FlameODE - Flame Response Modeling via Physically Interpretable Differential Equations
by Gregor Doehner and Camilo F. Silva
Motivation
Combustion instabilities remain a challenging obstacle on the road towards ever more efficient and environmentally friendly combustion systems. To predict the occurrence of such instabilities, a flame model, which links relative fluctuations in burner inlet velocity (input) to relative heat release rate fluctuations (output) is generally required. The 'linear' flame response to a given input is traditionally characterized by a finite impulse response (FIR). In the context of numerical simulations, such an FIR model is typically derived by applying broadband forcing to the system's inlet velocity.
To model limit cycles, which are characteristic of unstable thermoacoustic systems, a 'nonlinear' flame response is essential. To date, Wiener-Hammerstein (WH) models are commonly used to characterize such nonlinear behavior. However, these models are typically effective only within a predefined frequency range where the limit cycle is expected to occur. To overcome this limitation, data-driven machine learning (ML) methods, such as Multi-Layer Perceptrons (MLPs) and Long Short-Term Memory (LSTM) networks, have been employed to evaluate the nonlinear flame response over a wide frequency band. While these models are successful, they lack physical interpretability and function as black-box solutions.
Objectives and Strategy
This project aims at utilizing an alternative modeling approach. Here, we aim at describing the flame by a physical substitute system. This system can be described via a parsimonious system of ordinary differential equations (ODEs) that replicates the modeling capabilities of MLPs and LSTMs while remaining compact, especially compared to the thousands of parameters typically required by neural networks (NNs). At its current state, the project already demonstrated that a simple mass-spring-damper with nonlinear damping is able to recover key flame dynamics quantitatively in the linear and qualitatively in the nonlinear regime for laminar and turbulent flames. By modifying the architecture to enforce unidirectional information flow (from mass one to mass two only), the model becomes physically interpretable, with each oscillator targeting a specific mechanism of the overall flame response.
The resulting ODE model is highly flexible and can be trained on time series or frequency domain data alike. Below, we present a comparison of model prediction and reference data from CFD simulations. In both cases, the flame is forced with broadband forcing, but between the two cases, the forcing amplitude is significantly raised from 5% to 150%, where for larger forcing amplitudes, the flame behaves increasingly nonlinear. We can observe that the excellent quantitative fit of ODE prediction to CFD reference for the 5% forcing case transforms into a qualitative fit for larger forcing amplitudes, highlighting that the underlying ODE is still able to capture the majority of dynamics at play but also missing out on crucial highly nonlinear effects.
Current Work: Model Reduction via Universal Differential Equations
Current work focuses on deriving additional nonlinearities which are required to describe the energy transfer between different frequencies (harmonics), an essential feature to adequately describe more complex limit cycles. To do so, we leverage the frameworks of Universal Differential Equations (UDEs) and Sparse Identification of Nonlinear Dynamics (SINDy). The general workflow is presented in the above figure. Starting from known physics, represented by a differential equation, we add a neural network (NN) inside the differential equation forming a Universal Differential Equation (UDE). By construction, the NN learns the ‘missing’ dynamics which are required to close the existing differential equation. As the network is a) structurally constrained (by placement inside the differential equation) and b) supported by existing knowledge about the physics at hand, which is provided by the underlying differential equations, very small NNs suffice to significantly improve the performance w.r.t the original differential equation.
If one's primary interest lies in identifying a sparse, robust and physics based nonlinear dynamical model, one can simply employ the trained NN within the known differential equation framework and use it as dynamical model. As the employed networks are however comparatively small, trying to obtain an analytical expression that recovers the NN dynamics via regression methods such as PySINDy merits an attempt.
To demonstrate how the introduced workflow can be employed within the flameODE context, results of two variants which were investigated are presented. Note, depending on how one places the NN inside a given UDE, different results can be achieved. The choice of where to place the NN inside the ODE, which inputs to use and where to apply the output within the ODE significantly alters the reuslting UDE. Below, we present two possible placement variants and the corresponding results on time series data.
Variant 1:
The NN can account for the “missing” dynamics as nonlinear force acting on the element of the ODE that is associated with the flame. Variant testing of different NN architectures (no. of hidden layers, total neurons, …) yields that a very small NN comprising 3 neurons in total suffices to achieve the maximum possible performance increase for the chosen integration of the NN inside the existing ODE.
Variant 2:
This variant significantly increases the potency of the UDE by introducing a third element, whose dynamics are entirely governed by the NN. The position of said third element is added to the output equation and the size of the NN is increased to 7 neurons in total.
As evident from the results presented above, placement of the NN remains a critical design choice and has a large impact on the overall performance of the UDE as well as the success of the subsequent regression.
Below you can find a recording of a talk discussing the UDE framework in the context of flame dynamics, which was presented at the “Euromech Colloquium on Data-Driven Fluid Dynamics” in London, April 2025.
Please note that the actual interpretation of the flame ODE model has been subject to change since 2025. We expect to publish an updated version going into significantly more detail on the meaning of each state of the ODE by the end of 2026.
If you are a student or researcher and you are interested in collaborating, please contact mailto:gregor.doehner@tum.de
Let's connect!
Selected References
G. Doehner, Data Driven Modeling of the Laminar Flame Response Using Universal Differential Equations, TU München, München, 2023, http://dx.doi.org/10.13140/RG.2.2.30348.49285
Gregor Doehner, Alexander J. Eder, Camilo F. Silva, A parsimonious system of ordinary differential equations for the response modeling of turbulent swirled flames, Combustion and Flame, 2024, http://dx.doi.org/10.1016/j.combustflame.2024.113408
Doehner G, Haeringer M, Silva CF. Nonlinear flame response modelling by a parsimonious set of ordinary differential equations, International Journal of Spray and Combustion Dynamics, 2022, http://dx.doi.org/10.1177/17568277221094760