Solving Inverse Problems of Chaotic Systems with Bidirectional Conditional Flow Matching
For researchers studying chaotic dynamics, this method enables accurate and efficient inference of initial conditions from final states, addressing a long-standing open problem.
Bi-CFM solves inverse problems in chaotic systems by learning bidirectional mappings between initial and final state distributions, achieving over 100x speedup and improved distribution metrics across Lorenz, Circuit, Lorenz 96, and three-body systems, with conservation errors comparable to ground truth on real globular cluster observations spanning 10 Gyr.
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.