# Learning Anisotropic Interaction Rules from Individual Trajectories in a Heterogeneous Cellular Population

arXiv: https://arxiv.org/abs/2204.14141 · HTML: https://mlsanity.com/paper/2204.14141/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 78 references that are in the map.
- First posted on arXiv 29 Apr 2022.

## Builds on (most cited first, 10 of 10)

- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [Learning Mean-Field Equations from Particle Data Using WSINDy](https://mlsanity.com/paper/2110.07756/) (2021)
- [Learning interaction kernels in heterogeneous systems of agents from multiple trajectories](https://mlsanity.com/paper/1910.04832/) (2019)
- [Numerical differentiation of noisy data: A unifying multi-objective optimization framework](https://mlsanity.com/paper/2009.01911/) (2020)
- [Online Weak-form Sparse Identification of Partial Differential Equations](https://mlsanity.com/paper/2203.03979/) (2022)
- [Data-driven Discovery of Emergent Behaviors in Collective Dynamics](https://mlsanity.com/paper/1912.11123/) (2019)
- [Learning and Interpreting Potentials for Classical Hamiltonian Systems](https://mlsanity.com/paper/1907.11806/) (2019)

## Built on by (most cited first, 11 of 11)

- [Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamics](https://mlsanity.com/paper/2302.13271/) (2023)
- [Weak-Form Latent Space Dynamics Identification](https://mlsanity.com/paper/2311.12880/) (2023)
- [Asymptotic consistency of the WSINDy algorithm in the limit of continuum data](https://mlsanity.com/paper/2211.16000/) (2022)
- [Physics-informed active learning with simultaneous weak-form latent space dynamics identification](https://mlsanity.com/paper/2407.00337/) (2024)
- [Coarse-Graining Hamiltonian Systems Using WSINDy](https://mlsanity.com/paper/2310.05879/) (2023)
- [The Weak Form Is Stronger Than You Think](https://mlsanity.com/paper/2409.06751/) (2024)
- [Weak-Form Inference for Hybrid Dynamical Systems in Ecology](https://mlsanity.com/paper/2405.20591/) (2024)
- [Weak Form Scientific Machine Learning: Test Function Construction for System Identification](https://mlsanity.com/paper/2507.03206/) (2025)
- [Weak Form Learning for Mean-Field Partial Differential Equations: an Application to Insect Movement](https://mlsanity.com/paper/2510.07786/) (2025)
- [Learning Multi-type heterogeneous interacting particle systems](https://mlsanity.com/paper/2602.03954/) (2026)
- [Time-Dependent PDE-Constrained Optimization via Weak-Form Latent Dynamics](https://mlsanity.com/paper/2605.20639/) (2026)

## Abstract (arXiv)

Interacting particle system (IPS) models have proven to be highly successful for describing the spatial movement of organisms. However, it has proven challenging to infer the interaction rules directly from data. In the field of equation discovery, the Weak form Sparse Identification of Nonlinear Dynamics (WSINDy) methodology has been shown to be very computationally efficient for identifying the governing equations of complex systems, even in the presence of substantial noise. Motivated by the success of IPS models to describe the spatial movement of organisms, we develop WSINDy for second order IPSs to model the movement of communities of cells. Specifically, our approach learns the directional interaction rules that govern the dynamics of a heterogeneous population of migrating cells. Rather than aggregating cellular trajectory data into a single best-fit model, we learn the models for each individual cell. These models can then be efficiently classified according to the active classes of interactions present in the model. From these classifications, aggregated models are constructed hierarchically to simultaneously identify different species of cells present in the population and determine best-fit models for each species. We demonstrate the efficiency and proficiency of the method on several test scenarios, motivated by common cell migration experiments.