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Astrophysical observations taken from Earth are subject to weather, environmental, and scientific constraints that lead to sparse, irregular light curves. On the eve of the Vera C. Rubin Observatory Legacy Survey of Space and Time, its massive dataset offers unprecedented opportunities for transient science. Yet, a key challenge remains its cadence, which will be sparse and irregular across six bands, limiting scientific inference. Interpolating light curves helps mitigate this, with Gaussian Processes being the standard, but they struggle with cross-band correlations, require an a priori kernel specification, and must be fit to each light curve individually and hence scale poorly. Here, we introduce the neural process family for light curve reconstruction, combining the probabilistic framework of Gaussian Processes with the scalability of deep learning. By meta-learning on diverse simulated transients, Attentive Neural Processes shift the bulk of the computational cost to training, enabling rapid, amortized inference with a single, class-agnostic model. Evaluated on realistic Rubin cadences across 15 transient classes, Attentive Neural Processes consistently outperform all benchmarks - a suite of Gaussian Processes and neural networks on every tested metric, spanning regression quality, astrophysical feature recovery, and probabilistic calibration. Our model interpolates all bands simultaneously in microseconds, over four orders of magnitude faster than the next-best neural benchmark and five faster than Gaussian Processes, making them suitable for the nightly LSST alert stream. Attentive Neural Processes avoid the overconfidence of standard neural networks and the underconfidence of Gaussian Processes, delivering sharp, well-calibrated uncertainties. This work establishes the neural process family as a scalable, probabilistic foundation for real-time transient science in the Rubin era.