Physical AI at the edge -- enabling autonomous systems to understand and predict real-world dynamics in real time -- requires hardware-efficient learning and inference. Model recovery (MR), which identifies governing equations from sensor data, is a key primitive for safe and explainable monitoring in mission-critical autonomous systems operating under strict latency, compute, and power constraints. However, state-of-the-art MR methods (e.g., EMILY and PINN+SR) rely on Neural ODE formulations that require iterative solvers and are difficult to accelerate efficiently on edge hardware. We present \textbf{MERINDA} (Model Recovery in Reconfigurable Dynamic Architecture), an FPGA-accelerated MR framework designed to make physical AI practical on resource-constrained devices. MERINDA replaces expensive Neural ODE components with a hardware-friendly formulation that combines (i) GRU-based discretized dynamics, (ii) dense inverse-ODE layers, (iii) sparsity-driven dropout, and (iv) lightweight ODE solvers. The resulting computation is structured for streaming parallelism, enabling critical kernels to be fully parallelized on the FPGA. Across four benchmark nonlinear dynamical systems, MERINDA delivers substantial gains over GPU implementations: \textbf{114$\times$ lower energy} (434~J vs.\ 49{,}375~J), \textbf{28$\times$ smaller memory footprint} (214~MB vs.\ 6{,}118~MB), and \textbf{1.68$\times$ faster training}, while matching state-of-the-art model-recovery accuracy. These results demonstrate that MERINDA can bring accurate, explainable MR to the edge for real-time monitoring of autonomous systems.