# CF-GODE: Continuous-Time Causal Inference for Multi-Agent Dynamical Systems

arXiv: https://arxiv.org/abs/2306.11216 · HTML: https://mlsanity.com/paper/2306.11216/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 58 references that are in the map.
- First posted on arXiv 20 Jun 2023.

## Builds on (most cited first, 30 of 32)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Domain-Adversarial Training of Neural Networks](https://mlsanity.com/paper/1505.07818/) (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)
- [Estimating Counterfactual Treatment Outcomes over Time Through Adversarially Balanced Representations](https://mlsanity.com/paper/2002.04083/) (2020)
- [Graph Neural Ordinary Differential Equations](https://mlsanity.com/paper/1911.07532/) (2019)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Causal Transformer for Estimating Counterfactual Outcomes](https://mlsanity.com/paper/2204.07258/) (2022)
- [The Blessings of Multiple Causes](https://mlsanity.com/paper/1805.06826/) (2018)
- [Continuously Indexed Domain Adaptation](https://mlsanity.com/paper/2007.01807/) (2020)
- [BrainGB: A Benchmark for Brain Network Analysis with Graph Neural Networks](https://mlsanity.com/paper/2204.07054/) (2022)
- [Learning Continuous System Dynamics from Irregularly-Sampled Partial Observations](https://mlsanity.com/paper/2011.03880/) (2020)
- [Continuous-Time Modeling of Counterfactual Outcomes Using Neural Controlled Differential Equations](https://mlsanity.com/paper/2206.08311/) (2022)
- [Learning Individual Causal Effects from Networked Observational Data](https://mlsanity.com/paper/1906.03485/) (2019)
- [Integrating Expert ODEs into Neural ODEs: Pharmacology and Disease Progression](https://mlsanity.com/paper/2106.02875/) (2021)
- [Identification and estimation of treatment and interference effects in observational studies on networks](https://mlsanity.com/paper/1609.06245/) (2016)
- [Graph Infomax Adversarial Learning for Treatment Effect Estimation with Networked Observational Data](https://mlsanity.com/paper/2106.02881/) (2021)
- [Neural Dynamics on Complex Networks](https://mlsanity.com/paper/1908.06491/) (2019)
- [Causal Inference under Networked Interference and Intervention Policy Enhancement](https://mlsanity.com/paper/2002.08506/) (2020)
- [A probabilistic model for the numerical solution of initial value problems](https://mlsanity.com/paper/1610.05261/) (2016)
- [Using Embeddings to Correct for Unobserved Confounding in Networks](https://mlsanity.com/paper/1902.04114/) (2019)
- [Learning Causal Effects on Hypergraphs](https://mlsanity.com/paper/2207.04049/) (2022)
- [Predicting the impact of treatments over time with uncertainty aware neural differential equations](https://mlsanity.com/paper/2202.11987/) (2022)
- [Neural Ordinary Differential Equations for Intervention Modeling](https://mlsanity.com/paper/2010.08304/) (2020)
- [Policy Analysis using Synthetic Controls in Continuous-Time](https://mlsanity.com/paper/2102.01577/) (2021)
- [A Bayesian Nonparametric Approach for Estimating Individualized Treatment-Response Curves](https://mlsanity.com/paper/1608.05182/) (2016)

## Built on by (most cited first, 16 of 16)

- [Generalizing Graph ODE for Learning Complex System Dynamics across Environments](https://mlsanity.com/paper/2307.04287/) (2023)
- [Automatic Failure Attribution and Critical Step Prediction Method for Multi-Agent Systems Based on Causal Inference](https://mlsanity.com/paper/2509.08682/) (2025)
- [Graph Neural Flows for Unveiling Systemic Interactions Among Irregularly Sampled Time Series](https://mlsanity.com/paper/2410.14030/) (2024)
- [Graph ODEs and Beyond: A Comprehensive Survey on Integrating Differential Equations with Graph Neural Networks](https://mlsanity.com/paper/2503.23167/) (2025)
- [Physics-Informed Regularization for Domain-Agnostic Dynamical System Modeling](https://mlsanity.com/paper/2410.06366/) (2024)
- [Causal Graph ODE: Continuous Treatment Effect Modeling in Multi-agent Dynamical Systems](https://mlsanity.com/paper/2403.00178/) (2024)
- [Causal Contrastive Learning for Counterfactual Regression Over Time](https://mlsanity.com/paper/2406.00535/) (2024)
- [Modeling Time-evolving Causality over Data Streams](https://mlsanity.com/paper/2502.08963/) (2025)
- [Architecture-Aware Learning Curve Extrapolation via Graph Ordinary Differential Equation](https://mlsanity.com/paper/2412.15554/) (2024)
- [TANGO: Time-Reversal Latent GraphODE for Multi-Agent Dynamical Systems](https://mlsanity.com/paper/2310.06427/) (2023)
- [COSTAR: Improved Temporal Counterfactual Estimation with Self-Supervised Learning](https://mlsanity.com/paper/2311.00886/) (2023)
- [Causal GNNs: A GNN-Driven Instrumental Variable Approach for Causal Inference in Networks](https://mlsanity.com/paper/2409.08544/) (2024)
- [Estimating Treatment Effects in Networks under Unknown Exposure Mappings](https://mlsanity.com/paper/2510.21457/) (2025)
- [Treatment Effect Estimation with Differentiated Networked Effect on Graph Data](https://mlsanity.com/paper/2605.24358/) (2026)
- [Compass: Navigating Global Marine Lead Data Integration through Expert-Guided LLM Agent](https://mlsanity.com/paper/2605.29966/) (2026)
- [INDEQS: Informed Neural controlled Differential EQuationS](https://mlsanity.com/paper/2606.19138/) (2026)

## Abstract (arXiv)

Multi-agent dynamical systems refer to scenarios where multiple units interact with each other and evolve collectively over time. To make informed decisions in multi-agent dynamical systems, such as determining the optimal vaccine distribution plan, it is essential for decision-makers to estimate the continuous-time counterfactual outcomes. However, existing studies of causal inference over time rely on the assumption that units are mutually independent, which is not valid for multi-agent dynamical systems. In this paper, we aim to bridge this gap and study how to estimate counterfactual outcomes in multi-agent dynamical systems. Causal inference in a multi-agent dynamical system has unique challenges: 1) Confounders are time-varying and are present in both individual unit covariates and those of other units; 2) Units are affected by not only their own but also others' treatments; 3) The treatments are naturally dynamic, such as receiving vaccines and boosters in a seasonal manner. We model a multi-agent dynamical system as a graph and propose CounterFactual GraphODE (CF-GODE), a causal model that estimates continuous-time counterfactual outcomes in the presence of inter-dependencies between units. To facilitate continuous-time estimation, we propose Treatment-Induced GraphODE, a novel ordinary differential equation based on GNN, which incorporates dynamical treatments as additional inputs to predict potential outcomes over time. To remove confounding bias, we propose two domain adversarial learning based objectives that learn balanced continuous representation trajectories, which are not predictive of treatments and interference. We further provide theoretical justification to prove their effectiveness. Experiments on two semi-synthetic datasets confirm that CF-GODE outperforms baselines on counterfactual estimation. We also provide extensive analyses to understand how our model works.