Neural Control: Adjoint Learning Through Equilibrium Constraints

Many physical AI tasks, such as bending a deformable linear object (DLO) to a target shape, involve implicit equilibrium: an agent actuates boundary degrees of freedom while the rest settle by minimizing total potential energy. These systems exhibit strongly nonlinear behavior due to multi-stability, where the same boundary conditions can yield multiple equilibrium shapes depending on the actuation trajectory. Conventional learning and control approaches are brittle because the actuation-to-configuration map is defined implicitly, and naive backpropagation through iterative equilibrium solvers is memory- and compute-intensive.

Neural Control addresses this by deriving memory-efficient proxy gradients from the equilibrium conditions via an adjoint formulation, completely avoiding unrolling of solver iterations. These sensitivities are then integrated into a receding-horizon model predictive control (MPC) scheme that repeatedly re-anchors optimization to realized equilibria, mitigating basin-switching in multi-stable regimes. The team evaluated the framework in simulation and on physical robots manipulating DLOs, demonstrating improved performance over gradient-free baselines such as SPSA and CEM. This work opens new possibilities for robust, compute-efficient control of highly nonlinear deformable systems.