Nonlinear Model Order Reduction of Dynamical Systems in Process Engineering: Review and Comparison

Computationally cheap yet accurate dynamical models are a key requirement for real-time capable nonlinear optimization and model-based control. When given a computationally expensive high-order prediction model, a reduction to a lower-order simplified model can enable such real-time applications. Herein, we review nonlinear model order reduction methods and provide a comparison of method characteristics. Additionally, we discuss both general-purpose methods and tailored approaches for chemical process systems and we identify similarities and differences between these methods. As machine learning manifold-Galerkin approaches currently do not account for inputs in the construction of the reduced state subspace, we extend these methods to dynamical systems with inputs. In a comparative case study, we apply eight established model order reduction methods to an air separation process model: POD-Galerkin, nonlinear-POD-Galerkin, manifold-Galerkin, dynamic mode decomposition, Koopman theory, manifold learning with latent predictor, compartment modeling, and model aggregation. Herein, we do not investigate hyperreduction, i.e., reduction of floating point operations. Based on our findings, we discuss strengths and weaknesses of the model order reduction methods.

💡 Research Summary

The paper provides a comprehensive review and comparative study of nonlinear model order reduction (MOR) techniques aimed at enabling real‑time optimization and model‑based control for high‑dimensional process models. After introducing the mathematical formulation of MOR for both ordinary differential equations (ODEs) and index‑one differential‑algebraic equations (DAEs) with external inputs, the authors classify reduction methods into intrusive (requiring access to the full‑order model’s equations) and non‑intrusive (data‑driven) categories.

Intrusive methods are discussed in depth. Linear subspace projection via Proper Orthogonal Decomposition (POD)‑Galerkin is presented as the baseline approach, followed by its nonlinear extensions: nonlinear‑POD‑Galerkin, manifold‑Galerkin, and a newly proposed input‑aware manifold‑Galerkin that incorporates control inputs directly into the manifold learning stage (MFL‑ANN with inputs). The paper explains how these techniques construct a reduced basis, perform Galerkin projection, and optionally residualize fast dynamics, highlighting the role of invariant manifolds and singular perturbation theory in justifying low‑dimensional representations.

Non‑intrusive techniques include Dynamic Mode Decomposition (DMD) and its control‑aware variant DMDc, Koopman operator‑based linearization, and purely data‑driven strategies such as compartment modeling and model aggregation. The authors note that while these methods avoid any need for the original equations, they often rely on linear approximations of inherently nonlinear dynamics, which can limit accuracy in strongly nonlinear or stiff regimes.

A case study on an air‑separation unit (ASU) is used to benchmark eight representative MOR approaches: POD‑Galerkin, nonlinear‑POD‑Galerkin, manifold‑Galerkin, DMD, DMDc, Koopman‑Wiener, manifold learning with latent predictor (MFL‑ANN), compartment modeling, and model aggregation. For each method the authors evaluate root‑mean‑square error (RMSE) of state reconstruction, computational time, stability across operating points, and performance during transient events such as start‑up and shut‑down. Results show that linear POD‑Galerkin offers fast simulation and reasonable average accuracy but suffers large errors during rapid transients. Nonlinear POD‑Galerkin and manifold‑Galerkin capture the slow manifold more faithfully, delivering stable predictions even in highly transient regimes. The input‑aware manifold‑Galerkin further improves robustness to varying control trajectories, demonstrating that incorporating inputs into the reduced subspace is crucial for non‑autonomous processes. DMDc and Koopman‑based models provide compact linear representations but struggle to reproduce strong nonlinear reaction kinetics, leading to higher RMSE in the ASU case. Compartment and aggregation methods preserve physical interpretability and achieve substantial order reduction, yet they demand extensive data for parameter identification and may not generalize well beyond the training operating conditions.

The authors deliberately omit hyper‑reduction (the reduction of floating‑point operations) from the study, acknowledging that while MOR alone can lower the number of differential equations, additional techniques such as empirical interpolation or sparse regression are often required to achieve true real‑time performance.

In the concluding discussion, the paper synthesizes the strengths and weaknesses of each approach: linear projection methods are easy to implement and computationally cheap but limited by linear subspace assumptions; nonlinear manifold techniques offer higher fidelity for stiff, nonlinear dynamics and can handle input variations when extended appropriately; data‑driven non‑intrusive methods are attractive when the full model is inaccessible but may need large datasets and can suffer from stability issues. The authors provide practical guidelines for selecting MOR techniques based on model characteristics (degree of nonlinearity, stiffness, input dependence), desired accuracy, and computational budget, thereby offering process engineers a clear decision framework for deploying reduced‑order models in real‑time optimization and model predictive control environments.