Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Michael Hintermüller, Michael Hinze, Denis Korolev

TL;DR

This work casts neural network design as an optimal control problem over neural ODEs on a continuous depth horizon $[0,T]$, and introduces a layerwise, goal-oriented adaptivity driven by a dual-weighted residual error estimator. By discretizing with a discontinuous Galerkin Petrov–Galerkin scheme (DG(0) for state/adjoint) and a CG(1) gradient discretization, the authors derive an explicit forward-backward time-stepping framework and an error-driven mechanism to insert layers where needed. They prove existence, regularity, and discretization-consistency of the optimality system, and develop a posteriori estimators to guide adaptive mesh (layer) refinement. Numerical experiments on Swiss roll binary and Peaks multiclass tasks show that adaptive layer insertion yields faster convergence and smaller networks than non-adaptive or random-layer schemes, highlighting practical benefits for efficient and scalable neural ODE design.

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.

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

TL;DR

This work casts neural network design as an optimal control problem over neural ODEs on a continuous depth horizon , and introduces a layerwise, goal-oriented adaptivity driven by a dual-weighted residual error estimator. By discretizing with a discontinuous Galerkin Petrov–Galerkin scheme (DG(0) for state/adjoint) and a CG(1) gradient discretization, the authors derive an explicit forward-backward time-stepping framework and an error-driven mechanism to insert layers where needed. They prove existence, regularity, and discretization-consistency of the optimality system, and develop a posteriori estimators to guide adaptive mesh (layer) refinement. Numerical experiments on Swiss roll binary and Peaks multiclass tasks show that adaptive layer insertion yields faster convergence and smaller networks than non-adaptive or random-layer schemes, highlighting practical benefits for efficient and scalable neural ODE design.

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.
Paper Structure (20 sections, 7 theorems, 150 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 20 sections, 7 theorems, 150 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Neural ODEs
  3. Optimal control of neural ODEs
  4. Proof:
  5. Proof:
  6. Proof:
  7. Stationarity conditions
  8. Proof:
  9. Proof:
  10. Petrov-Galerkin discretization
  11. The dual-weighted residual error estimation
  12. Proof:
  13. Proof:
  14. Layerwise adaptive algorithm
  15. Numerical examples
  16. ...and 5 more sections

Key Result

Proposition 1

Suppose that $\theta \in L^{2}(I; \mathbb{R}^{n})$ and that $\boldsymbol{\sigma} \in C(\mathbb{R}^{d})$ is Lipschitz continuous, i.e., there exists $L_\sigma > 0$ such that Then the problem bResnet admits a unique solution $\boldsymbol{x} \in H^{1}(I; \mathbb{R}^{md})$. Moreover, there exist finite constants $C_1, C_2>0$ such that the following stability bound holds:

Figures (4)

  • Figure 1: Discrete components of the state, control and the adjoint, and their respective reconstructions. (A): $x_{\tau,j}^{i}$ ($j$-th component of the $i$-th equation) of the discrete state $\boldsymbol{x}_{\tau}$ and its piecewise linear reconstruction. (B): component $p_{\tau, j}^{i}$ of the discrete adjoint $\boldsymbol{p}_\tau$ and its piecewise linear reconstruction. (C): component $\theta^{i}$ of the discrete control $\theta_{\tau}$ and its piecewise quadratic reconstruction
  • Figure 2: (A): Swiss roll training data with two colors indicating two different classes. (B): neural network prediction on the Swiss roll test data. (C): Peaks function level set classes. (D): neural network prediction on the Peaks test data.
  • Figure 3: Binary classification: examples of time grids illustrating the new layer insertion positions in the interval $T=[0, 20]$ produced by the adaptive algorithm (Algorithm 1).
  • Figure 4: Multiclass classification: examples of time grids illustrating the new layer insertion positions in the interval $T=[0, 10]$ produced by the adaptive algorithm (Algorithm 1).

Theorems & Definitions (11)

  • Proposition 1
  • Remark 1
  • Proposition 2
  • Proposition 3: Existence of optimal controls
  • Proposition 4: Constraint qualification
  • Remark 2
  • Proposition 5
  • Remark 3
  • Remark 4
  • Proposition 6
  • ...and 1 more