# DICE: Discrete inverse continuity equation for learning population dynamics

arXiv: https://arxiv.org/abs/2507.05107 · HTML: https://mlsanity.com/paper/2507.05107/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 52 references that are in the map.
- First posted on arXiv 7 Jul 2025.

## Builds on (most cited first, 21 of 21)

- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [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)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [Supervised Training of Conditional Monge Maps](https://mlsanity.com/paper/2206.14262/) (2022)
- [Deep Momentum Multi-Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2303.01751/) (2023)
- [Sampling in Unit Time with Kernel Fisher-Rao Flow](https://mlsanity.com/paper/2401.03892/) (2024)
- [Liouville Flow Importance Sampler](https://mlsanity.com/paper/2405.06672/) (2024)
- [Fast and Smooth Interpolation on Wasserstein Space](https://mlsanity.com/paper/2010.12101/) (2020)
- [Learning diffusion at lightspeed](https://mlsanity.com/paper/2406.12616/) (2024)
- [Second order models for optimal transport and cubic splines on the Wasserstein space](https://mlsanity.com/paper/1801.04144/) (2018)
- [Trajectory Inference with Smooth Schr"odinger Bridges](https://mlsanity.com/paper/2503.00530/) (2025)
- [Parametric model reduction of mean-field and stochastic systems via higher-order action matching](https://mlsanity.com/paper/2410.12000/) (2024)
- [Sequential Ensemble Transform for Bayesian Inverse Problems](https://mlsanity.com/paper/1909.09591/) (2019)

## Built on by (most cited first, 8 of 8)

- [A Call to Lagrangian Action: Learning Population Mechanics from Temporal Snapshots](https://mlsanity.com/paper/2605.08550/) (2026)
- [Leveraging Gauge Freedom for Learning Non-Gradient Population Dynamics of Stochastic Systems](https://mlsanity.com/paper/2605.25107/) (2026)
- [Two-Parameter Flows for Learning Population Dynamics of Physical Systems](https://mlsanity.com/paper/2605.26285/) (2026)
- [Stochastic Lifting for Generating Trajectories of Stochastic Physical Systems](https://mlsanity.com/paper/2605.29194/) (2026)
- [On the Unique Recovery of Transport Maps and Vector Fields from Finite Measure-Valued Data](https://mlsanity.com/paper/2604.07671/) (2026)
- [One-Step Generative Surrogate Models via Block-Triangular Joint Drifting](https://mlsanity.com/paper/2609.26435/) (2026)
- [Simulation-Free Learning of Population Dynamics with Wasserstein Lagrangian Residuals](https://mlsanity.com/paper/2610.03679/) (2026)
- [Discrete Action Matching: Learning Stochastic Dynamics from Samples via State Graphs](https://mlsanity.com/paper/2610.05071/) (2026)

## Abstract (arXiv)

We introduce the Discrete Inverse Continuity Equation (DICE) method, a generative modeling approach that learns the evolution of a stochastic process from given sample populations at a finite number of time points. Models learned with DICE capture the typically smooth and well-behaved population dynamics, rather than the dynamics of individual sample trajectories that can exhibit complex or even chaotic behavior. The DICE loss function is developed specifically to be invariant, even in discrete time, to spatially constant but time-varying spurious constants that can emerge during training; this invariance increases training stability and robustness. Generating a trajectory of sample populations with DICE is fast because samples evolve directly in the time interval over which the stochastic process is formulated, in contrast to approaches that condition on time and then require multiple sampling steps per time step. DICE is stable to train, in situations where other methods for learning population dynamics fail, and DICE generates representative samples with orders of magnitude lower costs than methods that have to condition on time. Numerical experiments on a wide range of problems from random waves, Vlasov-Poisson instabilities and high-dimensional chaos are included to justify these assertions.