Learning Effective SDEs from Brownian Dynamics Simulations of Colloidal Particles

The paper empirically derives and validates an EM-based Gaussian loss for learning parameter-dependent eSDEs in a Diffusion Maps latent space, compares it to Kramers–Moyal (KM) estimation, observes that the neural network is typically more accurate with fewer data, and shows free-energy trends with increasing voltage; all of these appear correct within the stated scope. The model’s solution supplies a rigorous quasi-MLE interpretation of the EM loss, KM consistency and rate sketches, a coupling/BDG–Grönwall trajectory-discrepancy bound, and sufficient conditions for parameter monotonicity and free-energy scaling—results consistent with and extending the paper’s claims. The two accounts align but differ in rigor: the paper is empirical and qualitative; the model adds formal assumptions and theoretical guarantees.