Data-driven optimal control with neural network modeling of gradient flows

TL;DR

This paper introduces Optimal Control Neural Networks (OCN), a data-driven method to learn vector fields in dynamical systems without assuming their analytical form, providing error bounds and demonstrating effectiveness on canonical systems like Lorenz.

Contribution

The paper presents a novel neural network framework combined with optimal control to learn dynamical laws directly from data, with theoretical error bounds and validation on complex systems.

Findings

OCN effectively learns vector fields from trajectory data.

Error bounds depend on training error and data sampling interval.

Demonstrates successful application to chaotic Lorenz system.

Abstract

Extracting physical laws from observation data is a central challenge in many diverse areas of science and engineering. We propose Optimal Control Neural Networks (OCN) to learn the laws of vector fields in dynamical systems, with no assumption on their analytical form, given data consisting of sampled trajectories. The OCN framework consists of a neural network representation and an optimal control formulation. We provide error bounds for both the solution and the vector field. The bounds are shown to depend on both the training error and the time step between the observation data. We also demonstrate the effectiveness of OCN, as well as its generalization ability, by testing on several canonical systems, including the chaotic Lorenz system.