# Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data

arXiv: https://arxiv.org/abs/2205.13545 · HTML: https://mlsanity.com/paper/2205.13545/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 112 references that are in the map.
- First posted on arXiv 26 May 2022.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Backpropagation Algorithms and Reservoir Computing in Recurrent Neural Networks for the Forecasting of Complex Spatiotemporal Dynamics](https://mlsanity.com/paper/1910.05266/) (2019)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Multiscale Simulations of Complex Systems by Learning their Effective Dynamics](https://mlsanity.com/paper/2006.13431/) (2020)
- [Data-Driven Deep Learning of Partial Differential Equations in Modal Space](https://mlsanity.com/paper/1910.06948/) (2019)
- [Deep Neural Networks for Data-Driven Turbulence Models](https://mlsanity.com/paper/1806.04482/) (2018)
- [Coarse-scale PDEs from fine-scale observations via machine learning](https://mlsanity.com/paper/1909.05707/) (2019)
- [Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Space](https://mlsanity.com/paper/2106.03603/) (2021)
- [Using Machine Learning to Augment Coarse-Grid Computational Fluid Dynamics Simulations](https://mlsanity.com/paper/2010.00072/) (2020)
- [Discovering Nonlinear PDEs from Scarce Data with Physics-encoded Learning](https://mlsanity.com/paper/2201.12354/) (2022)
- [Numerical Aspects for Approximating Governing Equations Using Data](https://mlsanity.com/paper/1809.09170/) (2018)
- [Data-driven Discovery of Closure Models](https://mlsanity.com/paper/1803.09318/) (2018)
- [Linking Gaussian Process regression with data-driven manifold embeddings for nonlinear data fusion](https://mlsanity.com/paper/1812.06467/) (2018)
- [Accelerating PDE-constrained Inverse Solutions with Deep Learning and Reduced Order Models](https://mlsanity.com/paper/1912.08864/) (2019)
- [Learning emergent PDEs in a learned emergent space](https://mlsanity.com/paper/2012.12738/) (2020)

## Built on by (most cited first, 18 of 18)

- [AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identification](https://mlsanity.com/paper/2310.01433/) (2023)
- [RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operators](https://mlsanity.com/paper/2406.05470/) (2024)
- [Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learning](https://mlsanity.com/paper/2309.07946/) (2023)
- [Nonlinear Discrete-Time Observers with Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.12360/) (2024)
- [GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Laws](https://mlsanity.com/paper/2410.22193/) (2024)
- [Discovering Symbolic Differential Equations with Symmetry Invariants](https://mlsanity.com/paper/2505.12083/) (2025)
- [Some of the variables, some of the parameters, some of the times, with some physics known: Identification with partial information](https://mlsanity.com/paper/2304.14214/) (2023)
- [Data-driven and Physics Informed Modelling of Chinese Hamster Ovary Cell Bioreactors](https://mlsanity.com/paper/2305.03257/) (2023)
- [Tasks Makyth Models: Machine Learning Assisted Surrogates for Tipping Points](https://mlsanity.com/paper/2309.14334/) (2023)
- [A physics-informed neural network method for the approximation of slow invariant manifolds for the general class of stiff systems of ODEs](https://mlsanity.com/paper/2403.11591/) (2024)
- [Active search for Bifurcations](https://mlsanity.com/paper/2406.11141/) (2024)
- [Towards Coordinate- and Dimension-Agnostic Machine Learning for Partial Differential Equations](https://mlsanity.com/paper/2505.16549/) (2025)
- [Next Generation Equation-Free Multiscale Modelling of Crowd Dynamics via Machine Learning](https://mlsanity.com/paper/2508.03926/) (2025)
- [Tipping Points of Evolving Epidemiological Networks: Machine Learning-Assisted, Data-Driven Effective Modeling](https://mlsanity.com/paper/2311.00797/) (2023)
- [Fredholm Neural Networks for inverse problems in elliptic PDEs](https://mlsanity.com/paper/2507.06038/) (2025)
- [Machine Learning for the identification of phase-transitions in interacting agent-based systems: a Desai-Zwanzig example](https://mlsanity.com/paper/2310.19039/) (2023)
- [Using Linearized Optimal Transport to Predict the Evolution of Stochastic Particle Systems](https://mlsanity.com/paper/2408.01857/) (2024)
- [CBINNS: Cancer Biology-Informed Neural Network for Unknown Parameter Estimation and Missing Physics Identification](https://mlsanity.com/paper/2510.17920/) (2025)

## Abstract (arXiv)

We propose a machine learning framework for the data-driven discovery of macroscopic chemotactic Partial Differential Equations (PDEs) -- and the closures that lead to them -- from high-fidelity, individual-based stochastic simulations of E.coli bacterial motility. The fine scale, detailed, hybrid (continuum - Monte Carlo) simulation model embodies the underlying biophysics, and its parameters are informed from experimental observations of individual cells. We exploit Automatic Relevance Determination (ARD) within a Gaussian Process framework for the identification of a parsimonious set of collective observables that parametrize the law of the effective PDEs. Using these observables, in a second step we learn effective, coarse-grained "Keller-Segel class" chemotactic PDEs using machine learning regressors: (a) (shallow) feedforward neural networks and (b) Gaussian Processes. The learned laws can be black-box (when no prior knowledge about the PDE law structure is assumed) or gray-box when parts of the equation (e.g. the pure diffusion part) is known and "hardwired" in the regression process. We also discuss data-driven corrections (both additive and functional) of analytically known, approximate closures.