Improving Reduced-Order Rotating Detonation Engine Models with Data Assimilation and Machine Learning

We propose an efficient retraining strategy for a parameterized Reduced Order Model (ROM) that attains accuracy comparable to full retraining while requiring only a fraction of the computational time and relying solely on sparse observations of the full system. The architecture employs an encode-process-decode structure: a Variational Autoencoder (VAE) to perform dimensionality reduction, and a transformer network to evolve the latent states and model the dynamics. The ROM is parameterized by an external control variable, the Reynolds number in the Navier-Stokes setting, with the transformer exploiting attention mechanisms to capture both temporal dependencies and parameter effects. The probabilistic VAE enables stochastic sampling of trajectory ensembles, providing predictive means and uncertainty quantification through the first two moments. After initial training on a limited set of dynamical regimes, the model is adapted to out-of-sample parameter regions using only sparse data. Its probabilistic formulation naturally supports ensemble generation, which we employ within an ensemble Kalman filtering framework to assimilate data and reconstruct full-state trajectories from minimal observations. We further show that, for the dynamical system considered, the dominant source of error in out-of-sample forecasts stems from distortions of the latent manifold rather than changes in the latent dynamics. Consequently, retraining can be limited to the autoencoder, allowing for a lightweight, computationally efficient adaptation procedure with very sparse fine-tuning data.

Reduced-order modeling of high-dimensional dynamical systems is often hindered by closure effects arising from unresolved variables, which can introduce non-Markovian dependence into the resolved dynamics. Motivated by the history-dependent memory term arising in the Mori--Zwanzig formalism, we recast non-Markovian closure modeling as a sequence modeling problem and propose the Mamba-Assisted Closure (MAC) framework. MAC employs a Mamba-based sequence model to predict the closure from the resolved trajectory and couples the learned closure with the reduced-order governing equations through a numerical integrator to advance the resolved variables in time. During training, the selective scan mechanism in Mamba enables efficient parallel sequence processing with linear scaling in sequence length, while autoregressive inference proceeds through recurrent state updates at essentially constant per-step cost. We evaluate MAC on four benchmark systems with complementary characteristics: the viscous Burgers' equation, the chaotic two-scale Lorenz '96 system, the 3-bus DeMarco--Zheng power-grid system, and the dispersive Korteweg--de Vries equation. Across these benchmarks, MAC consistently improves predictive accuracy and long-time rollout stability relative to the comparison models, demonstrating an effective and computationally scalable approach to non-Markovian closure modeling.

Turbulence is ubiquitous in engineering and science, yet direct simulation is prohibitively expensive. The Reynolds-averaged Navier-Stokes (RANS) equations provide savings exceeding ten orders of magnitude but introduce unclosed terms (the closure problem). Offline-trained machine-learning (ML) closures suffer distribution shift in predictive simulations, while ML methods that bypass the governing equations struggle to generalise from scarce high-fidelity data. We develop a physics-derived deep learning closure model for RANS, the Deep Algebraic Reynolds Stress Model (DARSM), which can be trained on small datasets and accurately generalise across Reynolds numbers, to unseen geometries, and to different flow regimes. A neural network maps flow invariants to empirical parameters in an implicit algebraic Reynolds stress equation, derived from the Reynolds stress transport equations under the weak-equilibrium assumption, imposing physics-based structure on the ML closure. End-to-end optimisation through the governing PDEs and the coupled implicit closure eliminates distribution shift, but both unrolled and implicit automatic differentiation fail on the stiff coupled solver. We derive adjoint equations that exploit the solver's implicit-explicit structure for efficient optimisation. On canonical square-duct and periodic-hill benchmarks, DARSM reduces average test velocity error over baseline RANS by $2$-$4\times$ across Reynolds number, geometries, and flow regimes, with peak case-level reductions of $12\times$. The model trained on attached, anisotropy-dominated flows (square duct) accurately generalises without retraining to separated flows (periodic hills), a regime change in the underlying physics. DARSM also outperforms five established ML methods: offline training, tensor-basis neural networks, field-inversion machine learning, DeepONets, and physics-informed neural networks.