Structure-Aware Variational Learning of a Class of Generalized Diffusions

2026-04-22

AI Summary

This work addresses the challenge of learning the latent energy function of stochastic gradient systems from partial and noisy observations by proposing a novel approach grounded in energy-variational structure. By leveraging the energy dissipation law associated with the Fokker–Planck equation, the method constructs an energy-driven loss functional centered on the De Giorgi dissipation functional, jointly modeling the free energy and dissipation mechanism without explicitly fitting partial differential equations. This is the first framework to integrate energy–dissipation structure into the learning of diffusion processes, preserving the system’s variational structure while significantly enhancing robustness to noise, observation duration, and data volume. Numerical experiments across one- to three-dimensional systems demonstrate that the approach consistently reconstructs energy functions with high accuracy and stability under diverse conditions.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Learning the underlying potential energy of stochastic gradient systems from partial and noisy observations is a fundamental problem arising in physics, chemistry, and data-driven modeling. Classical approaches often rely on direct regression of governing equations or velocity fields, which can be sensitive to noise and external perturbations and may fail when observations are incomplete. In this work, we propose a structure-aware, energy-based learning framework for inferring unknown potential functions in generalized diffusion processes, grounded in the energetic variational approach. Starting from the energy-dissipation law associated with the Fokker-Planck equation, we construct loss functions based on the De Giorgi dissipation functional, which consistently couple the free energy and the dissipation mechanism of the system. This formulation avoids explicit enforcement of the governing partial differential equation and preserves the underlying variational structure of the dynamics. Through numerical experiments in one, two, and three dimensions, we demonstrate that the proposed energy-based loss exhibits enhanced robustness with respect to observation time, noise level, and the diversity and amount of available training data. These results highlight the effectiveness of energy-dissipation principles as a reliable foundation for learning stochastic diffusion dynamics from data.

Problem

• potential energy
• stochastic gradient systems
• generalized diffusion
• noisy observations
• data-driven modeling

Innovation

• energy-based learning
• variational structure
• De Giorgi dissipation
• generalized diffusion
• structure-aware modeling