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Data Driven Modeling of Nonlinear Dynamics in a Rotating Detonation Combustor via Finite Dimensional Approximations of the Koopman Operator

This paper presents a data-driven modeling approach for Rotating Detonation Combustors that utilizes time-delay embedded Dynamic Mode Decomposition to construct finite-dimensional Koopman operator approximations from flame luminosity video, enabling accurate reconstruction of nonlinear dynamics, standing wave patterns, and noise-mitigated insights into various operating modes.

Original authors: David Oexle, Tobias Breiten, Myles D. Bohon

Published 2026-07-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: David Oexle, Tobias Breiten, Myles D. Bohon

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

Imagine trying to predict the weather, but instead of clouds and wind, you are watching a ring of fire spinning so fast it creates its own thunder. This is the world of the Rotating Detonation Combustor (RDC), a high-tech engine part that promises to make rockets and power plants much more efficient. Unlike normal engines that burn fuel gently, an RDC uses a continuous, supersonic explosion that races around a circular chamber. The problem? These explosions are chaotic. They don't just spin in one direction; sometimes they crash into each other, form standing patterns, or split into multiple waves that dance in complex, non-linear ways. To engineers, this chaos is a puzzle: if they can't predict how these waves interact, they can't build a stable, efficient engine.

To solve this, scientists often use a mathematical tool called the "Koopman operator." Think of it as a magical pair of glasses. When you look at a chaotic, swirling system through normal eyes, it looks messy and unpredictable. But through Koopman glasses, that same mess transforms into a set of simple, straight lines. It turns a wild, non-linear dance into a predictable, linear rhythm. However, these glasses are usually too heavy and complicated to use on real-world data, which is often noisy and full of glitches. This is where the work of David Oexle, Tobias Breiten, and Myles D. Bohon comes in. They didn't invent a new kind of glass; instead, they figured out how to build a lightweight, custom version of these glasses specifically for the RDC, using a technique called "Dynamic Mode Decomposition" (DMD) combined with a clever trick called "time-delay embedding."

The team started by filming the RDC at the Technical University of Berlin. They used a high-speed camera to capture the natural glow of the flames at a blistering 87,500 frames per second. This gave them a massive amount of data, but it was also messy. The camera's view was distorted by the exhaust, and the sensors had a bit of static noise. When they tried to use the standard "DMD" method to analyze this footage, it failed. It was like trying to sort a pile of mixed-up LEGO bricks by color when the bricks were all the same shade of gray; the standard method got confused by the noise and couldn't tell the difference between a spinning wave and a glitch in the camera. The standard approach suggested the waves were dying out too fast or behaving in ways that didn't match the physics of a stable engine.

To fix this, the authors proposed a new, step-by-step recipe. First, they realized that the waves were moving, so they shifted their perspective. Imagine watching a race car on a track; if you stand still, the car zooms by. But if you hop in a car driving alongside it at the same speed, the race car looks like it's standing still. The team did this mathematically, shifting their "frame of reference" to match the speed of the spinning waves. This made the moving waves look stationary, which is much easier to analyze.

Next, they tackled the "non-linear" problem. In the RDC, waves can crash into each other and create new patterns, which standard math struggles to describe. The authors used a technique called "time-delay embedding." Imagine trying to understand a song by listening to just one note; you'd have no idea what the melody is. But if you listen to the note, then the note a split-second later, and the one after that, you can hear the tune. By feeding the computer not just the current image of the flame, but also a stack of previous images (time delays), they gave the model enough context to "hear" the melody of the explosion. They also used a special filter to clean up the sensor noise, ensuring the model wasn't learning from the static.

The results were a success. By combining these steps, the team could separate the "traveling waves" (the ones spinning around) from the "standing waves" (the ones that look like they are vibrating in place) and the messy "non-linear interactions" where they crash. They found that for a specific mode with two counter-rotating waves, the standard method missed the mark, but their new sequence could reconstruct the flame patterns with high accuracy. They identified specific frequencies, like a primary wave spinning at roughly 4 kHz and a secondary wave at 3.9 kHz, and even spotted a "beating frequency" of 0.2 kHz caused by the two waves interacting.

Crucially, the paper suggests that this method is a powerful new workflow for engineers. It doesn't claim to have solved the RDC problem forever, but it offers a reliable way to take messy, real-world video data and turn it into a clear, mathematical model. By separating the different types of wave behaviors, engineers can now better understand why certain engine modes are stable and others are not. The authors show that with the right combination of shifting perspectives, stacking time delays, and cleaning up the noise, even the most chaotic fire rings can be tamed and understood.

