Accurate real-time estimation of key epidemic parameters, particularly the effective reproduction number (Rₜ), is critically undermined by the pervasive and variable issue of case under-reporting in public health surveillance data, leading to biased models and flawed policy insights. To address this fundamental problem of unobserved true incidence, we develop a novel hybrid computational framework that synergistically integrates a mechanistic stochastic epidemiological model with a data-driven deep learning corrector. Our methodology first constructs a stochastic Susceptible–Exposed–Infectious–Removed (SEIR) model where the time-varying transmission rate is modeled as a flexible latent function using a Gaussian Process. The core innovation is the seamless coupling of this model to a Temporal Convolutional Network (TCN) module, which is trained jointly via amortized variational inference to learn the complex, non-stationary mapping between the model's simulated true incidence and the officially reported cases, thereby explicitly correcting for biases stemming from fluctuating testing capacity, healthcare access, and reporting behavior. Applied to COVID-19 case and mortality data from Italy, Germany, and France, our hybrid model significantly improved the accuracy and stability of Rₜ estimation, reducing 14-day-ahead forecast error for hospital admissions by 40% compared to a pure stochastic SEIR model and outperforming standard Bayesian filtering techniques. The TCN module provided interpretable, time-varying reporting probabilities, identifying periods of severe under-reporting (up to 70% correction) that correlated strongly with independent indicators of testing coverage. This paradigm successfully merges principled epidemiological theory with flexible machine learning, creating a robust tool for real-time situational awareness that provides reliable estimates of true transmission dynamics from incomplete data, with direct applicability to managing future outbreaks of emerging infectious diseases.