Solving Inverse Problems of Chaotic Systems with Bidirectional Conditional Flow Matching

2026-06-23

AI Summary

This work addresses the highly challenging inverse problem of chaotic systems, which suffers from ill-posedness, non-uniqueness, and exponential sensitivity to time reversal. The authors propose Bidirectional Conditional Flow Matching (Bi-CFM), the first method to introduce bidirectional flow matching into this domain, jointly learning mappings between initial and final state distributions to effectively suppress error accumulation over long-term evolution. By further incorporating conservation law constraints, they develop Conservation-constrained Bi-CFM (CBi-CFM). Evaluated on Lorenz, electronic circuit, and Lorenz 96 systems, the approach outperforms baseline methods across five distributional metrics and achieves a speedup of over two orders of magnitude. In three-body scattering problems, CBi-CFM yields conservation errors approaching ground truth and successfully enhances inference accuracy on observational data from globular clusters evolved over billions of years.

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.

Problem

• inverse problems
• chaotic systems
• initial condition inference
• ill-posedness
• time-reverse dynamics

Innovation

• Bidirectional Conditional Flow Matching
• Inverse Problems
• Chaotic Systems
• Conservation Laws
• Scalable Inference