Results show that explicit residual compensation improves stable autoregressive prediction of unsteady flows and achieves the lowest mean rollout error among all compared methods.
Abstract
Stable prediction of unsteady flows requires accurate multiscale spatial representation and robust temporal propagation. We introduce the Compensated Koopman U-shaped Neural Operator (CoKo-UNO), which combines a U-shaped spectral backbone with Koopman-dominated latent propagation. Finite-dimensional Koopman truncation produces a state-dependent residual that is repeatedly reinjected during autoregressive rollout. CoKo-UNO models this residual with a selective state-space model (SSM), a principled input-dependent compensation mechanism, together with resolution-adaptive compensatory skip connections and an overlapping-warmup rollout strategy. \NEW{Across four benchmark problems, CoKo-UNO achieves the lowest mean rollout error among all compared methods. Its largest gain is a $76.76\%$ reduction relative to the strongest baseline, while requiring about $41.40\%$ of RNO's training time.} These results show that explicit residual compensation improves stable autoregressive prediction of unsteady flows.
Neural spectral PDE solvers often learn an entire unresolved vector field even when an inexpensive approximate model can already capture most of the trajectory. Here we introduce Perturbative-NeuSA, a residual formulation that decomposes the target solution into a low-fidelity background and a high-resolution perturbation, so that only the unresolved dynamics is learned. Starting from the exact perturbation equation, the method combines a fixed spectral operator, a background-dependent correction, the background defect in the target PDE, and an optional neural closure. This construction makes the roles of physical structure and neural closure separately measurable. Across 2D Burgers, Klein-Gordon, and heterogeneous 2D wave equations, the deterministic structured solver outperforms the trained NeuSA baseline while requiring no neural-network training. The largest gains occur on Burgers, where the deterministic correction reduces training and extrapolation errors by factors of 24 and 44, respectively. In addition, a Klein-Gordon sweep over seven background resolutions shows that the effect of the closure is conditional: it improves a poor background by 3.6 times, becomes neutral at intermediate resolutions, and degrades a well-resolved background. For the wave equation, however, the closure provides an additional 18% reduction when the remaining residual is interface-localized. Multi-initial-condition diagnostics further show that the useful closure regime depends on the initial-condition spectrum and can disappear in extrapolation when structured correction already captures the dominant Burgers dynamics. Perturbative-NeuSA therefore reframes neural closure as a conditional, diagnosable correction governed by background fidelity, residual organization, and compatibility with the closure model.
Accurately capturing unsteady flow dynamics remains challenging for physics-informed neural networks because nonlinear convection, multiscale structures, boundary effects, and accumulated temporal errors can degrade long-time prediction accuracy. This work proposes a physics-informed convolutional leaky-integrator recurrent network for capturing time-dependent flow fields. By combining local convolutions with learnable leaky memory, the proposed method provides a compact recurrent time-marching model with fewer parameters than multi-gate recurrent architectures. To support long-time prediction, the temporal domain is decomposed into overlapping windows coupled by consistency constraints, governing-equation residuals are evaluated using finite-difference differentiation matrices, and prescribed boundary conditions are enforced through hard constraints. The framework is evaluated on incompressible Navier–Stokes flows, including forced two-dimensional flow and the pre-merger interaction of two co-rotating vortices in a no-slip square cavity. Numerical results show that the proposed method reconstructs velocity and vorticity fields with low errors and provides temporally consistent solutions over the tested intervals. For the co-rotating-vortex case, vortex-core kinematics, vorticity statistics, circulation, restricted enstrophy, and the second moment of vorticity are evaluated. The proposed method captures the mutual rotation and gradual approach of the vortex cores and reproduces the principal trends in vorticity redistribution and restricted-enstrophy decay. Compared with the Convolutional Long Short-Term Memory variant, the proposed method achieves errors of the same order of magnitude while using fewer trainable parameters and requiring shorter runtimes in the main recurrent comparisons. Additional tests in Appendix B illustrate its applicability to selected nonlinear evolutionary partial differential equations.
Ziyi Zhen, Yan Zhang, Hui Xu· The Physics of Fluids· 0 citations
Unsteady aerodynamic phenomena, such as gusts, turbulence, and fluid-structure interactions affect an aircraft during flight. For design, optimisation and certification, it is indispensable to quantify such unsteady aerodynamic effects. Industry-standard computational fluid dynamics methods, such as solving the unsteady Reynolds-averaged Navier-Stokes equations or the linearized frequency domain method, are either computationally expensive or restricted by assumptions like linearity. Once trained, machine learning methods are capable of computing non-linear relationships very fast, making them suitable as surrogate models. By autoregressively applying graph neural networks (GNNs), operating on a discretised spatial domain, spatio-temporal predictions can be made. However, autoregressive GNNs suffer from error accumulation leading to unstable rollouts over time. Here we show that combining GNNs with augmented Neural Ordinary Differential Equations yields temporally stable predictions of the surface forces on a pitching airfoil. We found that our approach, called GNODE, based on Graph Neural Ordinary Differential Equations, provides temporally more stable, spatially smoother, and overall more accurate results than an autoregressive GNN baseline. Tests are conducted on a dataset consisting of a simulations of a pitching airfoil, including transonic shocks, transient behaviour and dynamic non-linearities. Augmenting GNODEs with additional latent dimensions improves the expressivity and accuracy by capturing underlying history effects. The developed method demonstrates an approach that is suitable to model non-linear spatio-temporal systems with exogenous inputs.
