# Continuous-Time Modeling of Counterfactual Outcomes Using Neural Controlled Differential Equations

arXiv: https://arxiv.org/abs/2206.08311 · HTML: https://mlsanity.com/paper/2206.08311/

## Key facts

- Cited by 59 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 56 references that are in the map.
- First posted on arXiv 16 Jun 2022.

## Builds on (most cited first, 23 of 23)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Domain-Adversarial Training of Neural Networks](https://mlsanity.com/paper/1505.07818/) (2015)
- [Estimation and Inference of Heterogeneous Treatment Effects using Random Forests](https://mlsanity.com/paper/1510.04342/) (2015)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Learning Representations for Counterfactual Inference](https://mlsanity.com/paper/1605.03661/) (2016)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [Bayesian Inference of Individualized Treatment Effects using Multi-task Gaussian Processes](https://mlsanity.com/paper/1704.02801/) (2017)
- [Estimating Counterfactual Treatment Outcomes over Time Through Adversarially Balanced Representations](https://mlsanity.com/paper/2002.04083/) (2020)
- [Uncertainty Estimates and Multi-Hypotheses Networks for Optical Flow](https://mlsanity.com/paper/1802.07095/) (2018)
- [The Blessings of Multiple Causes](https://mlsanity.com/paper/1805.06826/) (2018)
- [Support and Invertibility in Domain-Invariant Representations](https://mlsanity.com/paper/1903.03448/) (2019)
- [Time Series Deconfounder: Estimating Treatment Effects over Time in the Presence of Hidden Confounders](https://mlsanity.com/paper/1902.00450/) (2019)
- [Generalization Bounds and Representation Learning for Estimation of Potential Outcomes and Causal Effects](https://mlsanity.com/paper/2001.07426/) (2020)
- [Identifying Causal-Effect Inference Failure with Uncertainty-Aware Models](https://mlsanity.com/paper/2007.00163/) (2020)
- [Approximation Capabilities of Neural ODEs and Invertible Residual Networks](https://mlsanity.com/paper/1907.12998/) (2019)
- [Neural Controlled Differential Equations for Online Prediction Tasks](https://mlsanity.com/paper/2106.11028/) (2021)
- [Neural Ordinary Differential Equations for Intervention Modeling](https://mlsanity.com/paper/2010.08304/) (2020)
- [Learning from Clinical Judgments: Semi-Markov-Modulated Marked Hawkes Processes for Risk Prognosis](https://mlsanity.com/paper/1705.05267/) (2017)
- [Policy Analysis using Synthetic Controls in Continuous-Time](https://mlsanity.com/paper/2102.01577/) (2021)
- [Hawkes Processes with Stochastic Excitations](https://mlsanity.com/paper/1609.06831/) (2016)
- [The additive hazard estimator is consistent for continuous-time marginal structural models](https://mlsanity.com/paper/1802.01946/) (2018)

