A PDE-constrained Optimization Approach to Optimal Trajectory Planning under Uncertainty via Reflected Schrödinger Bridges

TL;DR

This work addresses trajectory planning under uncertainty in domains with reflecting boundaries by formulating a Reflected Schrödinger Bridge Problem (RSBP) as a PDE-constrained optimization. The authors linearize the nonlinear optimality system via a Hopf–Cole transformation, obtaining a pair of forward–backward linear advection–diffusion equations for potentials $\varphi$ and $\hat{\varphi}$, which are solved with finite element discretization and a fixed-point iteration to enforce endpoint coupling. The method naturally handles reflecting boundaries through the weak FE formulation, preserves mass, and achieves fast convergence in challenging 3D maze geometries, even when incorporating a divergence-free prior drift. This yields a practical framework for robust, uncertainty-aware trajectory planning in complex geometries with obstacle-like boundaries and drift priors.

Abstract

A computational PDE-constrained optimization approach is proposed for optimal trajectory planning under uncertainty by means of an associated Schroedinger Bridge Problem (SBP). The proposed SBP formulation is interpreted as the mean-field limit associated to the energy-optimal evolution of a particle governed by a stochastic differential equation (SDE) with nonlinear drift and reflecting boundary conditions, constrained to initial and terminal densities for its state. The resulting mean-field system consists of a nonlinear Fokker-Planck equation coupled with a Hamilton-Jacobi-Bellman equation, subject to two-point boundary conditions in time and Neumann boundary conditions in space. Through the Hopf-Cole transformation, this nonlinear system is recast as a pair of forward-backward advection-diffusion equations, which are amenable to efficient numerical solution via standard finite element discretization. The weak formulation naturally enforces reflecting boundary conditions without requiring explicit particle-boundary collision detection, thus circumventing the computational difficulties inherent to particle-based methods in complex geometries. Numerical experiments on challenging 3D maze configurations demonstrate fast convergence, mass conservation, and validate the optimal controls computed through reflected SDE simulations.

Figures (7)

• Figure 1: Mesh discretization and Gaussian initial/terminal density setup for the RSBP over a spiral domain.
• Figure 2: Pure diffusion ($\nu=0$): temporal evolution of density $\rho(t,x)$ moving from bottom to top along the spiral channel.
• Figure 3: The optimal velocity field $u^*(t,x)$ along the spiral exhibits a strong vertical component aligned with the ascent.
• Figure 4: RSBP with prior drift: complete 3D temporal evolution of density $\rho(t,x)$ along the spiral channel.
• Figure 5: RSBP with prior drift: reflected SDE validation with 10 particles evolving through the controlled SDE \ref{['sde_part']}.