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Improving Reduced-Order Rotating Detonation Engine Models with Data Assimilation and Machine Learning
Ashwin Suriyanarayanan, Romit Maulik
TL;DR
High-fidelity RDE simulations resolve complex thermal structures but are computationally costly, while the reduced Koch-Kutz model misses some fine-scale features. The paper uses nudging to record the correction needed to synchronize the reduced solver, then trains a closure on that forcing. Without observations at deployment, the corrected model remains bounded and improves statistics relative to the baseline.
Problem
Method
The authors nudge the Koch-Kutz solver toward processed high-fidelity temperatures, record the corrective source, and train a Jacobian-regularized closure on that source.
Source: §1 · p. 2, §2 · p. 3, §3.3 · p. 7Results
The trained closure runs autonomously without explicit high-fidelity observations and improves the nudged reference statistics relative to the Koch-Kutz baseline while remaining bounded over rollouts beyond training.
Source: §5 · p. 11Takeaways & Limitations
The nudging source provides a systematic estimate of forcing that can diagnose reduced-model error and support recovery of missing physics in the reduced solver.
Source: §5 · p. 11The nudging and training cases are limited by the availability of high-fidelity datasets.
Source: §5 · p. 11
Abstract
from arXiv · showhide
Rotating detonation engines (RDEs) exhibit strongly nonlinear, multiscale wave dynamics that set the observed thermal field. High-fidelity simulations (DNS/LES) resolve these structures but remain computationally prohibitive, while low-order models such as the one-dimensional Koch-Kutz model capture circumferential wave motion yet lack the expressivity for high-frequency content. We use continuous data assimilation (nudging) to synchronize the Koch-Kutz solver with processed high-fidelity temperature data, introducing the prediction-observation mismatch as a relaxation source in the conserved energy equation; where observations are temporally sparse, interpolation supplies a target at every source update. As the nudging strength increases, the reduced model is progressively drawn onto the high-fidelity trajectory, and the forcing recorded along it provides an explicit, state-dependent estimate of the correction the model requires. We then train a Jacobian-regularized closure a priori on this recorded source. With the observation term removed, the corrected model advances autonomously, remains bounded, and recovers the temperature spectrum and the marginal statistics of the conserved variables relative to the baseline.
1 Introduction
High-fidelity RDE simulations are too costly for routine design and inference, while reduced descriptions remain limited by model-form deficits. This work uses observation-informed nudging to diagnose and correct the Koch-Kutz model’s mismatch with high-fidelity data.
1 Introduction
- Full-annulus high-fidelity RDE simulations are computationally intractable for routine design and inference, motivating reduced models that lump complex mechanisms into tractable circumferential dynamics.The Koch-Kutz model represents injection, heat release, and loss through source terms, but its phase, amplitude, and fine-scale temperature features may not stay synchronized with high-fidelity data.Source: §1 · p. 2
- Reduced descriptions face model-form deficits: projection-based models can degrade beyond their construction regime, while a-priori learned closures may become unstable at deployment.Stabilizing a-posteriori remedies can require differentiating through or repeatedly integrating the solver.Source: §1 · p. 2
- Continuous data assimilation adds feedback proportional to model-observation mismatch, drawing the reduced trajectory toward observations while leaving a residual dependent on observation sparsity and nudging strength.The residual can be interpreted as state-dependent forcing needed to retain finer-scale physics unavailable to the reduced model.Source: §1 · p. 2
- The paper records the nudging residual and trains a neural network to map the instantaneous conserved state to corrective temperature forcing, avoiding high-fidelity observations at deployment.The work analyzes synchronization at different nudging settings and aims to recover high-fidelity invariant properties in the reduced model.Source: §1 · p. 2
2 Koch-Kutz Model
The paper uses a one-dimensional periodic representation of an annular RDE and compares the Koch-Kutz model with processed high-fidelity temperature fields. Its PyTorch solver reproduces the released Koch-Kutz counterpart before assimilation is introduced.
