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.

翻译：能量模型（EBMs）已被证明是有限维空间密度建模中极具效力的方法。其通过组合将领域特定选择与约束融入模型结构的能力，使EBMs成为物理学、生物学、计算机视觉及其他领域应用中的理想候选方案。近期，针对\textit{无条件}可交换数据（如点云）建模随机过程的能量过程（EBP）被提出。本文提出一种新型EBP子类$\mathcal{F}$-EBM，用于处理\textit{条件}可交换数据，其能通过有限点处评估的函数样本学习函数分布（如曲线或曲面）。函数场景存在两个独特挑战：首先，训练数据往往并非沿着固定点集进行评估；其次，必须采取措施控制模型在评估点之间的行为，以缓解过拟合。所提模型是基于函数空间的能量模型，采用谱分解方式，通过高斯过程路径测度重新加权分布，以捕捉被建模底层过程的平滑特性。该模型不仅能利用不规则采样的训练数据，还可输出任意分辨率下的预测，为函数数据的升尺度分析提供了有效方法。我们通过多个数据集验证了该方法的有效性，包括标普500指数（S&P）和英国国家电网数据。