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Model order reduction via Lie groups

This paper introduces MORLie, a novel non-intrusive model order reduction framework that approximates high-dimensional dynamical systems on manifolds using low-dimensional systems on Lie groups, effectively handling non-equivariant dynamics and outperforming traditional linear-subspace methods in accuracy and computational efficiency across various applications.

Original authors: Yannik P. Wotte, Patrick Buchfink, Silke Glas, Federico Califano, Stefano Stramigioli

Published 2026-04-01

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CC BY 4.0

Original authors: Yannik P. Wotte, Patrick Buchfink, Silke Glas, Federico Califano, Stefano Stramigioli

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Problem: The "Too Many Details" Dilemma

Imagine you are trying to simulate how a jellyfish moves through the ocean. To do this accurately, a computer needs to track millions of tiny points on the jellyfish's body as it stretches, twists, and flows. This is the "Full Order Model" (FOM). It's incredibly accurate, but it's like trying to count every single grain of sand on a beach to predict the tide. It takes forever, requires a supercomputer, and is too slow to be useful for real-time decisions (like a robot avoiding an obstacle).

Scientists want a "Reduced Order Model" (ROM)—a simplified version that captures the essence of the movement without the millions of details. It's like describing the jellyfish's motion as "a smooth wave moving forward" rather than tracking every cell.

The Old Way: The "Flat Sheet" Limitation

For a long time, the best way to simplify these systems was Linear Subspace Methods (like POD).

The Analogy: Imagine the jellyfish's movement is a complex 3D dance. The old method tries to flatten this dance onto a 2D sheet of paper.
The Problem: If the jellyfish just wiggles a little, the 2D sheet works fine. But if the jellyfish spins, stretches, and moves in a circle, you can't flatten that 3D motion onto a flat sheet without tearing it or losing huge amounts of information.
The Result: The math hits a "wall" (called the Kolmogorov N-width barrier). No matter how many lines you draw on your 2D sheet, you can't perfectly describe a 3D spinning object. You need too many lines, making the "simplified" model almost as slow as the original.

The New Way: MORLie (The "Dance Floor" Approach)

This paper introduces a new method called MORLie. Instead of flattening the movement onto a sheet, it realizes that many physical movements (like spinning, sliding, or stretching) follow specific geometric patterns called Lie Groups.

The Analogy: Think of a dance floor.
- Old Method: Tries to draw the dancer's path on a flat piece of paper.
- MORLie: Realizes the dancer is moving on a rotating turntable (a Lie Group). Instead of tracking every step, you just track how fast the turntable is spinning and how far it's moved.
- The Magic: By understanding the "rules of the dance floor" (the group action), you can predict the dancer's position with just a few numbers, even if the dancer is spinning wildly.

How It Works (The "Magic Recipe")

The authors propose a three-step recipe to simplify complex systems:

Find the Dance Floor (The Group): Identify the underlying symmetry. Is the object rotating? Sliding? Stretching? This defines the "Lie Group" (the rules of movement).
Map the Moves (The Action): Figure out how these rules apply to the specific object. If the object is a liver, the "dance floor" might be a rigid rotation. If it's a shearing fluid, the "dance floor" is a stretching motion.
Solve the Simple Equation: Instead of solving the million-point equation, you solve a tiny equation that tells you how the "dance floor" is moving. Then, you use the rules to reconstruct the full image.

Why Is This Better? (The "Non-Equivariant" Breakthrough)

Previous methods using Lie groups only worked if the system was perfectly symmetrical (like a perfect spinning top). But real life is messy. A liver doesn't spin perfectly; it deforms. A fluid doesn't flow perfectly; it gets turbulent.

MORLie's Superpower: It works even when the system is imperfect (non-equivariant). It can handle "noisy" data and approximate symmetries. It's like being able to predict the path of a drunk dancer who is mostly following the rhythm, even if they stumble a bit.