Technical Summary: Data Driven Modeling of Nonlinear Dynamics in a Rotating Detonation Combustor via Finite Dimensional Approximations of the Koopman Operator

Problem Statement
Rotating Detonation Combustors (RDCs) offer significant thermodynamic efficiency gains through Pressure Gain Combustion (PGC). However, their operation involves complex nonlinear dynamics, including counter-rotating detonation waves, standing wave patterns, and nonlinear amplification at interaction points. Understanding these dynamics is critical for assessing stability and efficiency. Traditional modeling approaches, such as first-order balance equations or purely data-driven deep learning, often struggle to capture these specific nonlinear interactions or require extensive computational resources. Furthermore, standard data-driven techniques like Dynamic Mode Decomposition (DMD) face specific challenges when applied to RDC data:

Traveling Waves: Standard DMD struggles to describe traveling wave structures effectively.
Sensor Noise: Measurement noise in luminosity and pressure data biases DMD eigenvalues, leading to inaccurate signal reconstruction (e.g., overestimating decay rates).
Nonlinearity: Standard DMD is a linear method and cannot inherently depict nonlinear phenomena without specific extensions.
Standing Waves: The interaction of counter-rotating waves can create standing patterns that are difficult to reconstruct accurately with standard methods.

Methodology
The authors propose a tailored algorithmic sequence based on finite-dimensional approximations of the Koopman operator to address these challenges. The workflow integrates several established methods into a cohesive pipeline:

Data Acquisition and Preprocessing: High-speed video data capturing flame luminosity and piezoelectric pressure sensor data are collected from an RDC at TU Berlin. A post-processing pipeline reduces the dimensionality of the image data (from $329 \times 329$ pixels) to azimuthal luminosity profiles ( $n=819$ ) by masking, polar coordinate transformation, and radial averaging, correcting for mirror distortions.
Frame of Reference (FR) Transformation: To handle traveling waves, the data is transformed from an inertial frame to a moving frame of reference (FR) aligned with the wave speed. The wave speed is determined via Fast Fourier Transform (FFT) of the pressure or luminosity signals. Circular shifts are applied to the state vectors to render the traveling waves stationary in the new FR.
Extended Dynamic Mode Decomposition (EDMD) with Time-Delay Embedding: Instead of using arbitrary dictionary functions, the method employs time-delay embeddings of the state as the dictionary. The number of embeddings ( $s$ ) is not chosen heuristically but is estimated based on the physics of the operating modes, specifically the periodicity of the data ( $s \approx 1 / (f_{base} \Delta t)$ ), where $f_{base}$ is the lowest dynamically relevant frequency (identified as the longitudinal mode $L4$ ).
Exact DMD (ExactDMD): This algorithm is used to compute an initial finite-dimensional approximation of the Koopman operator, leveraging the low-rank structure of the data.
Optimized DMD (OptDMD): To mitigate the bias introduced by sensor noise (which causes eigenvalues to deviate from the unit circle), OptDMD is applied as a post-processing step. It reframes the problem as an exponential fitting task, using the ExactDMD results as an initial guess to iteratively refine the eigenvalues and modes.
Optimal Thresholding: A rigorous criterion based on the aspect ratio of the data matrix is used to determine the truncation rank ( $r$ ), distinguishing true dynamical content from noise-dominated modes.
Decomposition and Reconstruction: The final sequence separates the data into traveling wave components (reconstructed in their respective moving FRs) and a remainder term containing the nonlinear interactions. Scaling coefficients are computed to reconstruct the original signal as a superposition of these components.

Key Contributions

Algorithmic Sequence: The paper introduces a specific workflow tailored to RDC data that combines EDMD, time-delay embedding, frame shifting, and noise mitigation (OptDMD).
Physics-Informed Embedding: A novel proposal to estimate the number of time-delay embeddings ( $s$ ) based on the periodicity of the operating modes (specifically the longitudinal wave frequency) rather than arbitrary selection.
Separation of Dynamics: The method successfully separates traveling wave structures from the dataset, allowing for the independent analysis of the remaining nonlinear dynamics (interactions and standing waves).
Noise Mitigation: The integration of OptDMD with ExactDMD addresses the specific issue of sensor noise bias in RDC luminosity data, which standard DMD fails to handle accurately.

Results
The methodology was applied to three distinct operating modes of the RDC:

2CR: Two counter-rotating waves of equal strength and speed (standing wave pattern).
2CRT: A dominant primary wave and a slower secondary wave with a precessing interaction point.
DS2: A primary wave and a set of two or more weaker, slower counter-rotating waves.

Comparisons demonstrated that standard DMD (ExactDMD) failed to accurately reconstruct the 2CR mode, producing eigenvalues that deviated significantly from the unit circle due to noise and failing to capture the standing wave structure. In contrast, the proposed sequence:

Successfully reconstructed the traveling wave structures in the moving frames of reference.
Accurately identified the counter-rotating wave frequencies and their harmonics.
Preserved the longitudinal mode frequency ( $L4$ ) across frame transformations.
Provided a clear decomposition of the signal into traveling wave components and the nonlinear remainder, offering insight into the modal structure of the counter-rotating waves.

Significance and Claims
The authors explicitly state that the methodological novelty of this work lies not in advancing the DMD method itself, but in the problem-specific composition of existing DMD variants to address the unique challenges of RDC data. The work provides a practical workflow for RDC practitioners to analyze the modal structure of counter-rotating detonation waves. By separating traveling waves from the data, the approach enables a clearer analysis of the nonlinear interactions that govern the stability and efficiency of the combustor. The paper concludes that this data-driven approach offers a viable path to understanding complex nonlinear dynamics in pressure-gain combustion systems without requiring the derivation of complex governing equations.

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