Inverse problems (IPs) deal with the task of reconstructing input variables from noisy observations, defined through a forward model. When the forward map is known, the reconstruction of input variables can be performed via numerous approaches, including optimization-based algorithms (e.g., maximum likelihood estimation) and sampling-based algorithms (e.g., Markov chain Monte Carlo). However, there are scenarios where: (1) we do not have perfect knowledge about the forward model; or (2) the forward model is expensive to simulate, which severely limits the implementation of optimization and sampling algorithms that may require multiple forward model simulations. Therefore, in these scenarios, it is essential to approximate the forward model by a parameterized surrogate model. We propose data-driven approaches that jointly learn the parameters of the surrogate model, and reconstruct input variables, from observation data alone. Data assimilation (DA) deals with the task of reconstructing temporally evolving hidden states from noisy time series observations, defined through a state space model (SSM). When the SSM is known, the reconstruction of states can be performed using optimization-based algorithms (e.g., 4DVAR) or sampling-based algorithms (e.g., particle filtering). However, similar to IPs, there are scenarios where: (1) we do not have perfect knowledge about the SSM; or (2) the SSM is expensive to simulate, which increases the computational cost of the reconstruction algorithms. Therefore, in these scenarios, it is essential to approximate the SSM with a parameterized surrogate model. We employ machine learning techniques and propose data-driven approaches that jointly learn the parameters of the surrogate model, and reconstruct the states, from observation data alone.