# Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approach

arXiv: https://arxiv.org/abs/2201.13323 · HTML: https://mlsanity.com/paper/2201.13323/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 97 references that are in the map.
- First posted on arXiv 31 Jan 2022.

## Builds on (most cited first, 12 of 12)

- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Multiscale Simulations of Complex Systems by Learning their Effective Dynamics](https://mlsanity.com/paper/2006.13431/) (2020)
- [Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2012.02895/) (2020)
- [Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equations](https://mlsanity.com/paper/1907.03507/) (2019)
- [Deep Neural Networks with Random Gaussian Weights: A Universal Classification Strategy?](https://mlsanity.com/paper/1504.08291/) (2015)
- [Inferring solutions of differential equations using noisy multi-fidelity data](https://mlsanity.com/paper/1607.04805/) (2016)
- [Coarse-scale PDEs from fine-scale observations via machine learning](https://mlsanity.com/paper/1909.05707/) (2019)
- [Partial observations and conservation laws: Grey-box modeling in biotechnology and optogenetics](https://mlsanity.com/paper/1909.04234/) (2019)

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- [Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learning](https://mlsanity.com/paper/2309.07946/) (2023)
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- [Discovering Symbolic Differential Equations with Symmetry Invariants](https://mlsanity.com/paper/2505.12083/) (2025)
- [Data-driven modelling of brain activity using neural networks, Diffusion Maps, and the Koopman operator](https://mlsanity.com/paper/2304.11925/) (2023)
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- [Physics-informed neural networks for high-dimensional solutions and snaking bifurcations in nonlinear lattices](https://mlsanity.com/paper/2507.09782/) (2025)
- [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)
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- [Fredholm Neural Networks for inverse problems in elliptic PDEs](https://mlsanity.com/paper/2507.06038/) (2025)
- [Equation-Free Coarse Control of Distributed Parameter Systems via Local Neural Operators](https://mlsanity.com/paper/2509.23975/) (2025)
- [HEATNETs: Explainable Random Feature Neural Networks for High-Dimensional Parabolic PDEs](https://mlsanity.com/paper/2511.00886/) (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)
- [Linear Stability Analysis of Physics-Informed Random Projection Neural Networks for ODEs](https://mlsanity.com/paper/2408.15393/) (2024)
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- [Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approach](https://mlsanity.com/paper/2603.21568/) (2026)

## Abstract (arXiv)

We address a three-tier data-driven approach to solve the inverse problem in complex systems modelling from spatio-temporal data produced by microscopic simulators using machine learning. In the first step, we exploit manifold learning and in particular parsimonious Diffusion Maps using leave-one-out cross-validation (LOOCV) to both identify the intrinsic dimension of the manifold where the emergent dynamics evolve and for feature selection over the parametric space. In the second step, based on the selected features, we learn the right-hand-side of the effective partial differential equations (PDEs) using two machine learning schemes, namely shallow Feedforward Neural Networks (FNNs) with two hidden layers and single-layer Random Projection Networks(RPNNs) which basis functions are constructed using an appropriate random sampling approach. Finally, based on the learned black-box PDE model, we construct the corresponding bifurcation diagram, thus exploiting the numerical bifurcation analysis toolkit. For our illustrations, we implemented the proposed method to construct the one-parameter bifurcation diagram of the 1D FitzHugh-Nagumo PDEs from data generated by D1Q3 Lattice Boltzmann simulations. The proposed method was quite effective in terms of numerical accuracy regarding the construction of the coarse-scale bifurcation diagram. Furthermore, the proposed RPNN scheme was ~ 20 to 30 times less costly regarding the training phase than the traditional shallow FNNs, thus arising as a promising alternative to deep learning for solving the inverse problem for high-dimensional PDEs.