# A physics-informed neural network method for the approximation of slow invariant manifolds for the general class of stiff systems of ODEs

arXiv: https://arxiv.org/abs/2403.11591 · HTML: https://mlsanity.com/paper/2403.11591/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 60 references that are in the map.
- First posted on arXiv 18 Mar 2024.

## Builds on (most cited first, 14 of 14)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Multiscale Simulations of Complex Systems by Learning their Effective Dynamics](https://mlsanity.com/paper/2006.13431/) (2020)
- [A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder](https://mlsanity.com/paper/2009.11990/) (2020)
- [β-Variational autoencoders and transformers for reduced-order modelling of fluid flows](https://mlsanity.com/paper/2304.03571/) (2023)
- [Coarse-scale PDEs from fine-scale observations via machine learning](https://mlsanity.com/paper/1909.05707/) (2019)
- [AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identification](https://mlsanity.com/paper/2310.01433/) (2023)
- [Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approach](https://mlsanity.com/paper/2201.13323/) (2022)
- [Molecular enhanced sampling with autoencoders: On-the-fly collective variable discovery and accelerated free energy landscape exploration](https://mlsanity.com/paper/1801.00203/) (2017)
- [A Comparison of Neural Network Architectures for Data-Driven Reduced-Order Modeling](https://mlsanity.com/paper/2110.03442/) (2021)
- [Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data](https://mlsanity.com/paper/2205.13545/) (2022)
- [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)
- [Data-driven modelling of brain activity using neural networks, Diffusion Maps, and the Koopman operator](https://mlsanity.com/paper/2304.11925/) (2023)

## Built on by (most cited first, 2 of 2)

- [Quantification of total uncertainty in the physics-informed reconstruction of CVSim-6 physiology](https://mlsanity.com/paper/2408.07201/) (2024)
- [Invariant Manifolds of Discrete-time Dynamical Systems with Nonlinear Exosystems via Hybrid Physics-Informed Neural Networks](https://mlsanity.com/paper/2506.13950/) (2025)

## Abstract (arXiv)

We present a physics-informed neural network (PINN) approach for the discovery of slow invariant manifolds (SIMs), for the most general class of fast/slow dynamical systems of ODEs. In contrast to other machine learning (ML) approaches that construct reduced order black box surrogate models using simple regression, and/or require a priori knowledge of the fast and slow variables, our approach, simultaneously decomposes the vector field into fast and slow components and provides a functional of the underlying SIM in a closed form. The decomposition is achieved by finding a transformation of the state variables to the fast and slow ones, which enables the derivation of an explicit, in terms of fast variables, SIM functional. The latter is obtained by solving a PDE corresponding to the invariance equation within the Geometric Singular Perturbation Theory (GSPT) using a single-layer feedforward neural network with symbolic differentiation. The performance of the proposed physics-informed ML framework is assessed via three benchmark problems: the Michaelis-Menten, the target mediated drug disposition (TMDD) reaction model and a fully competitive substrate-inhibitor(fCSI) mechanism. We also provide a comparison with other GPST methods, namely the quasi steady state approximation (QSSA), the partial equilibrium approximation (PEA) and CSP with one and two iterations. We show that the proposed PINN scheme provides SIM approximations, of equivalent or even higher accuracy, than those provided by QSSA, PEA and CSP, especially close to the boundaries of the underlying SIMs.