Physics-aware deep learning (PADL) enables rapid prediction of complex physical systems, yet current convolutional neural network (CNN) architectures struggle with highly nonlinear flows. While scaling model size addresses complexity in broader AI, this approach yields diminishing returns for physics modeling. Drawing inspiration from Hybrid Lagrangian-Eulerian (HLE) numerical methods, we introduce deformable physics-aware recurrent convolutions (D-PARC) to overcome the rigidity of CNNs. Across Burgers' equation, Navier-Stokes, and reactive flows, D-PARC achieves superior fidelity compared to substantially larger architectures. Analysis reveals that kernels display anti-clustering behavior, evolving into a learned "active filtration" strategy distinct from traditional h- or p-adaptivity. Effective receptive field analysis confirms that D-PARC autonomously concentrates resources in high-strain regions while coarsening focus elsewhere, mirroring adaptive refinement in computational mechanics. This demonstrates that physically intuitive architectural design can outperform parameter scaling, establishing that strategic learning in lean networks offers a more effective path forward for PADL than indiscriminate network expansion.