# Neural graphical modelling in continuous-time: consistency guarantees and algorithms

arXiv: https://arxiv.org/abs/2105.02522 · HTML: https://mlsanity.com/paper/2105.02522/

## Key facts

- Cited by 42 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 85 references that are in the map.
- First posted on arXiv 6 May 2021.

## Builds on (most cited first, 20 of 20)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Causality for Machine Learning](https://mlsanity.com/paper/1911.10500/) (2019)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [DYNOTEARS: Structure Learning from Time-Series Data](https://mlsanity.com/paper/2002.00498/) (2020)
- [Economy Statistical Recurrent Units For Inferring Nonlinear Granger Causality](https://mlsanity.com/paper/1911.09879/) (2019)
- [Learning unknown ODE models with Gaussian processes](https://mlsanity.com/paper/1803.04303/) (2018)
- [Interpretable Models for Granger Causality Using Self-explaining Neural Networks](https://mlsanity.com/paper/2101.07600/) (2021)
- [From Ordinary Differential Equations to Structural Causal Models: the deterministic case](https://mlsanity.com/paper/1408.2063/) (2014)
- [From Deterministic ODEs to Dynamic Structural Causal Models](https://mlsanity.com/paper/1608.08028/) (2016)
- [Consistent Feature Selection for Analytic Deep Neural Networks](https://mlsanity.com/paper/2010.08097/) (2020)
- [Neural Granger Causality](https://mlsanity.com/paper/1802.05842/) (2018)
- [On the Non-asymptotic and Sharp Lower Tail Bounds of Random Variables](https://mlsanity.com/paper/1810.09006/) (2018)
- [Policy Analysis using Synthetic Controls in Continuous-Time](https://mlsanity.com/paper/2102.01577/) (2021)
- [ODIN: ODE-Informed Regression for Parameter and State Inference in Time-Continuous Dynamical Systems](https://mlsanity.com/paper/1902.06278/) (2019)
- [Oracle Inequalities for High Dimensional Vector Autoregressions](https://mlsanity.com/paper/1311.0811/) (2013)
- [Network Reconstruction From High Dimensional Ordinary Differential Equations](https://mlsanity.com/paper/1610.03177/) (2016)
- [Learning Dynamic and Personalized Comorbidity Networks from Event Data using Deep Diffusion Processes](https://mlsanity.com/paper/2001.02585/) (2020)
- [LASSO-Driven Inference in Time and Space](https://mlsanity.com/paper/1806.05081/) (2018)
- [Inverse Problems for Matrix Exponential in System Identification: System Aliasing](https://mlsanity.com/paper/1605.06973/) (2016)

## Built on by (most cited first, 30 of 42)

- [Simulation-free Schr"odinger bridges via score and flow matching](https://mlsanity.com/paper/2307.03672/) (2023)
- [Causal Discovery from Temporal Data: An Overview and New Perspectives](https://mlsanity.com/paper/2303.10112/) (2023)
- [CUTS+: High-dimensional Causal Discovery from Irregular Time-series](https://mlsanity.com/paper/2305.05890/) (2023)
- [CausalTime: Realistically Generated Time-series for Benchmarking of Causal Discovery](https://mlsanity.com/paper/2310.01753/) (2023)
- [CUTS: Neural Causal Discovery from Irregular Time-Series Data](https://mlsanity.com/paper/2302.07458/) (2023)
- [DynGFN: Towards Bayesian Inference of Gene Regulatory Networks with GFlowNets](https://mlsanity.com/paper/2302.04178/) (2023)
- [Causal Modeling with Stationary Diffusions](https://mlsanity.com/paper/2310.17405/) (2023)
- [Hierarchical Graph Neural Networks for Causal Discovery and Root Cause Localization](https://mlsanity.com/paper/2302.01987/) (2023)
- [Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes](https://mlsanity.com/paper/2402.18477/) (2024)
- [Neural Structure Learning with Stochastic Differential Equations](https://mlsanity.com/paper/2311.03309/) (2023)
- [Deep Causal Learning: Representation, Discovery and Inference](https://mlsanity.com/paper/2211.03374/) (2022)
- [Causality-Aware Spatiotemporal Graph Neural Networks for Spatiotemporal Time Series Imputation](https://mlsanity.com/paper/2403.11960/) (2024)
- [Sparsity in Continuous-Depth Neural Networks](https://mlsanity.com/paper/2210.14672/) (2022)
- [Jacobian Regularizer-based Neural Granger Causality](https://mlsanity.com/paper/2405.08779/) (2024)
- [Identifiability and Asymptotics in Learning Homogeneous Linear ODE Systems from Discrete Observations](https://mlsanity.com/paper/2210.05955/) (2022)
- [CausalDynamics: A large-scale benchmark for structural discovery of dynamical causal models](https://mlsanity.com/paper/2505.16620/) (2025)
- [Generator Identification for Linear SDEs with Additive and Multiplicative Noise](https://mlsanity.com/paper/2310.19491/) (2023)
- [Identifiability Challenges in Sparse Linear Ordinary Differential Equations](https://mlsanity.com/paper/2506.09816/) (2025)
- [Identifiability Analysis of Linear ODE Systems with Hidden Confounders](https://mlsanity.com/paper/2410.21917/) (2024)
- [Inferring stochastic dynamics with growth from cross-sectional data](https://mlsanity.com/paper/2505.13197/) (2025)
- [Towards Better Long-range Time Series Forecasting using Generative Forecasting](https://mlsanity.com/paper/2212.06142/) (2022)
- [Foundation Inference Models for Ordinary Differential Equations](https://mlsanity.com/paper/2602.08733/) (2026)
- [Adaptive Group Lasso Neural Network Models for Functions of Few Variables and Time-Dependent Data](https://mlsanity.com/paper/2108.10825/) (2021)
- [CTP:A Causal Interpretable Model for Non-Communicable Disease Progression Prediction](https://mlsanity.com/paper/2308.09735/) (2023)
- [Root-KGD: A Novel Framework for Root Cause Diagnosis Based on Knowledge Graph and Industrial Data](https://mlsanity.com/paper/2406.13664/) (2024)
- [LOCAL: Learning with Orientation Matrix to Infer Causal Structure from Time Series Data](https://mlsanity.com/paper/2410.19464/) (2024)
- [CausalStock: Deep End-to-end Causal Discovery for News-driven Stock Movement Prediction](https://mlsanity.com/paper/2411.06391/) (2024)
- [Causal Discovery for Irregularly Time Series with Consistency Guarantees](https://mlsanity.com/paper/2507.03310/) (2025)
- [Causal Structure Learning for Dynamical Systems with Theoretical Score Analysis](https://mlsanity.com/paper/2512.14361/) (2025)
- [Coarse-to-Fine Learning of Dynamic Causal Structures](https://mlsanity.com/paper/2602.22532/) (2026)

## Abstract (arXiv)

The discovery of structure from time series data is a key problem in fields of study working with complex systems. Most identifiability results and learning algorithms assume the underlying dynamics to be discrete in time. Comparatively few, in contrast, explicitly define dependencies in infinitesimal intervals of time, independently of the scale of observation and of the regularity of sampling. In this paper, we consider score-based structure learning for the study of dynamical systems. We prove that for vector fields parameterized in a large class of neural networks, least squares optimization with adaptive regularization schemes consistently recovers directed graphs of local independencies in systems of stochastic differential equations. Using this insight, we propose a score-based learning algorithm based on penalized Neural Ordinary Differential Equations (modelling the mean process) that we show to be applicable to the general setting of irregularly-sampled multivariate time series and to outperform the state of the art across a range of dynamical systems.