Solving Inverse Problems of Chaotic Systems with Bidirectional Conditional Flow Matching

Inverse problems in chaotic dynamics—inferring initial conditions from observed final states—are notoriously difficult due to the butterfly effect, where infinitesimal perturbations in initial states amplify exponentially over time. Traditional numerical methods, such as backward integration, often fail in these settings because they are unstable, ill-posed, and struggle with information loss (e.g., when components of a system vanish). This paper addresses these challenges by proposing a generative modeling approach to learn the mapping between initial and final state distributions.

The authors introduce Bidirectional Conditional Flow Matching (Bi-CFM), a framework that treats chaotic evolution as a stochastic process rather than a deterministic trajectory. By learning bidirectional mappings (forward and reverse) simultaneously, the model captures the global structure of the chaotic attractor and avoids the error accumulation inherent in iterative time-stepping. For systems governed by physical conservation laws (like energy or momentum), the authors extend this to Conservation-constrained Bi-CFM (CBi-CFM). This extension constrains the probability flow path to remain within a specific conservation manifold, ensuring that the generated initial states are physically valid.

Bi-CFM was evaluated across classic chaotic systems (Lorenz, Circuit, Lorenz 96) and complex physical scenarios, including three-body planetary scattering and the long-term evolution of globular clusters. The results demonstrate that Bi-CFM significantly outperforms traditional numerical solvers and standard deep learning baselines across multiple distribution-level metrics. Notably, the model achieves a speedup of over two orders of magnitude compared to traditional methods. In planetary dynamics, CBi-CFM successfully maintained conservation laws with errors comparable to ground-truth data, and in globular cluster simulations, it provided a more accurate reconstruction of historical states than state-of-the-art Monte Carlo methods.