Solving inverse problems in dynamical systems governed by high-dimensional coupled ordinary differential equations (ODEs) is a ubiquitous challenge in scientific machine learning. In many real-world applications, researchers seek to uncover unknown parameters or model unknown dynamics even as the underlying physics is only partially characterized, and observations are sparse and limited to specific measurable channels. While physics-informed neural networks (PINNs) are ideal for inverse inference under partial observability, existing PINNs typically rely on task-specific joint optimization, which suffers from optimization difficulties and poor generalization. In this paper, we propose a meta-inverse physics-informed neural network (MI-PINN) that reformulates inverse modeling as a two-stage meta-learning problem. MI-PINN first learns a physics-aware representation across multiple tasks, and then performs inverse modeling by optimizing task-specific unknowns while keeping the learned representation fixed. This two-stage formulation significantly reduces the parameter search dimension, thereby improving sample efficiency and enabling accurate inference. To handle multi-scale dynamics common in these high-dimensional ODE systems, we further introduce an adaptive clustering-based multi-branch learning scheme. We demonstrate the effectiveness of MI-PINN on whole-body physiologically based pharmacokinetic (PBPK) models with up to 33 coupled ODEs, using paracetamol and theophylline under intravenous and oral dosing scenarios. Experimental results show that MI-PINN enables accurate recovery of masked kinetic parameters and reconstruction of missing mechanistic terms despite limited clinical observations.