Neural Control: Adjoint Learning Through Equilibrium Constraints

Many physical AI tasks require sequential implicit computation: at each step, boundary controls are applied, and the resulting configuration is obtained by solving an equilibrium problem. This setting arises naturally in deformable object manipulation, where even bending a deformable linear object (DLO) to a target shape can be nonlinear and multistable: identical boundary conditions may produce different configurations depending on actuation history. Unlike explicit transition models, the control-to-configuration relation is implicit and history-dependent, making long-horizon learning and control brittle; backpropagating through iterative solves is also memory- and compute-intensive. We propose Neural Control, a boundary-control framework that propagates gradients through branch-dependent sequences of equilibrium solves rather than a single fixed point. Neural Control computes trajectory-dependent proxy gradients by differentiating equilibrium conditions with an adjoint formulation, avoiding solver unrolling while keeping forward rollouts on converged equilibria. Combined with receding-horizon continuation, Neural Control re-anchors optimization to realized equilibria and mitigates basin switching. We validate Neural Control on simulated and real DLO manipulation, compare against SPSA and iCEM, and demonstrate applicability to a learned DEQ-style implicit equilibrium model.

We develop control methods so that robot can control flexible physical objects, like elastic strips, cables, and soft structures. These systems are hard to control because the robot's control action can produce different shapes depending on the previous shape, so the robot cannot rely on simple "input gives one output" rule. The history of the motion of the robot and the deformation of the object is important. The paper proposed Neural Control, which lets the robot plan boundary motions while respecting physics (Newton's second law). Instead of slowly backpropagating through every solver step at an immense computational cost, it uses an adjoint method to estimate gradients efficiently. It also uses a receding-horizon strategy: optimize a short segment, execute it, then re-plan from the new shape. This helps avoid jumping to the wrong stable shape. Experiments show faster and more accurate control than black-box methods, including on real elastic-strip manipulation.