Trajectory-Based Inverse Dynamics for Neural Network Training

This early technical report introduces a trajectory-based formulation of neural network training in which learning is expressed as an inverse dynamics problem rather than as loss minimisation.
A feedforward neural network is treated as a discrete dynamical system: each layer applies a deterministic transition operator, and the forward pass generates a trajectory of internal states. Training is then posed as determining the transition operators that make this trajectory terminate at the desired output.

The report develops the mathematical foundations of this perspective.
We formalise neural computation as a sequence of state transitions, derive the inverse-dynamics problem for general architectures, and show how it connects to discrete variational principles and Euler–Lagrange equations. For linear layers, the resulting inverse problem admits closed-form solutions. For non-linear systems, we discuss structural properties such as existence, controllability, and non-uniqueness of solutions.

This framework shifts the focus of learning from error surfaces to internal computational paths. Instead of adjusting parameters to minimise a loss function, one directly solves for the operators that generate correct terminal states. The trajectory-level viewpoint provides new analytical tools for understanding neural computation and new opportunities for designing alternative training paradigms.

Planned future versions of this report will include numerical examples, algorithmic implementations, and empirical evaluations.