# Identifying Drift, Diffusion, and Causal Structure from Temporal Snapshots arXiv: https://arxiv.org/abs/2410.22729 ยท HTML: https://mlsanity.com/paper/2410.22729/ ## Key facts - Cited by 13 papers in the map (data through 14 Oct 2026). - Builds on 28 of its 124 references that are in the map. - First posted on arXiv 30 Oct 2024. ## Builds on (most cited first, 28 of 28) - [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019) - [SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations](https://mlsanity.com/paper/2108.01073/) (2021) - [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021) - [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017) - [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021) - [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020) - [Beware of the Simulated DAG! Causal Discovery Benchmarks May Be Easy To Game](https://mlsanity.com/paper/2102.13647/) (2021) - [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021) - [The Schr"odinger Bridge between Gaussian Measures has a Closed Form](https://mlsanity.com/paper/2202.05722/) (2022) - [A Variational Analysis of Stochastic Gradient Algorithms](https://mlsanity.com/paper/1602.02666/) (2016) - [Towards a mathematical theory of trajectory inference](https://mlsanity.com/paper/2102.09204/) (2021) - [Trajectory Inference via Mean-field Langevin in Path Space](https://mlsanity.com/paper/2205.07146/) (2022) - [DynGFN: Towards Bayesian Inference of Gene Regulatory Networks with GFlowNets](https://mlsanity.com/paper/2302.04178/) (2023) - [Stability of Entropic Optimal Transport and Schr"odinger Bridges](https://mlsanity.com/paper/2106.03670/) (2021) - [Causal machine learning for single-cell genomics](https://mlsanity.com/paper/2310.14935/) (2023) - [Learning diffusion at lightspeed](https://mlsanity.com/paper/2406.12616/) (2024) - [Causal Modeling with Stationary Diffusions](https://mlsanity.com/paper/2310.17405/) (2023) - [Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes](https://mlsanity.com/paper/2402.18477/) (2024) - [Necessary and sufficient graphical conditions for optimal adjustment sets in causal graphical models with hidden variables](https://mlsanity.com/paper/2102.10324/) (2021) - [Dynamic Structural Causal Models](https://mlsanity.com/paper/2406.01161/) (2024) - [An entropy minimization approach to second-order variational mean-field games](https://mlsanity.com/paper/1807.09078/) (2018) - [Inferring biological processes with intrinsic noise from cross-sectional data](https://mlsanity.com/paper/2410.07501/) (2024) - [Generator Identification for Linear SDEs with Additive and Multiplicative Noise](https://mlsanity.com/paper/2310.19491/) (2023) - [Learning the Causal Structure of Networked Dynamical Systems under Latent Nodes and Structured Noise](https://mlsanity.com/paper/2312.05974/) (2023) - [Joint trajectory and network inference via reference fitting](https://mlsanity.com/paper/2409.06879/) (2024) - [Learning Linear Non-Gaussian Graphical Models with Multidirected Edges](https://mlsanity.com/paper/2010.05306/) (2020) - [Deterministic Gibbs Sampling via Ordinary Differential Equations](https://mlsanity.com/paper/2106.10188/) (2021) - [Consistent diffusion matrix estimation from population time series](https://mlsanity.com/paper/2408.14408/) (2024) ## Built on by (most cited first, 13 of 13) - [Oh SnapMMD! Forecasting Stochastic Dynamics Beyond the Schr"odinger Bridge's End](https://mlsanity.com/paper/2505.16082/) (2025) - [A Call to Lagrangian Action: Learning Population Mechanics from Temporal Snapshots](https://mlsanity.com/paper/2605.08550/) (2026) - [Gradient-flow SDEs have unique transient population dynamics](https://mlsanity.com/paper/2505.21770/) (2025) - [Towards Identifiability of Interventional Stochastic Differential Equations](https://mlsanity.com/paper/2505.15987/) (2025) - [Wasserstein Residuals: Learning Gradient Flows from Population Dynamics](https://mlsanity.com/paper/2607.04738/) (2026) - [Robust SDE Parameter Estimation Under Missing Time Information Setting](https://mlsanity.com/paper/2601.20268/) (2026) - [Error Bounds for a Diffusion Model-Based Drift Estimator](https://mlsanity.com/paper/2606.02115/) (2026) - [Disentangling Continuous-Time Latent Dynamics: Identifiability of Latent SDEs via Diffusion Shifts](https://mlsanity.com/paper/2606.28228/) (2026) - [Identifiability Limits of Physics-Informed Inference for Spatial Stochastic Dynamics from Static Snapshots](https://mlsanity.com/paper/2607.01749/) (2026) - [KoopCell: Koopman-Based Generative Model for Learning Single-Cell Dynamics from Distribution Snapshots](https://mlsanity.com/paper/2609.33350/) (2026) - [Beyond Gradient Flow: Identifiability and Recovery from Distribution Snapshots](https://mlsanity.com/paper/2609.35060/) (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) Stochastic differential equations (SDEs) are a fundamental tool for modelling dynamic processes, including gene regulatory networks (GRNs), contaminant transport, financial markets, and image generation. However, learning the underlying SDE from data is a challenging task, especially if individual trajectories are not observable. Motivated by burgeoning research in single-cell datasets, we present the first comprehensive approach for jointly identifying the drift and diffusion of an SDE from its temporal marginals. Assuming linear drift and additive diffusion, we show that non-identifiability can only arise if the initial distribution possesses generalized rotational symmetries. We further prove that even if this condition holds, the drift and diffusion can almost always be recovered from the marginals. Additionally, we show that the causal graph of any SDE with additive diffusion can be recovered from the identified SDE parameters. To complement this theory, we adapt entropy-regularized optimal transport to handle anisotropic diffusion, and introduce APPEX (Alternating Projection Parameter Estimation from X0), an iterative algorithm designed to estimate the drift, diffusion, and causal graph of an additive noise SDE, solely from temporal marginals. We show that APPEX iteratively decreases Kullback-Leibler divergence to the true solution, and demonstrate its effectiveness on simulated data from linear additive noise SDEs.