Real-World Examples from the Paper

The authors tested this on three very different problems:

The "Sheering" Point Cloud:
- Scenario: Imagine a cloud of dots (like stars) that are being stretched and twisted like taffy.
- Result: The old method needed hundreds of lines to describe the stretch. MORLie figured out the "stretching rule" and described it with just a few numbers, doing it much faster and more accurately.
The "Rigid" Point Cloud:
- Scenario: Tracking a rigid object (like a robot arm) moving through space, but the camera data is noisy (fuzzy).
- Result: The old method got confused by the noise and needed a massive amount of data to figure out the motion. MORLie ignored the noise, recognized the "rigid body" dance floor, and reconstructed the motion perfectly in seconds.
The "Liver" Tracker (Medical Application):
- Scenario: A patient is breathing, and their liver is moving and deforming inside their body. Doctors need to track this in real-time for surgery.
- Result:
  - Old Way: Took hours on a massive computer cluster to train a model.
  - MORLie: Trained in minutes on a standard laptop.
  - Accuracy: For shallow breathing, it was nearly perfect. For deep breaths (where the liver squishes), it was still very good, though it noted that deep deformation requires even more complex "dance floors" (non-linear groups).

The Bottom Line

MORLie is a new way to simplify complex physics simulations.

Old Way: "Let's flatten the 3D world onto a 2D sheet." (Fails when things spin or twist).
MORLie: "Let's find the invisible dance floor the object is moving on, and just track the dance floor."

This allows scientists and engineers to simulate complex systems (like fluids, soft robots, or human organs) faster, with less computer power, and with higher accuracy, even when the data is messy. It turns a supercomputer problem into a laptop problem.

1. Problem Statement

Model Order Reduction (MOR) aims to approximate high-dimensional dynamical systems (Full Order Models or FOMs) with low-dimensional surrogate models (Reduced Order Models or ROMs) to reduce computational costs.

The Limitation of Linear Subspaces: Traditional MOR methods (e.g., Proper Orthogonal Decomposition - POD) approximate solutions within a fixed linear subspace. Their accuracy is bounded by the Kolmogorov $N$ -width, which measures the best possible approximation error by an $N$ -dimensional subspace.
The Kolmogorov Barrier: For systems dominated by transport, wave propagation, or large deformations (e.g., fluid dynamics, moving boundaries), the Kolmogorov $N$ -width decays very slowly. This means linear subspaces require a prohibitively large number of basis elements to capture the solution manifold accurately.
The Gap: Existing nonlinear MOR methods (e.g., adaptive bases, machine learning) often lack a unified geometric framework or are computationally expensive. Furthermore, previous Lie group-based MOR methods were restricted to equivariant systems (where the dynamics commute with the group action) and were often intrusive (requiring explicit knowledge of the governing equations).

Goal: Develop a general, non-intrusive MOR framework capable of handling non-equivariant and approximately equivariant dynamics by restricting the system's evolution to the tangent spaces of Lie group orbits rather than fixed linear subspaces.

2. Methodology: MORLie

The authors propose MORLie, a framework that approximates high-dimensional trajectories on a manifold $\mathcal{M}$ by trajectories on a lower-dimensional Lie group $G$ .

Core Concept

Instead of approximating the state $u_\mu(t)$ as a linear combination of fixed basis functions, MORLie approximates it via a group action $\Phi$ :
$u_\mu(t) \approx \Phi(g_\mu(t), u_{\mu,0})$
where:

$g_\mu(t) \in G$ is the state on the Lie group.
$\Phi: G \times \mathcal{M} \to \mathcal{M}$ is a group action.
$u_{\mu,0}$ is the initial condition.

The dynamics of $g_\mu(t)$ are governed by a reduced vector field $\rho_\mu: \mathcal{M} \to \mathfrak{g}$ (where $\mathfrak{g}$ is the Lie algebra of $G$ ):
$\dot{g}_\mu(t) = R_{g_\mu(t)*} \rho_\mu(\bar{x}_\mu(t))$
Here, $R_{g*}$ denotes the right translation, and $\bar{x}_\mu(t)$ is the reconstructed state. This formulation ensures the solution evolves along the orbit of the initial condition, preserving the initial state without projection.

Key Theoretical Components

Induced Distribution: The method induces a distribution $\Delta$ on the state manifold, spanned by the infinitesimal generators of the group action. The ROM dynamics are constrained to be tangent to this distribution.
Group Kolmogorov $N$ -width ( $d^{\Xi_N}$ ): The authors define a new error bound based on the distance of solution snapshots to the nearest group orbit.
- They prove that for linear transport problems, the Group Kolmogorov width can be zero (perfect approximation) with a 1D Lie group, whereas the linear Kolmogorov width decays slowly.
- This demonstrates that MORLie can break the "Kolmogorov barrier" inherent to linear methods.
Recovery of Existing Methods: The framework analytically recovers the "Method of Freezing" (a previous Lie group MOR technique) as a special case where the group is a product of the symmetry group and a vector space, showing MORLie's generality.