## Built on by (most cited first, 30 of 59)

- [Causal machine learning for predicting treatment outcomes](https://mlsanity.com/paper/2410.08770/) (2024)
- [Bayesian Neural Controlled Differential Equations for Treatment Effect Estimation](https://mlsanity.com/paper/2310.17463/) (2023)
- [Trajectory Flow Matching with Applications to Clinical Time Series Modeling](https://mlsanity.com/paper/2410.21154/) (2024)
- [Bounds on Representation-Induced Confounding Bias for Treatment Effect Estimation](https://mlsanity.com/paper/2311.11321/) (2023)
- [Model-agnostic meta-learners for estimating heterogeneous treatment effects over time](https://mlsanity.com/paper/2407.05287/) (2024)
- [Estimating Treatment Effects from Irregular Time Series Observations with Hidden Confounders](https://mlsanity.com/paper/2303.02320/) (2023)
- [Counterfactual Generative Models for Time-Varying Treatments](https://mlsanity.com/paper/2305.15742/) (2023)
- [CF-GODE: Continuous-Time Causal Inference for Multi-Agent Dynamical Systems](https://mlsanity.com/paper/2306.11216/) (2023)
- [Rough Transformers: Lightweight and Continuous Time Series Modelling through Signature Patching](https://mlsanity.com/paper/2405.20799/) (2024)
- [Accounting For Informative Sampling When Learning to Forecast Treatment Outcomes Over Time](https://mlsanity.com/paper/2306.04255/) (2023)
- [Automatically Learning Hybrid Digital Twins of Dynamical Systems](https://mlsanity.com/paper/2410.23691/) (2024)
- [ODE Discovery for Longitudinal Heterogeneous Treatment Effects Inference](https://mlsanity.com/paper/2403.10766/) (2024)
- [A Survey of Deep Causal Models and Their Industrial Applications](https://mlsanity.com/paper/2209.08860/) (2022)
- [Neural Laplace Control for Continuous-time Delayed Systems](https://mlsanity.com/paper/2302.12604/) (2023)
- [IGC-Net for conditional average potential outcome estimation over time](https://mlsanity.com/paper/2405.21012/) (2024)
- [Adversarial De-confounding in Individualised Treatment Effects Estimation](https://mlsanity.com/paper/2210.10530/) (2022)
- [medDreamer: Model-Based Reinforcement Learning with Latent Imagination on Complex EHRs for Clinical Decision Support](https://mlsanity.com/paper/2505.19785/) (2025)
- [Causal Graph ODE: Continuous Treatment Effect Modeling in Multi-agent Dynamical Systems](https://mlsanity.com/paper/2403.00178/) (2024)
- [G-Transformer: Counterfactual Outcome Prediction under Dynamic and Time-varying Treatment Regimes](https://mlsanity.com/paper/2406.05504/) (2024)
- [ChronoMedicalWorld: A Medical World Model for Learning Patient Trajectories from Longitudinal Care Data](https://mlsanity.com/paper/2605.21963/) (2026)
- [Causal Contrastive Learning for Counterfactual Regression Over Time](https://mlsanity.com/paper/2406.00535/) (2024)
- [Stabilized Neural Prediction of Potential Outcomes in Continuous Time](https://mlsanity.com/paper/2410.03514/) (2024)
- [Rough Transformers for Continuous and Efficient Time-Series Modelling](https://mlsanity.com/paper/2403.10288/) (2024)
- [Decoupled Marked Temporal Point Process using Neural Ordinary Differential Equations](https://mlsanity.com/paper/2406.06149/) (2024)
- [GST-UNet: A Neural Framework for Spatiotemporal Causal Inference with Time-Varying Confounding](https://mlsanity.com/paper/2502.05295/) (2025)
- [Balanced Neural ODEs: nonlinear model order reduction and Koopman operator approximations](https://mlsanity.com/paper/2410.10174/) (2024)
- [Transformer-Based Spatial-Temporal Counterfactual Outcomes Estimation](https://mlsanity.com/paper/2506.21154/) (2025)
- [TrajSurv: Learning Continuous Latent Trajectories from Electronic Health Records for Trustworthy Survival Prediction](https://mlsanity.com/paper/2508.00657/) (2025)
- [Counterfactual Probabilistic Diffusion with Expert Models](https://mlsanity.com/paper/2508.13355/) (2025)
- [On the Generalization and Approximation Capacities of Neural Controlled Differential Equations](https://mlsanity.com/paper/2305.16791/) (2023)

## Abstract (arXiv)

Estimating counterfactual outcomes over time has the potential to unlock personalized healthcare by assisting decision-makers to answer ''what-iF'' questions. Existing causal inference approaches typically consider regular, discrete-time intervals between observations and treatment decisions and hence are unable to naturally model irregularly sampled data, which is the common setting in practice. To handle arbitrary observation patterns, we interpret the data as samples from an underlying continuous-time process and propose to model its latent trajectory explicitly using the mathematics of controlled differential equations. This leads to a new approach, the Treatment Effect Neural Controlled Differential Equation (TE-CDE), that allows the potential outcomes to be evaluated at any time point. In addition, adversarial training is used to adjust for time-dependent confounding which is critical in longitudinal settings and is an added challenge not encountered in conventional time-series. To assess solutions to this problem, we propose a controllable simulation environment based on a model of tumor growth for a range of scenarios with irregular sampling reflective of a variety of clinical scenarios. TE-CDE consistently outperforms existing approaches in all simulated scenarios with irregular sampling.