[Paper Review] Implicit energy regularization of neural ordinary-differential-equation control

Lucas Böttcher, Nino Antulov-Fantulin+1 more|arXiv (Cornell University)|Mar 11, 2021

Model Reduction and Neural Networks1 references4 citations

TL;DR

This paper proposes Neural ODE Control (NODEC), a differentiable framework that learns optimal control signals for dynamical systems without explicit energy regularization. By leveraging neural ODEs and gradient descent, NODEC implicitly regularizes control energy, achieving performance comparable to analytical optimal control methods in linear and nonlinear systems, including Kuramoto oscillator synchronization.

ABSTRACT

Although optimal control problems of dynamical systems can be formulated within the framework of variational calculus, their solution for complex systems is often analytically and computationally intractable. In this Letter we present a versatile neural ordinary-differential-equation control (NODEC) framework with implicit energy regularization and use it to obtain neural-network-generated control signals that can steer dynamical systems towards a desired target state within a predefined amount of time. We demonstrate the ability of NODEC to learn control signals that closely resemble those found by corresponding optimal control frameworks in terms of control energy and deviation from the desired target state. Our results suggest that NODEC is capable to solve a wide range of control and optimization problems, including those that are analytically intractable.

Motivation & Objective

• To develop a neural ODE-based control framework that can steer complex dynamical systems toward target states without explicit energy regularization.
• To investigate whether implicit regularization in neural network training can yield control signals with energy consumption close to optimal control solutions.
• To demonstrate the framework's effectiveness on both linear and nonlinear networked dynamical systems, including synchronization of coupled oscillators.
• To compare NODEC's performance against analytical optimal control and adjoint-gradient methods in terms of control energy and convergence speed.
• To establish that neural ODEs can implicitly learn energy-regularized control policies through the interplay of initialization and gradient descent.

Proposed method

• Formulates control as an optimal control problem governed by a neural ODE, where the control input is parameterized by a neural network.
• Uses a loss function that minimizes final state deviation and does not include an explicit control energy penalty.
• Employs gradient descent on network weights to train the controller, relying on backpropagation through the ODE solver.
• Applies the adjoint sensitivity method to compute gradients through the ODE integration, enabling end-to-end training.
• Demonstrates the method on linear systems and nonlinear Kuramoto oscillators across various network topologies (complete, Erdős–Rényi, Watts–Strogatz, lattice).
• Compares learned control signals and energies against analytical optimal control and adjoint-gradient methods (AGM) using metrics like order parameter and control energy.

Experimental results

Research questions

• RQ1 Can a neural ODE-based control framework learn control signals with energy consumption comparable to optimal control without explicit energy regularization?
• RQ2 How does the implicit regularization in NODEC arise from the interplay of neural network initialization and gradient descent?
• RQ3 To what extent can NODEC match the performance of analytical optimal control in linear dynamical systems?
• RQ4 Can NODEC achieve near-optimal control energy and synchronization in nonlinear networked systems like Kuramoto oscillators?
• RQ5 How does NODEC’s performance compare to the adjoint-gradient method (AGM) in terms of convergence speed and energy efficiency across diverse network topologies?

Key findings

• NODEC successfully drives linear dynamical systems to the target state with control energy closely matching that of analytical optimal control solutions.
• For Kuramoto oscillator networks, NODEC achieves synchronization faster than the adjoint-gradient method (AGM), with comparable control energy across all tested network types.
• In all network topologies (complete, Erdős–Rényi, Watts–Strogatz, square lattice), NODEC achieved a final order parameter close to 1, indicating high synchronization, similar to AGM.
• Control energy for NODEC was within the same range as AGM across all network types, demonstrating that implicit regularization yields energy-efficient control signals.
• The framework generalizes well to large-scale systems, with successful control demonstrated on a 10,000-node square lattice in the supplementary material.
• The absence of explicit energy regularization in the loss function does not compromise performance, as implicit regularization emerges naturally from the training dynamics.