  Energy-Based Models (EBMs) have proven to be a highly effective approach for
modelling densities on finite-dimensional spaces. Their ability to incorporate
domain-specific choices and constraints into the structure of the model through
composition make EBMs an appealing candidate for applications in physics,
biology and computer vision and various other fields. Recently, Energy-Based
Processes (EBP) for modelling stochastic processes was proposed for
\textit{unconditional} exchangeable data (e.g., point clouds). In this work, we
present a novel subclass of EBPs, called $\mathcal{F}$-EBM for
\textit{conditional} exchangeable data, which is able to learn distributions of
functions (such as curves or surfaces) from functional samples evaluated at
finitely many points. Two unique challenges arise in the functional context.
Firstly, training data is often not evaluated along a fixed set of points.
Secondly, steps must be taken to control the behaviour of the model between
evaluation points, to mitigate overfitting. The proposed model is an energy
based model on function space that is decomposed spectrally, where a Gaussian
Process path measure is used to reweight the distribution to capture smoothness
properties of the underlying process being modelled. The resulting model has
the ability to utilize irregularly sampled training data and can output
predictions at any resolution, providing an effective approach to up-scaling
functional data. We demonstrate the efficacy of our proposed approach for
modelling a range of datasets, including data collected from Standard and
Poor's 500 (S\&P) and UK National grid.
