Enhancing computational efficiency in multiscale systems using deep learning of coordinates and flow maps

The balance between compactness and forecast accuracy of data-driven latent-space reduced-order models in controlled wake flows

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.

Physics-informed multi-scale spatiotemporal learning for transient incompressible flow prediction

Data-driven learning provides a promising route for accelerating computational fluid dynamics, yet transient flow prediction remains challenging because models must simultaneously capture evolving temporal dynamics and cross-scale spatial structures while generalizing across geometries and operating conditions. In this work, we study one-step-ahead prediction of two-dimensional incompressible transient flows from historical snapshots and propose Multi-Scale Spatiotemporal Flow (MuST-Flow), a physics-informed multi-scale spatiotemporal network for recurrent forecasting of velocity and pressure fields. The central idea of MuST-Flow is to couple cross-scale feature extraction with physical regularization, enabling the model to better resolve localized high-gradient flow structures while maintaining physically consistent evolution. Specifically, MuST-Flow adopts a multi-scale spatiotemporal convolutional design with dilated receptive fields to learn hierarchical flow representations and incorporates a hybrid loss derived from the residuals of the incompressible Navier–Stokes equations to encourage mass and momentum consistency during training. To evaluate generalization beyond fixed configurations, the proposed method is assessed on transient hydrofoil flow data under cross-geometry testing with varied operating conditions. Experimental results show that MuST-Flow achieves the lowest mean absolute error, highest structural similarity, and lowest speed and velocity-direction errors among both generic spatiotemporal prediction baselines and flow-oriented neural-operator baselines, while maintaining competitive divergence and Navier–Stokes residuals. These results demonstrate its effectiveness as an efficient surrogate modeling framework for two-dimensional incompressible transient flow prediction.

Spectral-Informed Neural Networks Outperform Spectral Methods in High-dimensional PDEs

For low-dimensional problems ($d\leq3$), spectral methods can achieve exceptionally high accuracy. For middle-dimensional problems ($4 \leq d \lesssim 10$), spectral methods remain feasible through specific techniques such as sparse grids or hyperbolic cross. However, for high-dimensional problems ($d\gg 10$), spectral methods suffer frome the curse of dimensionality. Physics-informed neural networks (PINNs) have emerged as a promising approach to overcome this challenge, offering scalability to high dimensions, but often suffer from limited accuracy and efficiency. Recently proposed spectral-informed neural networks (SINNs) combine spectral methods with PINNs, operating directly in the spectral domain to avoid spatial derivative computations and to reduce memory consumption. In this work, we introduce Modified SINNs, which integrate coefficient decay scaling and basis embeddings motivated by harmonic analysis to enhance accuracy in high-dimensional problems and enable accurate approximation of unknown spectral coefficients. Numerical experiments on steady and time-dependent partial differential equations demonstrate that Modified SINNs outperform sparse grid spectral methods on middle-dimensional problems with incomplete spectral information and achieve superior accuracy compared to PINNs on high-dimensional problems.

Multi-scale physics-informed neural networks with Fourier features for approximating time-fractional PDEs

Decoder-free operator autoencoder for reduced-order modeling of dynamical systems

A Fourier-enhanced operator autoencoder for decoder-free reconstruction and latent learning of dynamical systems and achieves accuracy comparable to or better than classical AE-based reduced-order models while providing a more efficient latent-to-field reconstruction path.

Encoding multiphase dynamics to predict spatiotemporal evolution via latent-space operators

Modeling complex multiphase flows relies on solving partial-differential equations (PDEs) that capture the intricate transfers of mass, momentum, and energy among the interacting phases. These systems typically involve intricate couplings between gas, liquid, and solid phases, exhibiting dynamics that diverge from those of single-phase flows. We propose a two-stage decoupled training paradigm that independently optimizes a Kolmogorov–Arnold autoencoder for spatial reduced-order representation and a latent-space operator network for initial-to-future dynamics. Systematic comparisons of bubble rise, particle deposition, and fluidized bed demonstrate that the width of layer dominates encoder fidelity, whereas operator accuracy depends on balancing expressiveness and overfitting. In addition, compared with fully connected and convolutional baselines, MultiOKAN reduces the reconstruction error by a significant margin while maintaining a similar computational cost. Latent projection also endows the model with strong resilience to noise and modest data budgets, preserving accuracy even under substantial perturbations or when trained on a fraction of the original samples. The proposed reduced-order and fusion schemes for multiphase problems characterize the accuracy-efficiency trade-off. These schemes enable either lightweight training through shared latent reconstruction or higher-fidelity prediction via enhanced cross-phase coupling in latent dynamics.