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ABSTRACT: Inverse problems and terminal-time value matching often require repeated forward solves while updating unknown parameters or sources. This motivates fast surrogate models that learn the PDE solution operator. We model families of solution trajectories across parameter choices using a hyperPINN with a WGAN-based framework. For label-scarce inverse problems, we introduce Physics-Informed Deep Inverse Operator Networks (PI-DIONs), which use physics-based losses for label-free training while remaining competitive when supervision is available. For dynamical prediction on irregular observation grids, we propose GraphDeepONet, an autoregressive graph-based operator network that combines DeepONet with message passing to enable stable prediction on arbitrary irregular grids.