# Generalizing Graph ODE for Learning Complex System Dynamics across Environments

arXiv: https://arxiv.org/abs/2307.04287 · HTML: https://mlsanity.com/paper/2307.04287/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 50 references that are in the map.
- First posted on arXiv 10 Jul 2023.

## Builds on (most cited first, 30 of 33)

- [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)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Learning deep representations by mutual information estimation and maximization](https://mlsanity.com/paper/1808.06670/) (2018)
- [LightGCN: Simplifying and Powering Graph Convolution Network for Recommendation](https://mlsanity.com/paper/2002.02126/) (2020)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Neural Graph Collaborative Filtering](https://mlsanity.com/paper/1905.08108/) (2019)
- [Heterogeneous Graph Transformer](https://mlsanity.com/paper/2003.01332/) (2020)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [Learning Gradient Fields for Molecular Conformation Generation](https://mlsanity.com/paper/2105.03902/) (2021)
- [GraphSMOTE: Imbalanced Node Classification on Graphs with Graph Neural Networks](https://mlsanity.com/paper/2103.08826/) (2021)
- [A Compositional Object-Based Approach to Learning Physical Dynamics](https://mlsanity.com/paper/1612.00341/) (2016)
- [Variational Graph Recurrent Neural Networks](https://mlsanity.com/paper/1908.09710/) (2019)
- [CoPhy: Counterfactual Learning of Physical Dynamics](https://mlsanity.com/paper/1909.12000/) (2019)
- [Propagation Networks for Model-Based Control Under Partial Observation](https://mlsanity.com/paper/1809.11169/) (2018)
- [Learning Continuous System Dynamics from Irregularly-Sampled Partial Observations](https://mlsanity.com/paper/2011.03880/) (2020)
- [Learning Disentangled Representations via Mutual Information Estimation](https://mlsanity.com/paper/1912.03915/) (2019)
- [Neural Dynamics on Complex Networks](https://mlsanity.com/paper/1908.06491/) (2019)
- [Ab-Initio Potential Energy Surfaces by Pairing GNNs with Neural Wave Functions](https://mlsanity.com/paper/2110.05064/) (2021)
- [A probabilistic model for the numerical solution of initial value problems](https://mlsanity.com/paper/1610.05261/) (2016)
- [Learning Bias-Invariant Representation by Cross-Sample Mutual Information Minimization](https://mlsanity.com/paper/2108.05449/) (2021)
- [Multilingual Knowledge Graph Completion with Self-Supervised Adaptive Graph Alignment](https://mlsanity.com/paper/2203.14987/) (2022)

## Built on by (most cited first, 23 of 23)

- [Graph ODEs and Beyond: A Comprehensive Survey on Integrating Differential Equations with Graph Neural Networks](https://mlsanity.com/paper/2503.23167/) (2025)
- [BrainODE: Dynamic Brain Signal Analysis via Graph-Aided Neural Ordinary Differential Equations](https://mlsanity.com/paper/2405.00077/) (2024)
- [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)
- [TANGO: Time-Reversal Latent GraphODE for Multi-Agent Dynamical Systems](https://mlsanity.com/paper/2310.06427/) (2023)
- [Online Relational Inference for Evolving Multi-agent Interacting Systems](https://mlsanity.com/paper/2411.01442/) (2024)
- [A Generalizable Physics-Enhanced State Space Model for Long-Term Dynamics Forecasting in Complex Environments](https://mlsanity.com/paper/2507.10792/) (2025)
- [Learning Continuous Network Emerging Dynamics from Scarce Observations via Data-Adaptive Stochastic Processes](https://mlsanity.com/paper/2310.16466/) (2023)
- [R-ODE: Ricci Curvature Tells When You Will be Informed](https://mlsanity.com/paper/2405.17282/) (2024)
- [Graph Fourier Neural ODEs: Modeling Spatial-temporal Multi-scales in Molecular Dynamics](https://mlsanity.com/paper/2411.01600/) (2024)
- [Parameter-Efficient Conditioning for Material Generalization in Graph-Based Simulators](https://mlsanity.com/paper/2511.05456/) (2025)
- [Relaxing Continuous Constraints of Equivariant Graph Neural Networks for Physical Dynamics Learning](https://mlsanity.com/paper/2406.16295/) (2024)
- [Learning System Dynamics without Forgetting](https://mlsanity.com/paper/2407.00717/) (2024)
- [Spatio-Temporal Graph Deep Learning with Stochastic Differential Equations for Uncovering Alzheimer's Disease Progression](https://mlsanity.com/paper/2509.21735/) (2025)
- [ChaosNexus: A Foundation Model for ODE-based Chaotic System Forecasting with Hierarchical Multi-scale Awareness](https://mlsanity.com/paper/2509.21802/) (2025)
- [On the Convergence and Size Transferability of Continuous-depth Graph Neural Networks](https://mlsanity.com/paper/2510.03923/) (2025)
- [One-Step Graph-Structured Neural Flows for Irregular Multivariate Time Series Classification](https://mlsanity.com/paper/2605.10179/) (2026)
- [Variety Is the Spice of Life: Detecting Misinformation with Dynamic Environmental Representations](https://mlsanity.com/paper/2508.03420/) (2025)
- [Physics-Inspired Spatial Temporal Graph Neural Networks for Predicting Industrial Chain Resilience](https://mlsanity.com/paper/2508.16836/) (2025)
- [Physics-Informed High-order Graph Dynamics Identification Learning for Predicting Complex Networks Long-term Dynamics](https://mlsanity.com/paper/2510.09082/) (2025)
- [Generalizing Dynamics Modeling More Easily from Representation Perspective](https://mlsanity.com/paper/2603.22655/) (2026)
- [Learning Manifold and Ito Dynamics with Branched Neural Rough Differential Equations](https://mlsanity.com/paper/2606.05272/) (2026)
- [Shaping Persistent Representations from Independent Interactions](https://mlsanity.com/paper/2609.34604/) (2026)

## Abstract (arXiv)

Learning multi-agent system dynamics has been extensively studied for various real-world applications, such as molecular dynamics in biology. Most of the existing models are built to learn single system dynamics from observed historical data and predict the future trajectory. In practice, however, we might observe multiple systems that are generated across different environments, which differ in latent exogenous factors such as temperature and gravity. One simple solution is to learn multiple environment-specific models, but it fails to exploit the potential commonalities among the dynamics across environments and offers poor prediction results where per-environment data is sparse or limited. Here, we present GG-ODE (Generalized Graph Ordinary Differential Equations), a machine learning framework for learning continuous multi-agent system dynamics across environments. Our model learns system dynamics using neural ordinary differential equations (ODE) parameterized by Graph Neural Networks (GNNs) to capture the continuous interaction among agents. We achieve the model generalization by assuming the dynamics across different environments are governed by common physics laws that can be captured via learning a shared ODE function. The distinct latent exogenous factors learned for each environment are incorporated into the ODE function to account for their differences. To improve model performance, we additionally design two regularization losses to (1) enforce the orthogonality between the learned initial states and exogenous factors via mutual information minimization; and (2) reduce the temporal variance of learned exogenous factors within the same system via contrastive learning. Experiments over various physical simulations show that our model can accurately predict system dynamics, especially in the long range, and can generalize well to new systems with few observations.