# Simulation-Free Differential Dynamics through Neural Conservation Laws

arXiv: https://arxiv.org/abs/2506.18604 · HTML: https://mlsanity.com/paper/2506.18604/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 48 references that are in the map.
- First posted on arXiv 23 Jun 2025.

## Builds on (most cited first, 26 of 26)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modifications](https://mlsanity.com/paper/1701.05517/) (2017)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Design](https://mlsanity.com/paper/1902.00275/) (2019)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Neural Flows: Efficient Alternative to Neural ODEs](https://mlsanity.com/paper/2110.13040/) (2021)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Neural Spatio-Temporal Point Processes](https://mlsanity.com/paper/2011.04583/) (2020)
- [Neural Conservation Laws: A Divergence-Free Perspective](https://mlsanity.com/paper/2210.01741/) (2022)
- [Generalized Schr"odinger Bridge Matching](https://mlsanity.com/paper/2310.02233/) (2023)
- [Stochastic Optimal Control for Collective Variable Free Sampling of Molecular Transition Paths](https://mlsanity.com/paper/2207.02149/) (2022)
- [Accelerating Motion Planning via Optimal Transport](https://mlsanity.com/paper/2309.15970/) (2023)

## Built on by (most cited first, 3 of 3)

- [Wasserstein Residuals: Learning Gradient Flows from Population Dynamics](https://mlsanity.com/paper/2607.04738/) (2026)
- [Diverse Sampling in Diffusion Models with Marginal Preserving Particle Guidance](https://mlsanity.com/paper/2605.06553/) (2026)
- [Gaussian Flow Dynamics: Simulation-Free Neural SDE Learning Beyond One-Time Marginals](https://mlsanity.com/paper/2610.04390/) (2026)

## Abstract (arXiv)

We present a novel simulation-free framework for training continuous-time diffusion processes over very general objective functions. Existing methods typically involve either prescribing the optimal diffusion process -- which only works for heavily restricted problem formulations -- or require expensive simulation to numerically obtain the time-dependent densities and sample from the diffusion process. In contrast, we propose a coupled parameterization which jointly models a time-dependent density function, or probability path, and the dynamics of a diffusion process that generates this probability path. To accomplish this, our approach directly bakes in the Fokker-Planck equation and density function requirements as hard constraints, by extending and greatly simplifying the construction of Neural Conservation Laws. This enables simulation-free training for a large variety of problem formulations, from data-driven objectives as in generative modeling and dynamical optimal transport, to optimality-based objectives as in stochastic optimal control, with straightforward extensions to mean-field objectives due to the ease of accessing exact density functions. We validate our method in a diverse range of application domains from modeling spatio-temporal events to learning optimal dynamics from population data.