Nonlinear and Hierarchical Discovery of Slow Molecular Modes using Sequential Learning with Natural Constraints

Discovering the slow modes governing molecular dynamics can unveil new mechanistic understanding and provide collective variables along which to direct enhanced sampling. Time-lagged independent component analysis (tICA) is a well-developed method that discovers linear combination of molecular features as slow modes. The linearity of tICA, however, hampers its capacity to discover nonlinear modes. Nonlinear feature engineering can prove profitable but is typically reliant on human intuition or expensive preprocessing. Kernel tICA integrates tICA and kernel trick to effect nonlinear discovery, but is computationally expensive and selection and tuning of the kernel limits its generalizability. Time-lagged autoencoders and variational dynamics encoders are neural networks that can identify the slowest mode but are unable to resolve higher-order modes. In this work, we introduce a sequential learning method that we term hierarchical dynamics encoder (HDE) as a novel neural network that sequentially learns hierarchical nonlinear slow CVs. Each CV is orthogonal to all previously learned CVs, and orthogonality is imposed naturally without regularization. We demonstrate HDEs for several toy systems where the true slow modes are known, and in simulations of peptides and proteins.