Energy-Based Operator Learning in Function Space

We propose Energy-Based Operators (EBOs), an architecture-agnostic framework for learning conditional distributions over functions on continuous domains. An EBO defines a scalar energy over target functions given an input function and induces a probability model through a Gaussian reference. The resulting score field is obtained as the gradient of the parametrized energy to perform function-space iterative energy minimization (EM). Our model shows strong performance in function generation, such as super-resolution and forecasting, over various 1D function classes (oscillations, damping, and Izhikevich) in comparison with prediction operators and denoising operators over various architecture backbones. Moreover, it achieves strong performance over PDEs, namely Navier-Stokes, Darcy flow and Burgers. Notably, our model successfully detects anomalous functions by automatically assigning high energy without any supervision. It enables seizure detection and volatility prediction after learning neural dynamics and market microstructure dynamics without pre-defined labels during training, highlighting its effectiveness for both learning dynamical systems and detecting functional anomalies arising in scientific simulations.