ANNet: A first-principles neural network for forward and inverse dynamics

Biological and robotic systems must solve two related computations to move: inverse dynamics, which determines the forces or torques needed to produce a desired movement, and forward dynamics, which maps applied forces to motion. Although these computations are coupled by the same equations of motion, they are usually estimated or implemented as distinct inverse and forward mappings, in both model-based and data-driven formulations. This separation can obscure the shared structure that constrains both problems. Here, we present ANNet, a physics-informed neural network that places both computations on a common learned representation by learning a single scalar quantity from classical mechanics--Appell acceleration energy. The network maps kinematic state and candidate accelerations to this scalar function, and inverse dynamics is obtained by differentiating the learned energy function with respect to acceleration to recover joint torques. Forward dynamics is then calculated without retraining by embedding the same learned energy landscape in an optimization objective whose unconstrained minimum satisfies the Gibbs- Appell equation. The resulting accelerations are integrated forward in time. We evaluate ANNet on a double pendulum paradigm. In trials unseen by the network during training, inverse and optimization-based forward simulations are real-time accurate. Our results provide a first-principles route for using a single learned representation to support both prediction and control. SignificanceRobots and animals must solve two problems to move: computing the forces or torques needed for a desired motion (inverse dynamics) and determining the motion produced by applied forces (forward dynamics), which are usually modeled separately. We show that both problems can be expressed using a single scalar function from classical mechanics, Appell acceleration energy. A neural network trained so that the derivative of this learned function matches reference joint torques performs inverse dynamics. The same network then computes forward dynamics by minimizing an objective built from the learned energy landscape, without retraining. This framework provides a unified representation for prediction and control in both neuroscience and robotics.