Neural Control: Adjoint Learning Through Equilibrium Constraints

TL;DR

Neural Control introduces an adjoint-based method for differentiating implicit equilibrium constraints in physical AI tasks, enabling efficient and robust boundary control of multi-stable systems like deformable objects.

Contribution

It develops a memory-efficient, trajectory-dependent gradient computation framework using adjoint sensitivity analysis for equilibrium-based control tasks.

Findings

Outperforms gradient-free methods like SPSA and CEM in manipulation tasks.

Improves robustness in multi-stable regimes through receding-horizon MPC.

Successfully applied to physical robots manipulating deformable linear objects.

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.