Hybrid Physics-Aware Sparse Neural Networks (Hy-PAS): A Unified Framework for Learning and Solving PDEs
Abstract
We introduce Hybrid Physics-Aware Sparse Neural Networks (Hy-PAS), a framework for solving ordinary and partial differential equations that reinterprets classical meshless partition-of-unity approximation as a structurally sparse neural network. Each trainable parameter of Hy-PAS corresponds to a concrete physical or numerical quantity a node position, a kernel width, or a nodal amplitude so that the trained model is directly interpretable and its basis adapts to the solution during optimization. The construction recovers radial-basis-function collocation as a special case when node positions and widths are frozen, and connects to physics-informed neural networks (PINNs), the Deep Ritz method, and finite-basis PINNs. We implement Hy-PAS together with a dense PINN and the Sparse, Physics-based, and partially Interpretable Neural Network (SPINN) in a common PyTorch training and evaluation harness, and benchmark all three on seven problems spanning ordinary, elliptic, parabolic, hyperbolic, and nonlinear PDEs, including the lid-driven cavity flow at Re=100. Across three random seeds per configuration, Hy-PAS attains the lowest relative L2 error on four of the seven benchmarks using roughly an order of magnitude fewer trainable parameters than the dense PINN, while SPINN is most accurate on the Helmholtz and heat problems and PINN on the viscous Burgers problem. A parameter-scaling study shows Hy-PAS reaching ∼10−4 relative L2 error with a few hundred parameters and saturating,…
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