Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structures

TL;DR

This paper introduces a two-stage, data-driven framework for learning the governing laws of complex generalized diffusions with non-gradient components, effectively recovering pseudo-potentials and rotations from noisy data.

Contribution

It presents a novel two-stage method combining energy laws and first-moment evolution to identify non-gradient drift structures in generalized diffusions.

Findings

Successfully recovers pseudo-potentials and rotations in complex diffusion processes

Demonstrates effectiveness on noisy and complex data

Applicable to dissipation-rotation dynamics and rough pseudo-potentials

Abstract

This paper proposes a data-driven learning framework for identifying governing laws of generalized diffusions with non-gradient components. By combining energy dissipation laws with a physically consistent penalty and first-moment evolution, we design a two-stage method to recover the pseudo-potential and rotation in the pointwise orthogonal decomposition of a class of non-gradient drifts in generalized diffusions. Our two-stage method is applied to complex generalized diffusion processes including dissipation-rotation dynamics, rough pseudo-potentials and noisy data. Representative numerical experiments demonstrate the effectiveness of our approach for learning physical laws in non-gradient generalized diffusions.