An Adaptive Framework for Autoregressive Forecasting in CFD Using Hybrid Modal Decomposition and Deep Learning

TL;DR

This work tackles the high computational burden of long-horizon CFD simulations by introducing a fully data-driven adaptive framework that stabilizes autoregressive deep-learning forecasts through iterative retraining on newly generated data. It combines proper orthogonal decomposition with a forecasting DL model (POD-DL) to predict POD coefficients, and employs an adaptive loop with initial and retraining data pools, transfer learning, and explicit speedup metrics such as PPE and TCS. The approach is validated on laminar and turbulent flow regimes, including a cylinder wake and an isothermal jet, achieving substantial cost reductions (up to 95%) while maintaining physical fidelity, though performance degrades when distribution shifts are not captured in retraining. The work provides a practical path toward accelerating CFD workflows with data-driven surrogates, supported by open-source code and a clear framework for adapting to evolving flow dynamics across diverse applications.

Abstract

This work presents, to the best of the authors' knowledge, the first generalizable and fully data-driven adaptive framework designed to stabilize deep learning (DL) autoregressive forecasting models over long time horizons, with the goal of reducing the computational cost required in computational fluid dynamics (CFD) simulations.The proposed methodology alternates between two phases: (i) predicting the evolution of the flow field over a selected time interval using a trained DL model, and (ii) updating the model with newly generated CFD data when stability degrades, thus maintaining accurate long-term forecasting. This adaptive retraining strategy ensures robustness while avoiding the accumulation of predictive errors typical in autoregressive models. The framework is validated across three increasingly complex flow regimes, from laminar to turbulent, demonstrating from 30 \% to 95 \% reduction in computational cost without compromising physical consistency or accuracy. Its entirely data-driven nature makes it easily adaptable to a wide range of time-dependent simulation problems. The code implementing this methodology is available as open-source and it will be integrated into the upcoming release of the ModelFLOWs-app.

Figures (15)

• Figure 1: Variation of the learning rate using the CosineAnnealingWarmRestarts scheduler lr_schedule (left) and the corresponding loss function values (right) during a training process of 1500 epochs. The learning rate decays from a maximum value of $10^{-3}$ to a minimum value of $10^{-6}$.
• Figure 2: Visual representation of the adaptive framework proposed in this work. Initially, a set of $S_{0}$ snapshots is extracted from a precomputed dataset to train the POD-DL model. Within this model, POD is applied to obtain the corresponding POD coefficients, which are then used to train a deep learning model, for $E$ epochs. This trained model forecasts $P$ future snapshots, after which an additional set of $S_{1}$ snapshots is drawn from the dataset to retrain the POD-DL model, with the same number of epochs. During each retraining step, the DL model leverages transfer learning by initializing its weights with those obtained from the previous training phase. This adaptive procedure is repeated iteratively until predictions cover the entire temporal domain of interest.
• Figure 3: Percentage of energy contained, Eg$(k)$, when the truncation is done at the $k$, up to $20$, most energetic modes in the laminar flow past a cylinder.
• Figure 4: Snapshots comparing the ground truth (GT) vorticity, of the laminar flow past a cylinder, with the one predicted from POD-DL, at some representative time steps for test cases T1 (a), T2 (b) and T3 (c), respectively. Here $t_{S}$ indicates the snapshot index rather than the actual time $t$.
• Figure 5: The top row of all figures displays the probability of achieving an absolute vorticity prediction error below thresholds of 0.5 (orange) and 1 (green) for the three test cases: T1 (a), T2 (b), and T3 (c), in the laminar flow. The bottom row presents the temporal evolution of the spatially averaged streamwise velocity. These error thresholds are selected based on the typical range of vorticity values identified in the flow, as evident in Fig. \ref{['fig: cil_laminar_snap']}.