The paper "DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systems" addresses the challenge of learning dynamics from dissipative chaotic systems, which are inherently unstable due to their positive Lyapunov exponents. The authors propose a new framework called Dynamics Stable Learning by Invariant Measure (DySLIM) that targets learning both the invariant measure and the dynamics of the system. This approach is contrasted with typical methods that only focus on the misfit between trajectories, which often lead to divergence as trajectory lengths increase. DySLIM introduces a system-agnostic measure-matching regularization term into the loss function, which induces stable and accurate trajectories while enhancing short-term predictive power. The framework is used to propose a tractable and sample-efficient objective, DySLIM, that can be used with any existing learning objectives. The authors demonstrate that DySLIM can handle larger and more complex systems than competing probabilistic methods, up to a state dimension of 4,096 with complex 2D dynamics. The paper evaluates DySLIM on three increasingly complex and higher-dimensional problems: the Lorenz 63 system, the Kuramoto-Sivashinsky equation, and the Kolmogorov-Flow. The results show that DySLIM achieves better point-wise tracking and long-term statistical accuracy compared to other learning objectives. The authors also discuss the limitations of existing methods and the advantages of DySLIM in terms of stability and performance. The contributions of the paper are three-fold: first, the proposal of a probabilistic and scalable framework for learning chaotic dynamics using data-driven, ML-based methods; second, the introduction of DySLIM as a tractable and sample-efficient objective; and third, the demonstration of DySLIM's capability to handle larger and more complex systems.The paper "DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systems" addresses the challenge of learning dynamics from dissipative chaotic systems, which are inherently unstable due to their positive Lyapunov exponents. The authors propose a new framework called Dynamics Stable Learning by Invariant Measure (DySLIM) that targets learning both the invariant measure and the dynamics of the system. This approach is contrasted with typical methods that only focus on the misfit between trajectories, which often lead to divergence as trajectory lengths increase. DySLIM introduces a system-agnostic measure-matching regularization term into the loss function, which induces stable and accurate trajectories while enhancing short-term predictive power. The framework is used to propose a tractable and sample-efficient objective, DySLIM, that can be used with any existing learning objectives. The authors demonstrate that DySLIM can handle larger and more complex systems than competing probabilistic methods, up to a state dimension of 4,096 with complex 2D dynamics. The paper evaluates DySLIM on three increasingly complex and higher-dimensional problems: the Lorenz 63 system, the Kuramoto-Sivashinsky equation, and the Kolmogorov-Flow. The results show that DySLIM achieves better point-wise tracking and long-term statistical accuracy compared to other learning objectives. The authors also discuss the limitations of existing methods and the advantages of DySLIM in terms of stability and performance. The contributions of the paper are three-fold: first, the proposal of a probabilistic and scalable framework for learning chaotic dynamics using data-driven, ML-based methods; second, the introduction of DySLIM as a tractable and sample-efficient objective; and third, the demonstration of DySLIM's capability to handle larger and more complex systems.