Source-linked AI summary
Data-driven reduced-order models via regularized operator inference for a single-injector combustion process
Shane A. McQuarrie, Cheng Huang, Karen E. Willcox
TL;DR
The paper develops predictive reduced-order models for combustion dynamics where governing-equation structure is known but high-fidelity simulation codes are inaccessible. It regularizes and scales Operator Inference, then demonstrates accurate training-regime reconstruction, acceptable future prediction, and major computational savings on a single-injector combustion example.
Problem
Reduced-order models for complex combustion dynamics must balance predictive accuracy with computational cost when high-fidelity CFD codes are difficult to access and unconstrained inference can overfit noisy, misspecified data.
Method
The paper uses non-intrusive Operator Inference to learn structured reduced-order operators from simulation data, with variable transformations, regularization selected through an optimization problem, and a scalable learning algorithm.
Results
The models achieve high accuracy in the training regime, acceptable accuracy in future dynamics, close to a million times computational speedup, and the same or better accuracy than a state-of-the-art method at approximately one thousandth of its computational cost.
Takeaways & Limitations
Training-data quality and quantity, regularization, and informed learning-variable selection are central to obtaining predictive Operator Inference models for the combustion example.
Takeaways & Limitations
The regularization optimization has no built-in constraints, so the resulting reduced-order models are not guaranteed to preserve properties such as energy conservation.
Abstract
from arXiv · showhide
This paper derives predictive reduced-order models for rocket engine combustion dynamics via Operator Inference, a scientific machine learning approach that blends data-driven learning with physics-based modeling. The non-intrusive nature of the approach enables variable transformations that expose system structure. The specific contribution of this paper is to advance the formulation robustness and algorithmic scalability of the Operator Inference approach. Regularization is introduced to the formulation to avoid over-fitting. The task of determining an optimal regularization is posed as an optimization problem that balances training error and stability of long-time integration dynamics. A scalable algorithm and open-source implementation are presented, then demonstrated for a single-injector rocket combustion example. This example exhibits rich dynamics that are difficult to capture with state-of-the-art reduced models. With appropriate regularization and an informed selection of learning variables, the reduced-order models exhibit high accuracy in re-predicting the training regime and acceptable accuracy in predicting future dynamics, while achieving close to a million times speedup in computational cost. When compared to a state-of-the-art model reduction method, the Operator Inference models provide the same or better accuracy at approximately one thousandth of the computational cost.
1. Introduction
The paper addresses the practical limits of intrusive projection-based reduction by using Operator Inference as a non-intrusive, physics-informed alternative for known governing equations and inaccessible simulation codes.
- The motivating goal is to model complex physical phenomena while retaining several orders of magnitude of computational speedup over CFD.
- Projection-based model reduction seeks accurate, lower-complexity models on low-dimensional manifolds when governing equations and high-fidelity models are available.
- Intrusive methods can be difficult or impossible to implement with legacy or commercial CFD codes, limiting their practical impact.
- Operator Inference learns reduced-order models from simulation data while retaining structure from known governing equations and projection-based reduction theory.
- The paper’s glass-box setting assumes the targeted dynamics are known but internal access to the CFD code is unavailable.
2. Methodology
The methodology learns polynomial reduced-order models from simulation data without requiring access to full-order operators. It combines variable transformations, POD projection, regression, and regularization selection to improve predictive behavior and scalability.
- Model structure: The full-order dynamics are represented with constant, linear, quadratic, and input terms after spatial discretization.This polynomial structure may arise directly from governing equations, variable transformations, or lifting transformations.
- Operator learning: The learned operators are obtained by solving a least-squares regression that is linear in the unknown reduced operators.The regression matches the reduced model to projected snapshot data in a minimum-residual sense.
- Data preparation: The training dataset is formed by collecting high-fidelity snapshots, transforming variables, computing a POD basis, projecting snapshots, and estimating projected time derivatives.The transformation may be reversible or lifting-based, allowing variables different from those used in the original discretization.
- Operator Inference: Operator Inference learns reduced operators from projected simulation data without explicit access to the original high-dimensional operators.The approach retains the known polynomial structure while avoiding intrusive queries to the full-order model.
- Regularization: Regularization addresses noisy derivatives, model mismatch, and unresolved dynamics that can otherwise cause operator overfitting and poor long-time prediction.A Tikhonov regularizer penalizes inferred operator entries, with separate penalties for linear and quadratic terms because their scalings differ.
- Regularization selection: The regularization parameters are selected by balancing training error against bounded integrated POD coefficients over the full time domain.The resulting procedure selects a ROM with reasonable behavior over the specified time interval but does not guarantee properties such as energy conservation.
3. Results
The single-injector combustion study evaluates regularized Operator Inference ROMs learned from transformed CFD snapshots against training and prediction dynamics. The models reproduce training behavior well, capture several qualitative features beyond the training horizon, and remain challenged by nonlinear temperature and species dynamics.
- Learning variables: The learning variables augment pressure and temperature with specific volume because these transformations expose quadratic structure and directly target temperature prediction.Non-intrusive Operator Inference permits transformations of snapshot data without constructing transformed full-order spatial operators.
- Problem setup: The study uses GEMS CFD snapshots from a 38,523-cell domain, with 60,000 time steps spanning 0.015 s to 0.021 s.Each snapshot contains 308,184 entries, and generating the dataset costs approximately 1,200 CPU hours.
- Sensitivity to training data: The required basis size increases linearly with the number of training snapshots, while the Operator Inference data-matrix dimension grows quadratically with reduced dimension.This behavior indicates that the combustion application contains rich, challenging dynamics and motivates choosing k much larger than d(r,m).
- Sensitivity to training data: OpInf ROMs faithfully reconstruct training pressure and velocity traces, capture frequencies across the full time domain, but can misrepresent prediction-regime amplitudes.Increasing the amount of training data significantly changes predictions and generally improves them, although not uniformly.
- Sensitivity to training data: Temperature prediction errors increase after training because advective flow and local chemical reactions create nonlinear, multiscale behavior, although general trends and time-averaged profiles remain adequately captured.The ROM also accurately re-predicts integrated chemical concentrations in training and captures much of the overall system behavior during prediction, with slightly more training error as snapshot count increases.
- Comparison to POD-DEIM: Against POD-DEIM, both ROMs preserve pressure frequencies and maintain temperature ranges, while neither accurately predicts global species dynamics after training.OpInf reconstructs training data more faithfully; its bounded behavior comes from regularization, whereas POD-DEIM constrains temperature during online reconstruction.
4. Conclusions
The approach is broadly applicable when governing equations are known but high-fidelity simulation-code access is limited. The conclusions emphasize the importance of training data, regularization, predictive validation, variable choice, and physical scale.
- Training-data quality and quantity are critical to the method’s success.
- Regularization is essential to avoid overfitting.
- Low training error does not necessarily indicate good predictive capability over future dynamics.
- Pressure-like quantities with large-scale coherent structures are predicted more accurately than multiscale quantities such as temperature and species concentrations.
- The reduced model can use different learning variables because the approach operates on snapshot data rather than the CFD model itself.