This paper presents a novel neural network-based framework for compressing physical field data defined on finite element meshes, aimed at constructing efficient surrogate models for numerical simulations. It addresses a critical technical challenge: when spatially correlated mesh data with complex geometric boundaries are vectorized as input to a neural network, the disruption of local adjacency causes artificial high-frequency oscillations, which prevent conventional autoencoders from converging during training and result in prohibitively high reconstruction errors. Our method overcomes this by integrating a tailored pre-compression stage with an autoencoder. The core innovation is based on a key mathematical insight: while the raw vectorized data are oscillatory, their projection coefficients onto a Proper Orthogonal Decomposition (POD) basis exhibit rapid spectral decay. Applying an inverse Fast Fourier Transform (iFFT) to these spectrally decaying coefficients inherently transforms them into a low-frequency, smooth representation. We architect a multi-layer process to systematically exploit this principle. By iteratively applying POD, followed by zero-padding, windowing, and real/imaginary separation of the iFFT output, the method progressively conditions the data and transforms the problematic high-frequency input into a neural-network-friendly representation across multiple channels. The subsequent autoencoder then performs efficient nonlinear compression on this conditioned data. Our experiments demonstrate that the framework enables stable and convergent training, whereas directly applying standard architectures like MLP or Convolutional Autoencoders to raw vectorized mesh data fails to converge, yielding reconstruction errors close to 100%. Furthermore, it significantly outperforms conventional linear POD, reducing reconstruction errors by approximately 50% for both static and dynamic systems. This work not only provides an effective compression pathway for high-fidelity finite element simulation data with complex geometries but also establishes a crucial bridge between the inherently structured format of finite element mesh data and the vector-based processing paradigm of deep learning. It facilitates broader integration of scientific computing and neural network methodologies, and enables efficient surrogate modeling that depends on data compression.