Self-Consistent Neural Control: Learning Trajectories for Implicit Equilibrium

Abstract

One big challenge in physical AI is interacting with systems whose states are defined by implicit equilibrium. Learning in these systems is difficult because their responses can be multistable, path-dependent, and sensitive over long horizons. Adjoint learning provides an efficient way to optimize trajectories without unrolling iterative equilibrium solvers, but its gradient relies on the frozen-tangent approximation, which neglects higher-order sensitivity terms and can become inaccurate when the local equilibrium varies rapidly. In this work, we introduce Self-Consistent Neural Control (SCNC), optimizing trajectories along which sensitivity variations are explicitly controlled. We derive a mathematical bound on the resulting gradient error and develop a practical finite-difference approximation. We validate the framework on deformable objects, which are representative of implicit equilibrium systems, ranging from simple quadratic energy models such as elastic rods to complex systems including thin-walled tubes, measuring tapes, and Slinky-like structures. Compared with the adjoint learning and the gradient-free iCEM baselines, SCNC achieves faster convergence and more stable optimization by reducing switching to undesirable equilibrium branches, doubling the stable optimization rate over adjoint learning 83.3% vs. 41.7% and reducing the average loss by more than 1.5 orders of magnitude on thin-walled tubes, where sensitivity variations are pronounced.