The curse of dimensionality refers to the exponential growth of computational cost with the number of dimensions in traditional mesh-based numerical methods for partial differential equations. For a PDE in d spatial dimensions, a grid with N points per dimension requires N^d total grid points, making even moderate dimensions (d > 4-5) computationally infeasible. This is the central obstacle for high-dimensional problems arising in quantitative finance (portfolio-wide XVA with dozens of risk factors), optimal control (Hamilton-Jacobi-Bellman equations), quantum mechanics (many-electron Schrodinger equations), and kinetic theory (Boltzmann equations).
The deep BSDE method overcomes the curse of dimensionality by reformulating the PDE as a stochastic optimisation problem via the nonlinear Feynman-Kac formula, replacing spatial discretisation with Monte Carlo sampling of forward diffusion paths. The computational cost scales with the number of neural network parameters rather than with a spatial grid. The theoretical analysis surveyed in Han, Jentzen, E (2025) confirms that deep neural networks can approximate solutions of high-dimensional PDEs with the number of parameters growing only polynomially in both the reciprocal 1/epsilon of the accuracy and the dimension d. Specifically, for Black-Scholes PDEs, semilinear heat equations, and Kolmogorov PDEs with Lipschitz nonlinearities, the approximation error achieves epsilon-accuracy with poly(d, 1/epsilon) parameters.
However, a complete understanding of the overall error — including approximation, generalisation, and optimisation errors — remains an open problem even for one-dimensional PDEs. The optimisation error (whether SGD finds a good minimum of the non-convex loss landscape) poses an especially significant theoretical challenge.
Key Details
- Classical complexity: N^d grid points for d-dimensional PDE with N points per dimension; infeasible for d > 4-5
- Deep BSDE complexity: polynomial in d and 1/epsilon for the approximation error; proven for Black-Scholes, semilinear heat, and Kolmogorov PDEs
- Practical performance: the original Deep BSDE paper achieved sub-1% errors in 100 dimensions; the Deep xVA Solver scales to 200 dimensions
- Open problems: full convergence analysis (approximation + generalisation + optimisation) is open even in 1D; gradient-dependent nonlinearities (HJB) are theoretically harder
- Alternative approaches: PINNs with Hutchinson trace estimation, Deep Ritz method, weak adversarial networks — all aim to bypass the curse but with different theoretical guarantees