# FNODE: Flow-Matching for data-driven simulation of constrained multibody systems

arXiv: https://arxiv.org/abs/2509.00183 · HTML: https://mlsanity.com/paper/2509.00183/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion and flow policies for robotics, one of 14 directions in Diffusion models.
- Builds on 27 of its 82 references that are in the map.
- First posted on arXiv 29 Aug 2025.

## Placed in

- Diffusion models / [Diffusion and flow policies for robotics](https://mlsanity.com/diffusion/diffusion-and-flow-policies-for-robotics/)

## Builds on (most cited first, 27 of 27)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Neural Rough Differential Equations for Long Time Series](https://mlsanity.com/paper/2009.08295/) (2020)
- [Improving the Training of Rectified Flows](https://mlsanity.com/paper/2405.20320/) (2024)
- [Learning Dynamics of LLM Finetuning](https://mlsanity.com/paper/2407.10490/) (2024)
- [ANODE: Unconditionally Accurate Memory-Efficient Gradients for Neural ODEs](https://mlsanity.com/paper/1902.10298/) (2019)
- [Neural ODE Processes](https://mlsanity.com/paper/2103.12413/) (2021)
- [ChemNODE: A Neural Ordinary Differential Equations Approach for Chemical Kinetics Solvers](https://mlsanity.com/paper/2101.04749/) (2020)
- [On Numerical Integration in Neural Ordinary Differential Equations](https://mlsanity.com/paper/2206.07335/) (2022)
- [Learning Physical Constraints with Neural Projections](https://mlsanity.com/paper/2006.12745/) (2020)
- [Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraints](https://mlsanity.com/paper/2306.09739/) (2023)
- [From Fourier to Neural ODEs: Flow Matching for Modeling Complex Systems](https://mlsanity.com/paper/2405.11542/) (2024)
- [Simulation-Free Training of Neural ODEs on Paired Data](https://mlsanity.com/paper/2410.22918/) (2024)
- [Data-Driven Dynamics Modeling of Miniature Robotic Blimps Using Neural ODEs With Parameter Auto-Tuning](https://mlsanity.com/paper/2404.18580/) (2024)
- [MBD-NODE: Physics-informed data-driven modeling and simulation of constrained multibody systems](https://mlsanity.com/paper/2407.08664/) (2024)

## Built on by (most cited first, 1 of 1)

- [Learning the Right Abstraction: Neural Reduced Dynamics for Complex Robot Control](https://mlsanity.com/paper/2608.19375/) (2026)

## Abstract (arXiv)

Data-driven modeling of constrained multibody dynamics remains challenged by (i) the training cost of Neural ODEs, which typically require backpropagation through an ODE solver, and (ii) error accumulation in rollout predictions. We introduce a Flow-Matching Neural ODE (FNODE) framework that learns the acceleration mapping directly from trajectory data by supervising accelerations rather than integrated states, turning training into a supervised regression problem and eliminating the ODE-adjoint/solver backpropagation bottleneck. Acceleration targets are obtained efficiently via numerical differentiation using a hybrid fast Fourier transform (FFT) and finite-difference (FD) scheme. Kinematic constraints are enforced through coordinate partitioning: FNODE learns accelerations only for the independent generalized coordinates, while the dependent coordinates are recovered by solving the position-level constraint equations. We evaluate FNODE on single and triple mass-spring-damper systems, a double pendulum, a slider crank with and without friction, a vehicle model, and a cart-pole, and compare against MBD-NODE, LSTM, and fully connected baselines. Across these benchmarks, FNODE achieves improved prediction accuracy and training/runtime efficiency, while maintaining constraint satisfaction through the partitioning procedure. Our code and scripts are released as open source to support reproducibility and follow-on research.