Finite-time information erasure is governed by probability transport through configuration space, not only by final logical populations. We show that a two-dimensional Brownian memory can lower fast-erasure cost by reorganizing probability currents through a transverse degree of freedom. Using physics-informed routing control (PIRC), we optimize conservative protocols at fixed erasure time and final-error criterion, and compare them with tilt-only and static minimum-energy-path (MEP) constrained driving. Constrained scans over transverse-gate strength reveal two regimes: MEP and PIRC enter the target-error band where tilt remains infeasible and, when all three protocols have comparable final error, PIRC has lower entropy production than both baselines. The reduction is not obtained by following the static MEP alone, but by suppressing off-channel dissipation and reshaping the transient current field.

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• Brownian motion
• Entropy production
• Nonequilibrium statistical mechanics
• Stochastic thermodynamics
• Nonequilibrium systems
• Fokker–Planck equation