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

arXiv: https://arxiv.org/abs/2506.13950 · HTML: https://mlsanity.com/paper/2506.13950/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 104 references that are in the map.
- First posted on arXiv 16 Jun 2025.

## Builds on (most cited first, 13 of 13)

- [Deep learning to discover and predict dynamics on an inertial manifold](https://mlsanity.com/paper/2001.04263/) (2019)
- [Deep nets for local manifold learning](https://mlsanity.com/paper/1607.07110/) (2016)
- [Training Neural Networks is NP-Hard in Fixed Dimension](https://mlsanity.com/paper/2303.17045/) (2023)
- [Double Diffusion Maps and their Latent Harmonics for Scientific Computations in Latent Space](https://mlsanity.com/paper/2204.12536/) (2022)
- [Data-driven Control of Agent-based Models: an Equation/Variable-free Machine Learning Approach](https://mlsanity.com/paper/2207.05779/) (2022)
- [Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learning](https://mlsanity.com/paper/2309.07946/) (2023)
- [Car-Following Models: A Multidisciplinary Review](https://mlsanity.com/paper/2304.07143/) (2023)
- [Nonlinear dimensionality reduction then and now: AIMs for dissipative PDEs in the ML era](https://mlsanity.com/paper/2310.15816/) (2023)
- [Direct computation of nonlinear mapping via normal form for reduced-order models of finite element nonlinear structures](https://mlsanity.com/paper/2009.12145/) (2020)
- [Constrained nonlinear output regulation using model predictive control -- extended version](https://mlsanity.com/paper/2005.12413/) (2020)
- [A physics-informed neural network method for the approximation of slow invariant manifolds for the general class of stiff systems of ODEs](https://mlsanity.com/paper/2403.11591/) (2024)
- [Deep neural network approximations for the stable manifolds of the Hamilton-Jacobi-Bellman equations](https://mlsanity.com/paper/2007.15350/) (2020)
- [Multiscale control of generic second order traffic models by driver-assist vehicles](https://mlsanity.com/paper/2008.07439/) (2020)

## Abstract (arXiv)

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.