(241g) Data-Driven Bifurcation Diagrams for Neural (Integro-) Differential Equations | AIChE

(241g) Data-Driven Bifurcation Diagrams for Neural (Integro-) Differential Equations


Evangelou, N. - Presenter, Johns Hopkins University
Fabiani, G., Universitá degli Studi di Napoli Federico II
Cui, T., Johns Hopkins University
Giovanis, D. G., Johns Hopkins University
Kevrekidis, I. G., Princeton University
We present a machine learning framework for the systematic identification of macroscopic governing laws and parameter-dependent critical transitions of high-dimensional complex dynamical systems from data. The data may come from either real experiments or can be synthetic data produced by a high-detailed and high-dimensional surrogate model at a fine-scale level, such lattice Boltzmann [1] simulations, adaptive complex networks [2], or agent-based models [3-4].

We use coarse mean field variables, if available, but also discover latent data-driven variables that capture the emergent behavior with the manifold learning scheme Diffusion maps [5]. We assume that such high-dimensional data live (or are fast attracted) on a nonlinear slow invariant manifold that can be “unfolded” with Diffusion Maps.

We illustrate how deep learning frameworks inspired by numerical integrators for ordinary (e.g. Runge-Kutta), partial (e.g finite differences), and stochastic (e.g. Euler-Maruyama) differential equations can be used to identify effective coarse-grained equations [6-8]. Thus, we re-construct coarse-scale data-driven bifurcation diagrams for the identification of tipping points (e.g saddle-node, subcritical Hopf) and we discuss the validity of our results.

We successfully test our framework with several complex systems, including (a) an event-driven agent-based financial model (b) Lattice Boltzmann simulation of biological neural dynamics (c) a susceptible-infected-susceptible (SIS) model adaptive network (d) Desai-Zwanzig [4].


[1] Galaris, E., Fabiani, G., Gallos, I., Kevrekidis, I., & Siettos, C. (2022). Numerical bifurcation analysis of pdes from lattice Boltzmann model simulations: a parsimonious machine learning approach. Journal of Scientific Computing, 92(2), 34.

[2] Gross, T., & Kevrekidis, I. G. (2008). Robust oscillations in SIS epidemics on adaptive networks: Coarse graining by automated moment closure. Europhysics Letters, 82(3), 38004.

[3] Liu, P., Siettos, C. I., Gear, C. W., & Kevrekidis, I. G. (2015). Equation-free model reduction in agent-based computations: Coarse-grained bifurcation and variable-free rare event analysis. Mathematical Modelling of Natural Phenomena, 10(3), 71-90.

[4] Zagli, N., Pavliotis, G. A., Lucarini, V., & Alecio, A. (2023). Dimension reduction of noisy interacting systems. Physical Review Research, 5(1), 013078.

[5] Coifman, R. R., & Lafon, S. (2006). Diffusion maps. Applied and computational harmonic analysis, 21(1), 5-30.

[6] Rico-Martinez, R., Krischer, K., Kevrekidis, I. G., Kube, M. C., & Hudson, J. L. (1992). Discrete-vs. continuous-time nonlinear signal processing of Cu electrodissolution data. Chemical Engineering Communications, 118(1), 25-48.

[7] González-García, R., Rico-Martìnez, R., & Kevrekidis, I. G. (1998). Identification of distributed parameter systems: A neural net based approach. Computers & chemical engineering, 22, S965-S968.

[8] Dietrich, F., Makeev, A., Kevrekidis, G., Evangelou, N., Bertalan, T., Reich, S., & Kevrekidis, I. G. (2023). Learning effective stochastic differential equations from microscopic simulations: Linking stochastic numerics to deep learning. Chaos: An Interdisciplinary Journal of Nonlinear Science, 33(2), 023121.