Appellian Neural Networks

Abstract

The movement of articulated bodies, whether biological or robotic, involves two interrelated computational challenges. Inverse dynamics determines the forces or torques required to achieve a specified movement, while forward dynamics computes the resulting accelerations from given forces or torques. Although both computations are governed by the same equations of motion, they are typically implemented as separate inverse and forward mappings in both model-based and data-driven approaches. This division can obscure the underlying structure common to both problems. Here, we introduce the Appellian Neural Network (ANNet), a physics-informed neural network that unifies both computations through a shared learned representation by modeling a single scalar quantity from classical mechanics, specifically Appell's acceleration energy. Inverse dynamics is obtained by differentiating the learned energy function with respect to acceleration to recover joint torques. Forward dynamics is computed without retraining by embedding the same learned energy landscape into an optimization objective, where the unregularized minimum satisfies the Gibbs-Appell equation. The resulting accelerations are then integrated forward in time. ANNet is evaluated using a double pendulum paradigm. For trajectories not encountered during training, both inverse and forward simulations demonstrate accuracy and real-time capability. Both computations propagate input noise similarly to their analytical counterparts, and adaptation of the inverse dynamics to a modified pendulum generalizes to forward simulation without additional forward-specific training. These results establish a first-principles approach for employing a single learned representation to support both inverse and forward dynamic computations.