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This paper introduces Neural Control, a framework for learning boundary control of physical systems governed by implicit equilibrium constraints. It computes memory-efficient proxy gradients by differentiating equilibrium conditions via an adjoint formulation, avoiding the need to unroll iterative solvers. The method integrates these sensitivities into a receding-horizon MPC scheme to improve robustness, particularly in multi-stable regimes.
Differentiating through physical simulations just got a whole lot easier: Neural Control avoids unrolling iterative solvers by using an adjoint formulation, enabling memory-efficient gradient-based control.
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.
