Layerwise goal-oriented adaptivity for neural ODEs: an optimal control perspective

TL;DR

This paper introduces a layerwise adaptive method for neural ODEs using an optimal control framework, enhancing neural network architecture design through goal-oriented residual techniques and explicit Euler discretization.

Contribution

It presents a novel adaptive construction approach for neural networks based on dual-weighted residuals and optimal control, with implementation via DG(0) Galerkin discretization and steepest descent optimization.

Findings

Effective neural network construction for classification tasks

Demonstrated approach on well-known datasets

Improved adaptivity in neural ODE architectures

Abstract

In this work, we propose a novel layerwise adaptive construction method for neural network architectures. Our approach is based on a goal--oriented dual-weighted residual technique for the optimal control of neural differential equations. This leads to an ordinary differential equation constrained optimization problem with controls acting as coefficients and a specific loss function. We implement our approach on the basis of a DG(0) Galerkin discretization of the neural ODE, leading to an explicit Euler time marching scheme. For the optimization we use steepest descent. Finally, we apply our method to the construction of neural networks for the classification of data sets, where we present results for a selection of well known examples from the literature.