Transition Path Sampling Using Koopman Operators and Exit-Time Optimal Control

Studying the dynamical behavior of a system often depends on characterizing how it transitions between long-lived states. Because such transitions are rare, observing them usually requires specialized enhanced sampling techniques. Transition Path Sampling (TPS) is a well-established method for generating reactive trajectories, which is simple to implement and does not require the definition of a preconceived reaction coordinate. However, its efficiency is limited by its sequential nature and the resulting correlations between sampled paths. Previous work addressed this limitation by combining TPS with a sampling scheme based on conditioned Boltzmann Generators, a generative machine learning model capable of sampling a given target probability distribution. This approach produces uncorrelated transition paths but relies on an accurate reaction coordinate, which is rarely known in advance. Building on recent advances in committor learning, specifically on the Artificial Intelligence for Molecular Mechanism Discovery (AIMMD) method, in this work we introduce GenAIMMD, an iterative algorithm that actively and self-consistently learns the ideal reaction coordinate (the committor) and trains a conditioned Boltzmann Generator to sample from arbitrary bias windows along it. GenAIMMD thereby provides a correlation-free and fully parallelizable path sampling scheme that does not require prior knowledge of the system's transition mechanism. We apply GenAIMMD to a two-dimensional toy model and a higher-dimensional polymer system. In both cases, GenAIMMD succeeds in training the Boltzmann Generator and learning the committor. Benchmark results show a substantial increase in performance compared to standard TPS.

This paper presents an efficient model predictive path integral (MPPI) control framework for systems with complex nonlinear dynamics. To improve the computational efficiency of classic MPPI while preserving control performance, we replace the nonlinear dynamics used for trajectory propagation with a learned linear deep Koopman operator (DKO) model, enabling faster rollout and more efficient trajectory sampling. The DKO dynamics are learned directly from interaction data, eliminating the need for analytical system models. The resulting controller, termed MPPI-DK, is evaluated in simulation on pendulum balancing and surface vehicle navigation tasks, and validated on hardware through reference-tracking experiments on a quadruped robot. Experimental results demonstrate that MPPI-DK achieves control performance close to MPPI with true dynamics while substantially reducing computational cost, enabling efficient real-time control on robotic platforms.

This paper proposes a novel data-driven algorithm to approximate the dominant eigenfunctions (aka.~modes) of the Koopman operator of nonlinear dynamical systems using neural networks. The relevance of learning the dominant Koopman modes is to approximate nonlinear dynamics by linear ones in a lifted space, thereby enabling simplified control and analysis. To fight the curse of dimensionality arising from using expressive templates (here neural networks) for the mode approximation, the proposed method leverages a power-iteration scheme that directly learns the dominant Koopman modes without explicitly constructing the projection of the Koopman operator on the template of functions. Our approach connects to other approaches in the literature that avoid the curse of dimensionality by learning small dictionaries of functions, but differs from them in that we do not require ``anti-collapse mechanisms'' to ensure that the learned dictionary is expressive enough to approximate the Koopman operator since our power-iteration scheme is designed to converge toward the dominant modes of the projected Koopman operator. The approach is fully data-driven, requiring only sampled state transitions. Theoretical guarantees are provided, showing convergence under increasing sample size and network width (in connection with the neural tangent kernel theorem). Numerical experiments demonstrate that the method achieves accurate and smooth approximations of dominant modes while avoiding the limitations of traditional techniques such as extended dynamic mode decomposition.