Testing the Semigroup Property of Generative Models for Dynamical Systems - Developing a test based on the Chapman–Kolmogorov equation

Abstract

Surrogate models for molecular dynamics, particularly those based on generative artificial intelligence, offer an efficient way to model molecular systems across timescales that may be difficult to access through simulation. However, such models should remain consistent with the underlying physics. For Markovian dynamics, the Chapman Kolmogorov equation is a cornerstone of this consistency, describing how transition dynamics across different timescales should relate to each other. One such surrogate model, the Implicit Transfer Operator (ITO) framework, learns transition dynamics across multiple timescales, making it natural to question whether the learned dynamics remain consistent. Existing methods to assess this quantitatively use comparisons of distributions in the molecular space, while the test proposed in this work instead evaluates distributions in latent space, enabling metrics that were previously unavailable.

In this thesis, we develop and evaluate a Chapman–Kolmogorov test for ITO models operating in the latent space of the model. The test is evaluated on both a one-dimensional model trained on the dynamics from a potential well and a three-dimensional transferable model trained on molecular dynamics data. The one dimensional model passes the test consistently, while the three-dimensional model gives more uncertain results, leading to a discussion about both the model and the multivariate version of the test. We further show that the CK-test’s performance improves alongside the learning of correct dynamics during training, suggesting that the semigroup property is learned rather than being inherent to the model architecture. However, passing the test does not guarantee that the model has learned the correct dynamics, as models with poor dynamical accuracy can still satisfy the CK-test.