Invariant Manifolds of Discrete-time Dynamical Systems with Nonlinear Exosystems via Hybrid Physics-Informed Neural Networks

TL;DR

This paper introduces a hybrid physics-informed neural network framework to approximate invariant manifolds in discrete-time dynamical systems with exogenous inputs, combining polynomial series and neural networks for improved accuracy.

Contribution

The paper develops a novel hybrid approach that integrates polynomial expansions and neural networks to effectively learn invariant manifolds in systems with exogenous dynamics.

Findings

Hybrid method outperforms standalone neural networks and polynomial schemes in accuracy.

The approach demonstrates convergence and efficiency on benchmark problems.

Universal approximation theorem established for neural network-based invariant manifold approximation.

Abstract

We propose a hybrid physics-informed machine learning framework to approximate invariant manifolds (IMs) of discrete-time dynamical systems driven by exogenous autonomous dynamics (exosystems). Such systems appear in applications ranging from control theory to modeling collective multi-agent behavior (e.g., bird flocks, traffic dynamics) under hierarchical leadership. The IM learning problem is formulated as solving nonlinear functional equations derived from the invariance equation, expressing the manifold as a relationship between exogenous and system states. The proposed approach combines polynomial series with shallow neural networks, leveraging their complementary strengths. We focus on low- to medium-dimensional manifolds where polynomial expansions remain tractable. Near equilibrium, polynomial series provide interpretability and convergence, while farther away neural networks capture global structure through their universal approximation capability. A continuity penalty enforces consistency between both representations at their interface, and training is performed using analytically derived derivatives within the Levenberg-Marquardt scheme. Naturally, depending on the dimensionality of the input-driven system, one may also employ a purely neural network-based IM approximation, for which we also establish a universal approximation theorem based on certain assumptions on system dynamics. The framework is evaluated on two benchmark problems: an enzymatic bioreactor and a leader-follower car-following model. We analyze convergence, approximation accuracy, and computational cost, and compare standalone neural networks, polynomial expansions, and the hybrid method. Results show that the hybrid approach achieves superior accuracy compared to standalone schemes.