DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systems This paper introduces DySLIM, a new framework for learning dynamics from chaotic systems by targeting the invariant measure of the system. Unlike traditional methods that focus on minimizing the misfit between trajectories, DySLIM aims to learn the invariant measure, which governs the long-term statistical behavior of the system. This approach enables more stable and accurate learning, particularly for long-term predictions. The paper discusses the challenges of learning dynamics from chaotic systems, which are inherently unstable due to their positive Lyapunov exponents. However, many such systems exhibit ergodicity and an attractor, a compact manifold towards which trajectories converge. These systems also support an invariant measure, a probability distribution that remains unchanged under the system's dynamics. DySLIM leverages this structure to propose a new objective function that can be used with any existing learning objective. The framework introduces a measure-matching regularization term into the loss, which induces stable and accurate trajectories that satisfy the long-term statistics while enhancing short-term predictive power. The paper demonstrates that DySLIM is capable of tackling larger and more complex systems than competing probabilistic methods, up to a state-dimension of 4,096 with complex 2D dynamics. It shows competitive results in three increasingly complex and higher dimensional problems: the Lorenz 63 system, the Kuramoto-Sivashinsky equation, and the Kolmogorov-Flow system. The paper also discusses the challenges of learning dynamics from chaotic systems, including the issue of distribution shift, where models trained on short-term data may fail to generalize to long-term behavior. DySLIM addresses this by focusing on the invariant measure of the system, which helps to stabilize the learning process and improve long-term accuracy. The paper presents experiments showing that DySLIM outperforms baselines in terms of stability and accuracy for the Lorenz 63, Kuramoto-Sivashinsky, and Kolmogorov-Flow systems. It also discusses the computational complexity of DySLIM, showing that it incurs relatively small overhead compared to the baselines. The paper concludes that DySLIM provides a tractable, scalable, and system-agnostic regularized training objective that can be used to stabilize real-world dynamical system models with slowly varying measures, such as those used in global weather prediction.DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systems This paper introduces DySLIM, a new framework for learning dynamics from chaotic systems by targeting the invariant measure of the system. Unlike traditional methods that focus on minimizing the misfit between trajectories, DySLIM aims to learn the invariant measure, which governs the long-term statistical behavior of the system. This approach enables more stable and accurate learning, particularly for long-term predictions. The paper discusses the challenges of learning dynamics from chaotic systems, which are inherently unstable due to their positive Lyapunov exponents. However, many such systems exhibit ergodicity and an attractor, a compact manifold towards which trajectories converge. These systems also support an invariant measure, a probability distribution that remains unchanged under the system's dynamics. DySLIM leverages this structure to propose a new objective function that can be used with any existing learning objective. The framework introduces a measure-matching regularization term into the loss, which induces stable and accurate trajectories that satisfy the long-term statistics while enhancing short-term predictive power. The paper demonstrates that DySLIM is capable of tackling larger and more complex systems than competing probabilistic methods, up to a state-dimension of 4,096 with complex 2D dynamics. It shows competitive results in three increasingly complex and higher dimensional problems: the Lorenz 63 system, the Kuramoto-Sivashinsky equation, and the Kolmogorov-Flow system. The paper also discusses the challenges of learning dynamics from chaotic systems, including the issue of distribution shift, where models trained on short-term data may fail to generalize to long-term behavior. DySLIM addresses this by focusing on the invariant measure of the system, which helps to stabilize the learning process and improve long-term accuracy. The paper presents experiments showing that DySLIM outperforms baselines in terms of stability and accuracy for the Lorenz 63, Kuramoto-Sivashinsky, and Kolmogorov-Flow systems. It also discusses the computational complexity of DySLIM, showing that it incurs relatively small overhead compared to the baselines. The paper concludes that DySLIM provides a tractable, scalable, and system-agnostic regularized training objective that can be used to stabilize real-world dynamical system models with slowly varying measures, such as those used in global weather prediction.