Developing neural surrogates for unsteady flow that generalize across complex geometries and maintain accuracy over extended rollouts remains a critical challenge for accelerating flow simulations. We present a time-dependent, geometry-aware Deep Operator Network that predicts velocity fields for moderate-Re flows around parametric and non-parametric shapes. The model encodes geometry via a signed distance field (SDF) trunk and flow history via a CNN branch, and is trained using 841 high-fidelity simulations from the FlowBench dataset (https://baskargroup.bitbucket.io/). The model attains $\sim 5\%$ relative L2 single-step error and up to 1000x speedups over CFD on 262 test geometries not used for training. We provide physics-centric rollout diagnostics, including probe-based phase lag and Strouhal frequency analysis, to quantify long-horizon fidelity. These reveal accurate near-term transients but systematic error accumulation in fine-scale wakes, most pronounced for sharp-cornered geometries. We analyze failure modes and outline practical mitigation strategies including physics-informed regularization and diffusion-based refinement. Code, splits, and scripts are openly released at: https://github.com/baskargroup/TimeDependent-DeepONet to support reproducibility and benchmarking.

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