# Solving Inverse Problems of Chaotic Systems with Bidirectional Conditional Flow Matching

arXiv: https://arxiv.org/abs/2606.24824 · HTML: https://mlsanity.com/paper/2606.24824/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 16 of its 120 references that are in the map.
- First posted on arXiv 23 Jun 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 16 of 16)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [On Distillation of Guided Diffusion Models](https://mlsanity.com/paper/2210.03142/) (2022)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [Inverse-design of nonlinear mechanical metamaterials via video denoising diffusion models](https://mlsanity.com/paper/2305.19836/) (2023)
- [Wavelet Diffusion Neural Operator](https://mlsanity.com/paper/2412.04833/) (2024)
- [Structural engineering from an inverse problems perspective](https://mlsanity.com/paper/2106.15177/) (2021)
- [A transformer-based generative model for planetary systems](https://mlsanity.com/paper/2509.07226/) (2025)

## Abstract (arXiv)

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 ~ 1010 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.