Henrik Lange, R. Thormann, P. Bekemeyer· 0 citations
Neural operators have shown strong potential for learning solution operators of partial differential equations (PDEs). However, long-horizon autoregressive prediction remains challenging: local errors accumulate as spectral inconsistency, phase misalignment, or mean drift. Existing methods mainly improve state representations and operator backbones, while leaving the repeatedly applied latent transition increment weakly structured, allowing spectral errors and unstable channel couplings to accumulate during rollout. To address these issues, we propose a geometry-aware incremental neural operator (GeoIncNO) for stable long-horizon PDE prediction. GeoIncNO predicts latent increments for residual advancement and uses lightweight low-rank projectors to regulate channel coupling within active frequency bands derived from the increment spectral energy distribution. To reduce physical-space reconstruction errors, GeoIncNO further introduces a mean--fluctuation decoupled reconstruction mechanism, where stable mean structures and dynamic fluctuations are fused separately, and phase correction is applied only to the zero-mean fluctuation component. Extensive experiments on six PDE benchmarks, covering 1D, 2D, and 3D dynamical systems, show that GeoIncNO achieves consistently strong prediction accuracy, improved rollout stability, and better spectral fidelity compared with competitive neural-operator baselines.
Jiaquan Zhang, Shuxu Chen, Haifan Meng et al.· 0 citations
Model-based active flow control requires predictive models that are accurate, stable, and fast enough for real-time optimisation. In controlled wake flows, this is often achieved through Reduced-Order Models (ROMs) that first compress high-dimensional velocity snapshots into a latent space and then learn a time- stepping predictor for the dynamics in the latent space. Here, we study how the choice of the spatial encoder affects the predictability of the resulting latent coordinates for wake flows under control inputs. Using two actuated 2D wake configurations, a simplified truck wake and the fluidic pinball, we compare Proper Orthogonal Decomposition (POD) against nonlinear Convolutional Autoencoders (CAEs) and two types of variational autoencoders for compression, and evaluate several temporal predictors based on Long Short-Term Memory networks. CAEs achieve higher compression efficiency and sharper short-term reconstructions, but they produce latent dynamics that are more irregular and with broadband spectral content. As a consequence, long-horizon forecasts degrade faster and show a higher probability of catastrophic divergence than POD-based models. POD yields smoother latent trajectories that are easier to learn and extrapolate, leading to more reliable predictions beyond the short- term regime. These results reveal a clear trade-off between compactness and forecast accuracy, and suggest that the stability of the latent dynamics prediction can outweigh maximal compression. This is particularly relevant for control strategies rooted in forecasts of the dynamics, such as model predictive control and reinforcement learning. The findings provide practical guidance for designing actuation-aware, hardware-feasible predictive ROMs for real-time flow control.
A. Solera-Rico, Patricia Garc'ia-Caspuenas, Carlos Sanmiguel Vila et al.· 1 citation
This paper presents an applied study of data-driven Model Predictive Control (MPC) based on the Koopman operator framework for a three-tank hydraulic benchmark. The central contribution is a compact, physics-informed lifting strategy: observable functions are chosen directly from Torricelli’s law governing turbulent orifice flow, yielding a seven-dimensional Extended Dynamic Mode Decomposition (EDMD) model that captures the dominant nonlinearities with fewer basis functions than generic dictionaries. The resulting Koopman-MPC replaces the nonconvex optimization of nonlinear MPC (NMPC) with a convex quadratic program, achieving comparable tracking accuracy (0.65 cm vs. 0.64 cm mean error) while reducing the average per-step computation time by 36.7× (50 ms vs. 1807 ms in Python/SciPy). A Moving Horizon Estimator (MHE) operating on the full nonlinear model reconstructs the unmeasured tank level with accuracy comparable to the 2 mm measurement noise floor. These results provide a quantitative benchmark for Koopman-based predictive control on a nonlinear hydraulic system with bidirectional inter-tank coupling and regime-dependent flow transitions.
Wilder Hernandez Manosalva, H. Ramirez-Murillo, D. Tellez-Castro· International Conference on...· 0 citations