Neural Control: Adjoint Learning Through Equilibrium Constraints

arXiv cs.RO / 5/6/2026

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Key Points

The paper addresses physical “physical AI” control problems where system configurations are determined implicitly by equilibrium constraints, leading to multi-stability and nonlinear behavior even under the same boundary actuation conditions.
It proposes Neural Control, which computes memory- and compute-efficient, trajectory-dependent proxy gradients by differentiating equilibrium conditions using an adjoint formulation rather than backpropagating through every iteration of an equilibrium solver.
To handle long-horizon robustness in multi-stable systems, the method integrates these sensitivity estimates into a receding-horizon MPC framework that re-anchors optimization to the actually realized equilibria and reduces basin-switching.
Experiments in both simulation and physical robot manipulation of deformable linear objects (DLOs) show improved performance over gradient-free approaches 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.