Continuous-time functional diffusion processes demonstrated great potential in the generation of resolution-invariant data and in generalising diffusion models to different data types. However, training these models is challenging due to the high number of hyperparameters and the instability of the training process. Meta-learning is the dominant approach for training this kind of models. Two sets of parameters are used, where one specialises to the task at hand, while the other is computed at inference time to adapt the network for the current datum through few iterations of stochastic gradient descent. In this thesis, we propose a more stable approach to infer functional representations of data and avoid the pitfalls of meta-learning.