EVODMs: variational learning of PDEs for stochastic systems via diffusion models with quantified epistemic uncertainty

We present Epistemic Variational Onsager Diffusion Models (EVODMs), a machine learning framework that integrates Onsager's variational principle with diffusion models to enable thermodynamically consistent learning of free energy and dissipation potentials (and associated evolution equations) from noisy, stochastic data in a robust manner. By further combining the model with Epinets, EVODMs quantify epistemic uncertainty with minimal computational cost. The framework is validated through two examples: (1) the phase transformation of a coiled-coil protein, modeled via a stochastic partial differential equation, and (2) a lattice particle process (the symmetric simple exclusion process) modeled via Kinetic Monte Carlo simulations. In both examples, we aim to discover the thermodynamic potentials that govern their dynamics in the deterministic continuum limit. EVODMs demonstrate a superior accuracy in recovering free energy and dissipation potentials from noisy data, as compared to traditional machine learning frameworks. Meanwhile, the epistemic uncertainty is quantified efficiently via Epinets and knowledge distillation. This work highlights EVODMs' potential for advancing data-driven modeling of non-equilibrium phenomena and uncertainty quantification for stochastic systems.