture for solving even more complex PDEs or extending the framework to domains requiring integration with other stochastic processes. The imperative remains in leveraging deep learning paradigms to handle the complexities inherent in high-dimensional PDE solutions.

In summary, this paper provides a robust framework integrating FBSDEs with modern deep learning techniques, thereby offering a computationally feasible solution to high-dimensional PDEs with potential broad applications in scientific and financial domains.