Convolutional autoencoder-augmented sparse identification of noisy nonlinear fluid dynamics from unlabelled data

Understanding complex flow dynamics from noisy, unlabeled data remains a fundamental challenge in fluid mechanics. This study introduces a method called convolutional autoencoder-augmented sparse identification of noisy nonlinear dynamics (CAE-SINNDy) to address this issue. CAE-SINNDy combines a convolutional autoencoder (CAE) with SINNDy in a closed-loop architecture. The CAE uses deep two-dimensional convolutional layers to extract dominant flow structures from high-dimensional, noise-contaminated input data, producing compact latent representations. These representations are then used by the SINNDy module to identify simple, physically interpretable equations that govern the underlying dynamics. Both components are trained jointly: SINNDy guides the CAE toward extracting dynamics-relevant features, while the CAE provides denoised latent variables that improve sparse model recovery. An additional loss term is introduced into the optimization objective to further align the latent dynamics with the observed flow fields. This regularization promotes a balanced trade-off between spatial reconstruction accuracy and dynamical fidelity. The inherent sparsity of the SINNDy component also contributes to denoising by filtering residual noise that was not eliminated during the CAE's encoding process. As a fully unsupervised approach, CAE-SINNDy requires no labeled data, making it well-suited for real-world flow diagnostics such as particle image velocimetry (PIV). Validation on both synthetic and experimental noisy PIV datasets demonstrates that CAE-SINNDy can accurately identify key flow features even under high-noise conditions. Statistical analyses—such as proper orthogonal decomposition energy distributions and coefficient time evolutions—confirm the physical consistency and robustness of the learned dynamics. Overall, CAE-SINNDy provides a label-free, noise-resilient framework for interpretable flow modeling, with strong potential for applications in turbulence control and the design of real-world fluid systems.