A simulation-free diffusion-based method for solving nonlinear PDEs

Deep learning methods based on backward stochastic differential equations (BSDEs) provide an effective approach to solving nonlinear partial differential equations (PDEs), particularly in high dimensions. By exploiting stochastic representations of PDE solutions, these methods encode second-order differential operators through the dynamics of diffusion processes, avoiding the explicit computation of second-order derivatives. Their application, however, typically requires sequential path simulation and, on bounded domains, accounting for random exit times, which introduces additional numerical error. A key observation is that the stochastic representation extends beyond the Markovian setting, as Itô's formula remains valid for general continuous semimartingales. We exploit this observation using Brownian bridges, whose endpoints can be prescribed and whose samples can be generated in closed form. Choosing the terminal endpoint on the domain boundary eliminates the need to account for random exit times, while closed-form sampling enables fully parallel generation of the stochastic inputs. This yields a diffusion-based deep-learning method for nonlinear PDEs that avoids sequential path simulation and random exit times, while retaining the advantages of stochastic representations in high dimensions.