2 Koch-Kutz Model
- The physical domain is a one-dimensional periodic circumferential coordinate, and the processed temperature array places processed time in rows and circumferential position in columns.The processed high-fidelity field has Nt = 250 and Nx = 100; the solver temperature is unnormalized and in the laboratory frame.Source: §2 · pp. 2–3
- The solver is reconstructed in PyTorch using the released Cheap2Rich data and Koch-Kutz setting, with model parameters retained rather than recalibrated against high-fidelity data.The same setting is used to compare the unnudged baseline and subsequent assimilated trajectories.Source: §2 · pp. 3–4
- The high-fidelity field contains additional fine-scale structures and variations compared with the strongly structured Koch-Kutz field after traveling-frame alignment.Figure 1 juxtaposes the released Koch-Kutz field, the PyTorch solver field, and the processed high-fidelity target; each array is 250 × 100.Source: §2 · p. 3, Figure 1 · p. 3
- The Koch-Kutz model represents dominant azimuthal wave dynamics with a reactive Euler system augmented by source terms for injection, heat release, and loss.Its conserved state includes density, circumferential velocity, total energy density, and a progress variable.Source: §2 · pp. 3–4
3 Methodology
The methodology assimilates processed high-fidelity temperature into the Koch-Kutz conservation law, then learns an autonomous state-dependent closure from recorded nudging corrections. Solver validation and closure regularization address numerical fidelity and rollout stability.
3.1 Nudging-based data assimilation
- Processed temperature observations are mapped to solver scale and laboratory coordinates, then inserted as a continuous source in the conserved PDE.Because observations are temporally sparse, the 250 frames are linearly interpolated to evaluate the source at every update; temperature is the only feedback, and synchronization is treated empirically because the reduced and high-fidelity models differ.Source: §3.1 · pp. 4–5
3.2 Numerical methods and solver validation
- The finite-volume solver uses 100 periodic cells on a length-24 domain and reproduces the released processed temperature field with correlation 0.9990.Comparisons use the same preprocessing shift, without independent trajectory alignment, so they measure phase agreement in a common display frame.Source: §3.2 · pp. 5–6
3.3 Model training and deployment
- Nudged trajectories provide state-correction pairs for an a-priori network that replaces the observation source and evolves from its own state.The recorded correction captures missing physics and solver discretization effects; deployment requires neither high-fidelity observations nor differentiation through the solver.Source: §3.3 · p. 6
- The closure loss combines mean-squared error with a binned spectral-power loss to represent both energy-containing scales and small-scale residual content.The spectral loss compares prediction and target power after grouping 51 wavenumbers into eight contiguous bins.Source: §3.3 · pp. 6–7
- A soft Jacobian penalty limits instantaneous closure-induced perturbation growth, but a good a-priori fit or bounded penalty does not guarantee a bounded rollout.The penalty measures the largest instantaneous amplification contributed by the closure, not the full operator or discrete solver; the paper notes that rollout divergence can occur.Source: §3.3 · p. 7
4 Results
The autonomous closure stayed bounded over the tested horizon and recovered reference spectral and marginal-statistical features, while pointwise tracking beyond assimilation remains unresolved. In the nudged system, stronger feedback improved field agreement and temperature occupancy relative to the high-fidelity target.
4.1 Synchronization of the nudged system
- For µ > 10, framewise synchronization settled onto a residual floor faster than for weaker nudging cases.The time-resolved diagnostics track framewise normalized-temperature error and spatial correlation over 250 processed frames.Source: §4.1 · p. 8, Figure 4 · p. 9
- As nudging strength increased, global field correlation rose and normalized-temperature RMSE fell against the processed high-fidelity target.Both metrics were computed across all 25,000 processed field entries; Table 1 and Figure 3 report the sweep.Source: Table 1 · p. 8, §4.1 · p. 8, Figure 3 · p. 9
- At µ = 100, the nudged field reproduced the high-fidelity target’s large-scale bands and oblique fine-scale features, while the residual view exposed remaining discrepancy.Figure 2 shows the progression across µ values; Figure 5 compares the µ = 100 field, target, and residual, which is clipped at its 99th percentile.Source: §4.1 · pp. 7–10, Figure 2 · p. 8, Figure 5 · p. 10
- Stronger nudging brought the marginal temperature distribution close to the high-fidelity distribution, correcting the baseline’s underpopulation of high temperatures.This PDF diagnostic measures temperature-state occupancy without preserving spatial or temporal ordering; the recorded µ = 25 residual is organized along detonation-front trajectories.Source: §4.1 · pp. 9–10
4.2 Autonomous deployment
- During autonomous rollout, the closure’s local feedback-gain measure remained between 3 and 16 without further growth, but positive values do not certify perturbation damping.The trajectory remained bounded over the tested horizon as a separate empirical observation.Source: §4.2 · pp. 10–11
- Without high-fidelity input, the closure stayed bounded over the tested horizon and recovered reference spectra and marginal distributions, but evaluation is constrained by 250 training snapshots and pointwise tracking remains unresolved.The rollout also carried surplus energy at higher wavenumbers and a modestly broader temperature peak.Source: §4.2 · pp. 10–11
5 Conclusions and Future Work
The paper uses continuous data assimilation to synchronize the reduced Koch-Kutz solver with processed high-fidelity temperature data, then interprets the recorded forcing as an estimate of missing model corrections. A neural closure trained on that forcing advances autonomously without high-fidelity observations, while the considered nudging and training cases remain limited by dataset availability.
