Since Newton’s era, we have been constructing mathematical models for dynamical systems (ordinary, partial, and stochastic differential equations) from first principles. Those closed-form models allowed our understanding of the physical world to advance for the past centuries. However, accurately deriving computationally efficient, and explainable models for systems like the financial market, the climate or an epidemic disease remains a challenging problem. This indicates the need for a different approach to construct mathematical models that circumvent explicit analytical formulation.

In my talk, I will discuss, how the construction of models directly from observations/data is possible for stochastic differential equations. I will portray that deep learning architectures based on numerical stochastic integrators, such as the Euler-Maruyama, can be used to learn/identify data-driven stochastic differential equations directly from data [1]. I will illustrate how identified effective stochastic differential equations can be constructed in physical-interpretable coarse variables when they are available, but also on latent data-driven observables using the manifold learning scheme Diffusion Maps [2]. I will provide examples to illustrate this approach for (a) electric-field mediated colloidal crystallization using data obtained from Brownian Dynamics Simulations [3,4] (b) a Susceptible-Infected-Susceptible adaptive network [5] and (c) an event-driven agent-based financial model [6,7].

References:
[1] 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.

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

[3] Evangelou, N., Dietrich, F., Bello-Rivas, J. M., Yeh, A., Stein, R., Bevan, M. A., & Kevekidis, I. G. (2022). Learning Effective SDEs from Brownian Dynamics Simulations of Colloidal Particles. arXiv preprint arXiv:2205.00286.

[4] Yang, Y., Thyagarajan, R., Ford, D. M., & Bevan, M. A. (2016). Dynamic colloidal assembly pathways via low dimensional models. The Journal of chemical physics, 144(20), 204904.

[5] 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.

[6] Omurtag, A., & Sirovich, L. (2006). Modeling a large population of traders: Mimesis and stability. Journal of Economic Behavior & Organization, 61(4), 562-576.

[7] 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.