# Learning Interpretable Network Dynamics via Universal Neural Symbolic Regression

arXiv: https://arxiv.org/abs/2411.06833 · HTML: https://mlsanity.com/paper/2411.06833/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 93 references that are in the map.
- First posted on arXiv 11 Nov 2024.

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

- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Interpretable Scientific Discovery with Symbolic Regression: A Review](https://mlsanity.com/paper/2211.10873/) (2022)
- [Symbolic Regression is NP-hard](https://mlsanity.com/paper/2207.01018/) (2022)
- [Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree search](https://mlsanity.com/paper/2205.13134/) (2022)
- [The physics of higher-order interactions in complex systems](https://mlsanity.com/paper/2110.06023/) (2021)
- [SymFormer: End-to-end symbolic regression using transformer-based architecture](https://mlsanity.com/paper/2205.15764/) (2022)
- [Epidemic processes in complex networks](https://mlsanity.com/paper/1408.2701/) (2014)
- [Deep symbolic regression for physics guided by units constraints: toward the automated discovery of physical laws](https://mlsanity.com/paper/2303.03192/) (2023)
- [Neural Dynamics on Complex Networks](https://mlsanity.com/paper/1908.06491/) (2019)
- [Implicit energy regularization of neural ordinary-differential-equation control](https://mlsanity.com/paper/2103.06525/) (2021)
- [Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers](https://mlsanity.com/paper/2410.08304/) (2024)
- [Deep learning of contagion dynamics on complex networks](https://mlsanity.com/paper/2006.05410/) (2020)
- [Do We Need an Encoder-Decoder to Model Dynamical Systems on Networks?](https://mlsanity.com/paper/2305.12185/) (2023)
- [Learning Continuous Network Emerging Dynamics from Scarce Observations via Data-Adaptive Stochastic Processes](https://mlsanity.com/paper/2310.16466/) (2023)

## Built on by (most cited first, 4 of 4)

- [Discovering Generalizable Governing Equations for Graph Dynamical Systems with Interpretable Neural Networks](https://mlsanity.com/paper/2508.18173/) (2025)
- [Turning Time Series into Algebraic Equations: Symbolic Machine Learning for Interpretable Modeling of Chaotic Time Series](https://mlsanity.com/paper/2603.07261/) (2026)
- [Predicting Dynamics of Ultra-Large Complex Systems by Inferring Governing Equations](https://mlsanity.com/paper/2604.00599/) (2026)
- [OrthoReg: Orthogonal Regularization for Hybrid Symbolic-Neural Dynamical Systems](https://mlsanity.com/paper/2606.19145/) (2026)

## Abstract (arXiv)

Discovering governing equations of complex network dynamics is a fundamental challenge in contemporary science with rich data, which can uncover the mysterious patterns and mechanisms of the formation and evolution of complex phenomena in various fields and assist in decision-making. In this work, we develop a universal computational tool that can automatically, efficiently, and accurately learn the symbolic changing patterns of complex system states by combining the excellent fitting ability from deep learning and the equation inference ability from pre-trained symbolic regression. We conduct intensive experimental verifications on more than ten representative scenarios from physics, biochemistry, ecology, epidemiology, etc. Results demonstrate the outstanding effectiveness and efficiency of our tool by comparing with the state-of-the-art symbolic regression techniques for network dynamics. The application to real-world systems including global epidemic transmission and pedestrian movements has verified its practical applicability. We believe that our tool can serve as a universal solution to dispel the fog of hidden mechanisms of changes in complex phenomena, advance toward interpretability, and inspire more scientific discoveries.