Abstract
High-dimensional partial differential equations (PDEs) pose significant challenges for numerical computation due to the curse of dimensionality, which limits the applicability of traditional mesh-based methods. Since 2017, the Deep BSDE method has introduced deep learning techniques that enable the effective solution of nonlinear PDEs in very high dimensions. This innovation has sparked considerable interest in using neural networks for high-dimensional PDEs, making it an active area of research. In this short review, we briefly sketch the Deep BSDE method, its subsequent developments, and future directions for the field.
Summary
This review paper by the original authors of the deep BSDE method surveys the method and its impact since its introduction in 2017. The Deep BSDE method solves high-dimensional semilinear parabolic PDEs by reformulating them as backward stochastic differential equations via the nonlinear Feynman-Kac formula. The unknown gradient of the PDE solution is approximated by neural networks at each time step, and the entire system is trained end-to-end by minimising the terminal condition mismatch. The paper reviews four main classes of subsequent methods that have emerged: BSDE-based methods, least-squares formulations (physics-informed neural networks), Ritz formulations (Deep Ritz method), and Galerkin formulations (weak adversarial networks). The review also surveys theoretical advances showing that deep neural networks can approximate PDE solutions without the curse of dimensionality, with network size growing only polynomially in dimension and accuracy.
Key Contributions
- Concise presentation of the Deep BSDE method as a stochastic optimal control problem: given the forward process X_t, choose Y_0 and control Z_t to match the terminal condition g(X_T)
- Taxonomy of four neural-network-based PDE formulations (BSDE, least-squares, Ritz, Galerkin) and their relative strengths
- Survey of theoretical results proving polynomial complexity in dimension for deep neural network PDE approximation (no curse of dimensionality)
- Identification of open problems: full convergence analysis (including optimisation error), gradient-dependent nonlinearities (HJB equations), and time-dependent Schrodinger equations
Methodology
The core method reformulates the semilinear parabolic PDE (eq. 2.1) with terminal condition u(T,x) = g(x) into a variational problem via BSDEs: inf_{Y_0, {Z_t}} E|g(X_T) - Y_T|^2, where X_t evolves as a forward SDE and Y_t is propagated forward using the BSDE dynamics. The key idea is to approximate Y_0 by a network psi_0 and Z_{t_n} = [sigma(t_n, X_{t_n})]^* grad u(t_n, X_{t_n}) at each discrete time step by sub-networks phi_n(X_{t_n}). Stacking these sub-networks produces a residual neural network architecture (cf. Solving High-Dimensional Partial Differential Equations Using Deep Learning). The loss function requires no pre-generated training data — the initial condition and Brownian paths serve as data generated on-the-fly, making it naturally suited to stochastic gradient descent.
Key Findings
- The Deep BSDE method was the first modern deep learning approach to effectively solve general nonlinear PDEs in hundreds or thousands of dimensions
- The BSDE-based stochastic optimisation objective is naturally compatible with the SGD paradigm, extending to actor-critic methods for HJB equations, Nash equilibria in mean-field games, elliptic problems, and fully nonlinear PDEs via second-order BSDEs
- Physics-informed neural networks (PINNs) are more general but suffer from conditioning issues; the Deep Ritz method connects to variational Monte Carlo for quantum mechanics
- Theoretical results confirm deep neural networks overcome the curse of dimensionality for Black-Scholes, semilinear heat equations, and Kolmogorov PDEs, with approximation error requiring only polynomially many parameters in 1/epsilon and dimension d
- Complete convergence analysis including optimisation error remains open even in one dimension
Important References
- Solving High-Dimensional Partial Differential Equations Using Deep Learning — Han, Jentzen, E (2018), the original Deep BSDE paper (PNAS, 1940 citations)
- Gnoatto, Picarelli, Reisinger (2023) — Deep xVA Solver, neural network counterparty credit risk framework (41 citations)
- Physics-informed neural networks — Raissi, Perdikaris, Karniadakis (2019), least-squares PDE formulation (15440 citations)
- Deep backward schemes — Hure, Pham, Warin (2020), backward time-stepping for high-dimensional nonlinear PDEs (232 citations)
- Pardoux, Peng (1992) — foundational BSDE theory connecting to quasilinear parabolic PDEs (878 citations)
Atomic Notes
- deep BSDE method
- nonlinear Feynman-Kac formula
- forward deep BSDE vs backward deep BSDE
- curse of dimensionality in PDE computation
- stochastic control formulation of PDEs