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

Sammanfattning

Surrogate models for molecular dynamics, particularly those based on generative artificial intelligence, offer an efficient way to model molecular systems across timescalesthat may be difficult to access through simulation. However, such models should remain consistent with the underlying physics. For Markovian dynamics, the ChapmanKolmogorov equation is a cornerstone of this consistency, describing how transitiondynamics across different timescales should relate to each other. One such surrogatemodel, the Implicit Transfer Operator (ITO) framework, learns transition dynamicsacross 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 workinstead evaluates distributions in latent space, enabling metrics that were previouslyunavailable.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 botha one-dimensional model trained on the dynamics from a potential well and athree-dimensional transferable model trained on molecular dynamics data. The onedimensional model passes the test consistently, while the three-dimensional modelgives more uncertain results, leading to a discussion about both the model and themultivariate version of the test. We further show that the CK-test’s performanceimproves alongside the learning of correct dynamics during training, suggesting thatthe semigroup property is learned rather than being inherent to the model architecture. However, passing the test does not guarantee that the model has learnedthe correct dynamics, as models with poor dynamical accuracy can still satisfy theCK-test.