# Continuous-Time Trajectory Generation from Discrete Observations with Stochasticity

arXiv: https://arxiv.org/abs/2609.32026 · HTML: https://mlsanity.com/paper/2609.32026/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 47 of its 56 references that are in the map.
- First posted on arXiv 25 Sep 2026.

## Builds on (most cited first, 30 of 47)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Taming Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2012.09841/) (2020)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context](https://mlsanity.com/paper/1901.02860/) (2019)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [Towards Accurate Generative Models of Video: A New Metric & Challenges](https://mlsanity.com/paper/1812.01717/) (2018)
- [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)
- [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)
- [VideoGPT: Video Generation using VQ-VAE and Transformers](https://mlsanity.com/paper/2104.10157/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Unsupervised Learning of Video Representations using LSTMs](https://mlsanity.com/paper/1502.04681/) (2015)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [MCVD: Masked Conditional Video Diffusion for Prediction, Generation, and Interpolation](https://mlsanity.com/paper/2205.09853/) (2022)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Unsupervised Learning for Physical Interaction through Video Prediction](https://mlsanity.com/paper/1605.07157/) (2016)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [Stochastic Adversarial Video Prediction](https://mlsanity.com/paper/1804.01523/) (2018)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Do generative video models understand physical principles?](https://mlsanity.com/paper/2501.09038/) (2025)

## Abstract (arXiv)

Physical systems evolve continuously in time, yet their states are typically observed only at discrete times. Generating trajectories consistent with their probability densities from such observations therefore requires capturing the continuous-time evolution rather than only learning transition mappings between consecutive observations. We propose PhiBE-Flow, a framework that directly estimates the probability velocity field induced by the stochastic differential equation (SDE) which governs this continuous-time distributional evolution. PhiBE-Flow learns from discrete observations using a model-free approach requiring neither known SDE coefficients nor score estimation. We establish convergence guarantees for the method, accounting for both time-discretization and finite-sample errors. We evaluate PhiBE-Flow on systems of increasing complexity, from controlled stochastic numerical systems to Navier--Stokes dynamics and real-world videos. Our results show that PhiBE-Flow accurately recovers probability flows of stochastic dynamics, preserves multiscale physical statistics, and improves video generation performance over representative baselines. The code is available at https://github.com/R1fe/PhiBE-Flow.