# Causal Graph ODE: Continuous Treatment Effect Modeling in Multi-agent Dynamical Systems

arXiv: https://arxiv.org/abs/2403.00178 · HTML: https://mlsanity.com/paper/2403.00178/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 44 references that are in the map.
- First posted on arXiv 29 Feb 2024.

## Builds on (most cited first, 26 of 26)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Heterogeneous Graph Transformer](https://mlsanity.com/paper/2003.01332/) (2020)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [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)
- [Causal Transformer for Estimating Counterfactual Outcomes](https://mlsanity.com/paper/2204.07258/) (2022)
- [Variational Graph Recurrent Neural Networks](https://mlsanity.com/paper/1908.09710/) (2019)
- [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)
- [Neural Dynamics on Complex Networks](https://mlsanity.com/paper/1908.06491/) (2019)
- [Temporal Domain Generalization with Drift-Aware Dynamic Neural Networks](https://mlsanity.com/paper/2205.10664/) (2022)
- [A probabilistic model for the numerical solution of initial value problems](https://mlsanity.com/paper/1610.05261/) (2016)
- [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)
- [Generalizing Graph ODE for Learning Complex System Dynamics across Environments](https://mlsanity.com/paper/2307.04287/) (2023)
- [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)
- [Assessing the Causal Impact of COVID-19 Related Policies on Outbreak Dynamics: A Case Study in the US](https://mlsanity.com/paper/2106.01315/) (2021)
- [CF-GODE: Continuous-Time Causal Inference for Multi-Agent Dynamical Systems](https://mlsanity.com/paper/2306.11216/) (2023)
- [Concept2Box: Joint Geometric Embeddings for Learning Two-View Knowledge Graphs](https://mlsanity.com/paper/2307.01933/) (2023)
- [TANGO: Time-Reversal Latent GraphODE for Multi-Agent Dynamical Systems](https://mlsanity.com/paper/2310.06427/) (2023)

## Built on by (most cited first, 8 of 8)

- [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)
- [Architecture-Aware Learning Curve Extrapolation via Graph Ordinary Differential Equation](https://mlsanity.com/paper/2412.15554/) (2024)
- [Online Relational Inference for Evolving Multi-agent Interacting Systems](https://mlsanity.com/paper/2411.01442/) (2024)
- [On the Convergence and Size Transferability of Continuous-depth Graph Neural Networks](https://mlsanity.com/paper/2510.03923/) (2025)
- [Generalizing Dynamics Modeling More Easily from Representation Perspective](https://mlsanity.com/paper/2603.22655/) (2026)
- [From Static Risk to Dynamic Trajectories: Toward World-Model-Inspired Clinical Prediction](https://mlsanity.com/paper/2605.16927/) (2026)
- [From Single- to Cross-Document: Benchmarking Multi-Granularity Event Analysis of Large Language Models](https://mlsanity.com/paper/2607.27654/) (2026)

## Abstract (arXiv)

Real-world multi-agent systems are often dynamic and continuous, where the agents co-evolve and undergo changes in their trajectories and interactions over time. For example, the COVID-19 transmission in the U.S. can be viewed as a multi-agent system, where states act as agents and daily population movements between them are interactions. Estimating the counterfactual outcomes in such systems enables accurate future predictions and effective decision-making, such as formulating COVID-19 policies. However, existing methods fail to model the continuous dynamic effects of treatments on the outcome, especially when multiple treatments (e.g., "stay-at-home" and "get-vaccine" policies) are applied simultaneously. To tackle this challenge, we propose Causal Graph Ordinary Differential Equations (CAG-ODE), a novel model that captures the continuous interaction among agents using a Graph Neural Network (GNN) as the ODE function. The key innovation of our model is to learn time-dependent representations of treatments and incorporate them into the ODE function, enabling precise predictions of potential outcomes. To mitigate confounding bias, we further propose two domain adversarial learning-based objectives, which enable our model to learn balanced continuous representations that are not affected by treatments or interference. Experiments on two datasets (i.e., COVID-19 and tumor growth) demonstrate the superior performance of our proposed model.