AI summaryⓘ
The authors tackle the difficult problem of predicting the starting conditions of chaotic systems by looking at their end states, which is usually hard because small changes can cause big differences over time. They introduce a method called Bidirectional Conditional Flow Matching (Bi-CFM) that learns how initial and final states relate to each other in both directions, making predictions more stable and accurate. For systems where certain quantities must be conserved, they improve their method to ensure these rules are followed. Their approach works well on several chaotic models and real astronomical data, running much faster and with better respect for physical laws than previous methods. This could help better understand complex systems that change unpredictably over long timescales.
chaotic systemsinverse problemsinitial conditionsBi-CFMconservation lawsLorenz systemthree-body problemplanetary dynamicsglobular clustersdistribution-level metrics
Authors
Peiyan Hu, Jian Zhang, Jiashu Pan, Ruiqi Feng, Tao Zhang, Zhi-Ming Ma, Yuan-Sen Ting, Gongjie Li, Tailin Wu
Abstract
Modeling chaotic systems is crucial yet challenging. Inverse problems in chaotic dynamics, namely inferring initial conditions from final states, remain largely unsolved because of ill-posedness, non-uniqueness, instability, and potentially chaotic time-reverse dynamics. We address this open problem with Bidirectional Conditional Flow Matching (Bi-CFM), which learns bidirectional mappings between distributions of initial and final states to capture the stochasticity of chaotic evolution and mitigate exponential error accumulation over time. Furthermore, for systems with conservation laws, we extend it to Conservation-constrained Bi-CFM (CBi-CFM). Across the classic Lorenz, Circuit, and high-dimensional Lorenz 96 systems, Bi-CFM improves five distribution-level metrics over baselines while achieving a speedup of more than two orders of magnitude. In the three-body planet-planet scattering problem in planetary dynamics, CBi-CFM better respects conservation laws, with conservation errors comparable to those of the ground truth. Finally, on real observations of globular clusters, collisional million-body systems shaped by $\sim 10^{10}$ years (10 Gyr) of evolution, our method represents an advance in accuracy, establishing a scalable route to solving inverse problems of long-timescale real-world chaotic dynamics.