Neural Control: Adjoint Learning Through Equilibrium Constraints

2026-05-04

AI Summary

This work addresses the challenge of efficiently learning and controlling actuator-to-configuration mappings defined by implicit equilibrium relations in physical AI tasks, particularly in multistable systems. The authors propose a boundary control framework that, for the first time, integrates the adjoint method into implicit equilibrium systems. By differentiating the equilibrium conditions, the approach yields trajectory-dependent surrogate gradients without unrolling iterative solvers, enabling memory-efficient and trajectory-sensitive gradient estimation. Coupled with receding-horizon model predictive control (MPC), the method substantially enhances long-horizon control robustness and effectively mitigates issues associated with switching between metastable energy basins. In manipulation tasks involving deformable linear objects, both simulation and real-world experiments demonstrate significant performance improvements over gradient-free baselines such as SPSA and CEM.

Abstract

Many physical AI tasks are governed by implicit equilibrium: an agent actuates a subset of degrees of freedom (boundary DoFs), while the remaining free DoFs settle by minimizing a total potential energy. Even seemingly basic tasks such as bending a deformable linear object (DLO) to a target shape can exhibit strongly nonlinear behavior due to multi-stability: the same boundary conditions may yield multiple equilibrium shapes depending on the actuation trajectory. However, learning and control in such systems is brittle because the actuation-to-configuration map is defined only implicitly, and naive backpropagation through iterative equilibrium solvers is memory- and compute-intensive. We propose Neural Control, a boundary-control framework that computes trajectory-dependent, memory-efficient proxy gradients by differentiating equilibrium conditions via an adjoint formulation, avoiding unrolling of solver iterations. To improve robustness over long horizons, we integrate these sensitivities into a receding-horizon MPC scheme that repeatedly re-anchors optimization to realized equilibria and mitigates basin-switching in multi-stable regimes. We evaluate Neural Control in simulation and on physical robots manipulating DLOs, and show improved performance over gradient-free baselines such as SPSA and CEM.

Problem

• implicit equilibrium
• multi-stability
• boundary control
• trajectory-dependent response
• actuation-to-configuration mapping

Innovation

• adjoint method
• equilibrium constraints
• deformable linear object
• receding-horizon MPC
• implicit differentiation