Optimization Strategies

The paper formulates the identification of $(G, \Phi, \rho)$ as an optimization problem.

Intrusive Approach: Minimizes the difference between the FOM vector field and the induced ROM vector field using the explicit system equations.
Non-Intrusive (Data-Driven) Approach:
- Velocity-based: Uses measured velocities to minimize projection error.
- Velocity-free (Flow-matching): Minimizes the distance between the next state in the data and the state predicted by the group action over a time step $\Delta t$ . This is robust to noise and does not require derivative calculations.
Subalgebra Search: To further reduce dimensionality, the method uses Principal Component Analysis (PCA) on the reduced snapshot matrix (values of $\rho$ ) followed by Lie bracket closure to identify the smallest Lie subalgebra that captures the dominant dynamics.

3. Key Contributions

Novel Framework (MORLie): A geometric MOR framework that reduces dynamics to a Lie group, allowing for non-equivariant and approximate symmetries.
Theoretical Bounds: Introduction of the Group Kolmogorov $N$ -width, providing a theoretical lower bound for MORLie that is strictly lower than the linear Kolmogorov width for transport-dominated problems.
Non-Intrusive Algorithms: Development of optimization methods that work directly with data (snapshots) without requiring access to the governing PDEs, including a robust velocity-free cost function.
Generalization: Demonstrating that the "Method of Freezing" is a specific instance of this broader geometric framework.
Practical Validation: Successful application to synthetic and real-world data, showing superior performance over POD in terms of accuracy and dimensionality.

4. Results and Applications

The authors validated MORLie on three distinct examples:

A. Radial Oscillator (Analytic)

Task: Reduce a radial oscillator system.
Result: Successfully projected the vector field onto the distribution induced by $SO(2)$, recovering the exact dynamics with a 1D group.

B. Rigid Point Cloud (Synthetic)

Task: Reconstruct the motion of a rigid body from noisy point cloud data.
Comparison: MORLie vs. Naive POD.
Findings:
- POD: Required $9(N_j+1)+3$ basis elements (e.g., 93 elements for 9 trajectories) to capture the dynamics, failing to scale with the number of trajectories.
- MORLie: Identified the underlying $SE(3)$ symmetry (6 dimensions) regardless of the number of trajectories.
- Performance: MORLie achieved high accuracy with a fixed low dimension (6), while POD accuracy degraded or required massive basis sizes as data complexity increased.

C. Sheering Point Clouds (Synthetic)

Task: Handle multiple clusters of particles undergoing different affine deformations (shearing).
Result: The algorithm successfully clustered the data and identified a product group $Aff(3) \times Aff(3)$ (12 dimensions), outperforming linear methods which could not separate the distinct deformation modes efficiently.

D. Liver Respiration Tracking (Real-world Medical Data)

Task: Reconstruct liver motion from MRI edge-detection data during breathing.
Findings:
- For shallow breathing (mostly rigid motion), MORLie achieved state-of-the-art accuracy.
- Efficiency: Reduced training time from hours (on a computing cluster for traditional methods) to minutes (on a mobile workstation).
- Limitation: For deep breaths involving significant non-rigid deformation, the rigid group model ($SE(3)$) showed increased error, suggesting a need for nonlinear group actions in future work.

5. Significance

Breaking the Kolmogorov Barrier: MORLie provides a mathematically rigorous way to overcome the limitations of linear subspaces for transport-dominated problems, offering a path to low-dimensional models for systems previously considered "hard" to reduce.
Data-Driven & Non-Intrusive: The ability to learn group actions and reduced dynamics directly from data without knowing the underlying physics equations makes this highly applicable to real-world engineering and medical problems where models are unknown or too complex.
Computational Efficiency: The method drastically reduces the computational cost of training ROMs and the dimensionality of the resulting models, enabling real-time applications on standard hardware.
Geometric Unification: It unifies various existing techniques (shifted POD, method of freezing, transformation-based methods) under a single geometric umbrella, providing a clearer theoretical understanding of how symmetries and group actions can be exploited for model reduction.

In conclusion, MORLie represents a significant shift from "approximating states in a subspace" to "approximating the evolution of states along group orbits," offering a powerful tool for the next generation of reduced-order modeling.