A deep backward regression-based scheme for high-dimensional nonlinear partial differential equations
This is an incremental improvement for researchers in computational mathematics and machine learning, enhancing stability and generalization in PDE solvers.
The paper tackles solving high-dimensional nonlinear parabolic partial differential equations by proposing a deep backward regression-based scheme that reformulates loss functions and incorporates conditional expectations to reduce variance. It outperforms prior methods, maintaining accuracy up to d=200 for bounded solutions and up to d=20 for unbounded ones with errors under 9.7%.
A deep backward regression-based (DBR) scheme for solving high-dimensional nonlinear parabolic partial differential equations is proposed. Building upon the established DBDP method (Huré et al., 2020), our algorithm introduces a reformulation of the local loss functions that are sequentially optimized via backward induction at each time step. The core of this approach involves reformulating simulated backward stochastic difference equations into their conditional expectation representations, thereby recasting a projection-based stochastic optimization problem as a deterministic function-approximation task. By explicitly incorporating conditional expectations, the DBR scheme facilitates a denoising mechanism prior to loss evaluation. This architecture substantially mitigates numerical variance, resulting in enhanced training stability and superior generalization performance. Numerical results demonstrate that the DBR scheme consistently outperforms the DBDP1 method, maintaining accuracy up to d=200 for bounded solutions (see Table 1). Notably, for complex unbounded PDEs where the DBDP1 method fails beyond d=10, the DBR scheme remains robust up to $d=20$ with relative errors under 9.7% (see Table 6}). Theoretically, we derive rigorous upper error bounds and establish half-order convergence for the proposed scheme. Extensions to variational inequalities are also provided.