Conclusions
- Continuous data assimilation synchronizes the one-dimensional Koch-Kutz solver to processed high-fidelity temperature data, with its source estimating the forcing needed to recover unresolved physics.The authors present the source as a systematic diagnostic of model-form error and missing physics.Source: §5 · p. 11
- The closure is trained a priori to map the conserved state to recorded corrective forcing, using spectral-power and Jacobian penalties.The spectral loss targets high-frequency residuals, while the Jacobian penalty discourages locally amplifying closure feedback.Source: §5 · p. 11
- The trained closure advances the reduced model without high-fidelity observations and improves statistics relative to the Koch-Kutz baseline over rollouts beyond the training regime.The trajectory remains bounded, and the closure recovers temperature-spectrum and conserved-variable marginal statistics relative to the baseline.Source: §5 · p. 11
Future Work
- The nudging and training cases are limited by high-fidelity dataset availability, and extending closure development to experimental observations is future work.The stated future direction is to use experimental observations of detonation experiments.Source: §5 · p. 11
Appendix A Processing of the high-fidelity target
The appendix describes processing the high-fidelity temperature target into a solver-compatible field. It specifies temporal interpolation for nudging updates and notes that the target retains high-fidelity structure but not its original temperature magnitude.
Target processing
- The processed target is a normalized 250 × 100 temperature field, with a row-dependent periodic spatial shift and min-max normalization.The shift uses σ = −0.27 grid points per processed row.Source: Appendix A · p. 13
Temporal and spatial sampling
- The processed dataset is formed by cropping raw indices 100000–109999, retaining every 40th output for 250 frames over t ∈ [90, 98.964].The frames use processed indices j = 0, …, 249.Source: Appendix A · p. 13
Target scaling
- Because scaling uses the Koch crop rather than the high-fidelity field, assimilation reproduces target structure on the reduced model’s range, not the original high-fidelity magnitude or mean.The absolute amplitude and mean level of the high-fidelity temperature are not retained.Source: Appendix A · p. 14
Target processing
- Linear temporal interpolation supplies a target at each source update between processed frames, and no assimilation source is applied outside the crop window.The solver is driven between observation frames rather than only at the original processed-frame cadence.Source: Appendix A · p. 14
Appendix B Table of notations
The notation appendix defines symbols used for the RDE model, its state and sources, and the processed temperature data. It also lists the principal parameter values and operators.
State and physical quantities
- The conserved state is q = (ρ, ρu, E, ρz)T, with E denoting total energy per unit volume and z the progress variable.The notation also identifies density ρ, circumferential velocity u, and gas temperature T = p/ρ.Source: Appendix B · p. 14
Model and source notation
- The source notation distinguishes the Koch-Kutz physical source SK, data-assimilation source SDA, and learned deployment closure source SNN.SNN replaces SDA at deployment.Source: Appendix B · p. 14
Operators and parameters
- The notation defines µ as nudging strength, µ2(·) as the logarithmic norm, and It as the linear temporal interpolation operator for target frames.The listed penalty weights are λS = 0.5 and λJ = 10−3.Source: Appendix B · p. 15
Appendix C Conversion of the temperature relaxation to a conserved-energy source
This appendix connects temperature relaxation to a conserved-energy forcing and presents diagnostics of the nudged and autonomous model fields. The figures compare conserved-variable trajectories, residual structure, temperature spectra, and marginal distributions.
Temperature relaxation to energy source
- The temperature relaxation is converted into an energy forcing and then into the complete data-assimilation source vector.The described conversion holds density and momentum fixed for the data-assimilation contribution during a source substep.Source: Appendix C · p. 15
Nudged conserved variables
- Figure 6 compares unnudged and µ = 100 fields for density, momentum, energy, and progress variable across processed frames and spatial positions.The panels in each row share a color scale.Source: Figure 6 · p. 16
Recorded residual
- Figure 8 shows a residual organized along detonation-front trajectories, with a spectrum dominated by wavenumber three and its first harmonic at wavenumber six.The spectrum has a broadband floor that declines slowly toward the grid cut-off.Source: Figure 8 · p. 17
Autonomous deployment diagnostics
- Figure 11 compares time-averaged temperature spectra, while Figure 12 compares marginal distributions of temperature and the four conserved variables across the reference, closure, and baseline.The reference uses the assimilation window, whereas closure and baseline spectra use the full rollout.Source: Figure 11 · p. 19, Figure 12 · p. 20