Hybrid Physics-Aware Sparse Neural Networks (Hy-PAS): A Unified Framework For Learning and Solving PDES

We introduce a modified version of the Hybrid Physics-Aware Sparse Neural Network (Hy-PAS), designed to tackle both ordinary and partial differential equations (PDEs). The approach blends classical numerical reasoning with modern deep learning, offering a sparse and interpretable framework that respects the underlying physics. Rather than treating PDE solutions as purely data driven, Hy-PAS reinterprets traditional mesh-free representations through a neural network perspective. In doing so, it bridges the gap between dense neural formulations such as Physics-Informed Neural Networks (PINNs) and established mesh-free numerical schemes. What makes Hy-PAS distinctive is that its parameters correspond directly to physical quantities like node locations, kernel widths, and basis coefficients. This connection allows the model to represent mesh adaptivity naturally and to handle steep gradients or discontinuities with improved stability and accuracy. Hy-PAS is sparse, which means it needs a lot fewer trainable parameters than fully linked networks, which are often used to approximate PDEs. We also show that classical representations, including Fourier and wavelet expansions, emerge as special cases of the proposed architecture, situating Hy-PAS within a broader family of physics-structured neural operators. Extensive numerical experiments with elliptic, parabolic, hyperbolic, and nonlinear PDEs, as well as benchmarks in fluid dynamics, demonstrate the accuracy, robustness, and computational efficiency of Hy-PAS. The framework lays out a mathematically sound way to create neural solvers for scientific computing that are easy to understand and can handle large amounts of data.