Autoencoders with Energy-Based Kernels for dimensionality Reduction in Viscoelastic Flow
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Dimensionality reduction is widely applied to the study of Newtonian fluids, primarily through classical techniques like Principal Component Analysis (PCA), also referred to as Proper Orthogonal Decomposition (POD). However, its application to non-Newtonian fluids has been comparatively limited, despite the growing need to efficiently analyze the complex behaviors exhibited by these fluids. Recently, deep learning strategies such as Autoencoders and kernel methods have emerged as promising techniques for fluid dimensionality reduction. However, most research still focuses on Newtonian fluids. In this work, we combine Variational Autoencoders (VAEs) with kernel-based methods by incorporating a viscoelastic-specific kernel [1]. This kernel, derived from the viscoelastic model, defines an energy-based Reproducing Kernel Hilbert Space (RKHS) that preserves energy distances between simulations, offering a tailored approach for viscoelastic fluid analysis. We introduce mechanical energy as a metric for the reconstruction of the flow field to ensure physically coherent results. This method directs the autoencoder to focus on meaningful information during training, addressing a key limitation of traditional Mean Squared Error (MSE) loss, which can yield seemingly accurate reconstructions but is inaccurate energy-wise.
