# Coarse-scale PDEs from fine-scale observations via machine learning

arXiv: https://arxiv.org/abs/1909.05707 · HTML: https://mlsanity.com/paper/1909.05707/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Sep 2019.

## Built on by (most cited first, 30 of 35)

- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
- [Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approach](https://mlsanity.com/paper/2201.13323/) (2022)
- [Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEs](https://mlsanity.com/paper/2203.05337/) (2022)
- [Learning the temporal evolution of multivariate densities via normalizing flows](https://mlsanity.com/paper/2107.13735/) (2021)
- [Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data](https://mlsanity.com/paper/2205.13545/) (2022)
- [A Framework Based on Symbolic Regression Coupled with eXtended Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion from Data](https://mlsanity.com/paper/2305.10706/) (2023)
- [RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operators](https://mlsanity.com/paper/2406.05470/) (2024)
- [Data-driven Control of Agent-based Models: an Equation/Variable-free Machine Learning Approach](https://mlsanity.com/paper/2207.05779/) (2022)
- [Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operator](https://mlsanity.com/paper/2408.02965/) (2024)
- [Recurrent Neural Networks for Dynamical Systems: Applications to Ordinary Differential Equations, Collective Motion, and Hydrological Modeling](https://mlsanity.com/paper/2202.07022/) (2022)
- [Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learning](https://mlsanity.com/paper/2309.07946/) (2023)
- [Enabling Local Neural Operators to perform Equation-Free System-Level Analysis](https://mlsanity.com/paper/2505.02308/) (2025)
- [GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Laws](https://mlsanity.com/paper/2410.22193/) (2024)
- [Time Series Forecasting Using Manifold Learning](https://mlsanity.com/paper/2110.03625/) (2021)
- [Extracting Stochastic Governing Laws by Nonlocal Kramers-Moyal Formulas](https://mlsanity.com/paper/2108.12570/) (2021)
- [Adaptive Uncertainty-Guided Model Selection for Data-Driven PDE Discovery](https://mlsanity.com/paper/2308.10283/) (2023)
- [Discovering Symbolic Differential Equations with Symmetry Invariants](https://mlsanity.com/paper/2505.12083/) (2025)
- [Black and Gray Box Learning of Amplitude Equations: Application to Phase Field Systems](https://mlsanity.com/paper/2207.03954/) (2022)
- [Data-driven modelling of brain activity using neural networks, Diffusion Maps, and the Koopman operator](https://mlsanity.com/paper/2304.11925/) (2023)
- [Multi-Scale Simulation of Complex Systems: A Perspective of Integrating Knowledge and Data](https://mlsanity.com/paper/2306.10275/) (2023)
- [Characterization of partial wetting by CMAS droplets using multiphase many-body dissipative particle dynamics and data-driven discovery based on PINNs](https://mlsanity.com/paper/2307.09142/) (2023)
- [Learning particle swarming models from data with Gaussian processes](https://mlsanity.com/paper/2106.02735/) (2021)
- [Tasks Makyth Models: Machine Learning Assisted Surrogates for Tipping Points](https://mlsanity.com/paper/2309.14334/) (2023)
- [Learning Macroscopic Dynamics from Partial Microscopic Observations](https://mlsanity.com/paper/2410.23938/) (2024)
- [Learning Collective Behaviors from Observation](https://mlsanity.com/paper/2311.00875/) (2023)
- [Data-Driven Model Selections of Second-Order Particle Dynamics via Integrating Gaussian Processes with Low-Dimensional Interacting Structures](https://mlsanity.com/paper/2311.00902/) (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)
- [Next Generation Equation-Free Multiscale Modelling of Crowd Dynamics via Machine Learning](https://mlsanity.com/paper/2508.03926/) (2025)
- [Fredholm Neural Networks for inverse problems in elliptic PDEs](https://mlsanity.com/paper/2507.06038/) (2025)

## Abstract (arXiv)

Complex spatiotemporal dynamics of physicochemical processes are often modeled at a microscopic level (through e.g. atomistic, agent-based or lattice models) based on first principles. Some of these processes can also be successfully modeled at the macroscopic level using e.g. partial differential equations (PDEs) describing the evolution of the right few macroscopic observables (e.g. concentration and momentum fields). Deriving good macroscopic descriptions (the so-called "closure problem") is often a time-consuming process requiring deep understanding/intuition about the system of interest. Recent developments in data science provide alternative ways to effectively extract/learn accurate macroscopic descriptions approximating the underlying microscopic observations. In this paper, we introduce a data-driven framework for the identification of unavailable coarse-scale PDEs from microscopic observations via machine learning algorithms. Specifically, using Gaussian Processes, Artificial Neural Networks, and/or Diffusion Maps, the proposed framework uncovers the relation between the relevant macroscopic space fields and their time evolution (the right-hand-side of the explicitly unavailable macroscopic PDE). Interestingly, several choices equally representative of the data can be discovered. The framework will be illustrated through the data-driven discovery of macroscopic, concentration-level PDEs resulting from a fine-scale, Lattice Boltzmann level model of a reaction/transport process. Once the coarse evolution law is identified, it can be simulated to produce long-term macroscopic predictions. Different features (pros as well as cons) of alternative machine learning algorithms for performing this task (Gaussian Processes and Artificial Neural Networks), are presented and discussed.