This work presents a novel approach for adapting neural network architecture along the depth based on a posteriori error estimation. By formulating neural network training as a continuous-time optimal control problem, the authors derive rigorous error estimates that quantify how approximation error distributes across network layers.
- New layers are inserted at locations of maximum estimated error to capture complex, nonlinear variations.
- The framework treats weights and biases as piecewise linear functions varying across layers.
- Dual weighted residual methodology from finite element analysis is used to derive computable upper bounds on functional error.
- Explicit error bounds decompose total approximation error into interval-wise contributions for targeted refinement.
- The method was demonstrated on scientific datasets, including learning the observable-to-parameter map for the Navier-Stokes equation.
Numerical results reveal that this approach consistently outperforms existing architecture adaptation methods in terms of generalization performance.