An optimal control approach for neural network architecture adaptation with a posteriori error estimation

What the math gives to ML

The paper contains transferable machinery for dynamically refining a residual network's depth rather than choosing depth heuristically before training. Its key asset is the optimal-control interpretation of a residual block as a time-discretized controlled dynamical system, combined with dual-weighted residual indicators that estimate how much each depth interval contributes to objective error. A practical transfer is an adaptive-depth training loop that periodically estimates forward discretization residuals and adjoint sensitivities, then inserts layers only where the estimated loss contribution is largest. The extracted text omits the paper's full estimator equations, so the implementation below uses a standard computable dual-weighted-residual specialization of the stated framework.