Neural networks learn forward dynamics when freed from numerical integration

Abstract

Seamless interaction between humans and machines requires interfaces that remain robust to the variability inherent in biological signals and physical environments. Advanced human-machine interfaces (HMIs) increasingly rely on machine learning to predict or control limb dynamics. These systems must learn input-to-output mappings between control variables and limb state, such as the mapping from muscle forces or joint torques acting about segmented arm joints to limb posture over time. Such statistical input-to-output transformations can result in numerical instability of predicted musculoskeletal kinematics and dynamics. Achieving the robustness of biological motor control requires solving both forward and inverse dynamics problems; however, these problems are computationally asymmetric because they entail opposing operations, integration and differentiation. Since we have previously shown that neural networks solve the inverse dynamics problem when trained to map kinematic to dynamic signals during reaching, we hypothesized that representing separately the approximation of equations of motion (EOM) and their temporal numerical integration may capture the relevant computational structure of the forward dynamics problem. We tested this hypothesis by comparing a conventional direct-mapping neural network with a two-stage model, the artificial physics engine (APE). When predicting the state of a two-segment system under external perturbations not encountered during training, the direct-mapping, monolithic model produced large prediction errors inconsistent with the expected interaction torque, whereas the APE maintained low error and remained stable under novel initial conditions and perturbations. Mapping system dynamics in terms of the EOM imposes a causal, physics-based structure on forward models for HMI design.