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Multiscale simulations of complex systems by learning their effective dynamics

A preprint version of the article is available at arXiv.


Predictive simulations of complex systems are essential for applications ranging from weather forecasting to drug design. The veracity of these predictions hinges on their capacity to capture effective system dynamics. Massively parallel simulations predict the system dynamics by resolving all spatiotemporal scales, often at a cost that prevents experimentation, while their findings may not allow for generalization. On the other hand, reduced-order models are fast but limited by the frequently adopted linearization of the system dynamics and the utilization of heuristic closures. Here we present a novel systematic framework that bridges large-scale simulations and reduced-order models to learn the effective dynamics of diverse, complex systems. The framework forms algorithmic alloys between nonlinear machine learning algorithms and the equation-free approach for modelling complex systems. Learning the effective dynamics deploys autoencoders to formulate a mapping between fine- and coarse-grained representations and evolves the latent space dynamics using recurrent neural networks. The algorithm is validated on benchmark problems, and we find that it outperforms state-of-the-art reduced-order models in terms of predictability, and large-scale simulations in terms of cost. Learning the effective dynamics is applicable to systems ranging from chemistry to fluid mechanics and reduces the computational effort by up to two orders of magnitude while maintaining the prediction accuracy of the full system dynamics. We argue that learning the effective dynamics provides a potent novel modality for accurately predicting complex systems.

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Fig. 1: Multiscale-LED.
Fig. 2: FitzHugh–Nagumo model.
Fig. 3: Kuramoto-Sivashinsky equation.
Fig. 4: Iterative latent forecasting in flow past a cylinder.
Fig. 5: Multiscale LED in flow past a cylinder.

Data availability

All the data analysed in this paper were produced with open-source software described in the code availability statement. Reference data and the scripts used to produce the data figures, as well as instructions to launch training and inference (evaluation of trained models) scripts for LED, are available on the GitHub repository:

Code availability

Simulations of the KS equation have been performed with a spectral fourth-order solver for stiff PDEs developed in Python. Simulation of the FHN equation has been performed with an LB method developed in Python. The LED software is implemented in Python, utilizing the PyTorch library for the neural networks. All codes are made readily available in the GitHub repository: Direct numerical simulations were performed with the flow solver CubismUP 2D:


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We thank N.Kallikounis (ETH Zurich) for helpful discussions on the LB method, P. Weber and M. Chatzimanolakis (ETH Zurich) for help with the simulations of the flow past a cylinder and Y. Kevrekidis (Johns Hopkins University) and K. Spiliotis (University of Rostock) for providing code to reproduce data for the FHN equation. We acknowledge the infrastructure and support of CSCS, providing the necessary computational resources under project s929.

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Authors and Affiliations



P.K. conceived the project; P.R.V., G.A., C.U. and P.K. designed and performed research; P.R.V. and G.A. contributed new analytic tools; P.R.V., G.A. and P.K. analysed data and P.R.V., G.A. and P.K. wrote the paper.

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Correspondence to Petros Koumoutsakos.

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Nature Machine Intelligence thanks Harsh Bhatia and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Methods E and F, Comparison Measures A–D, Results A–E, Figs. 1–14 and Tables 1–17.

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Vlachas, P.R., Arampatzis, G., Uhler, C. et al. Multiscale simulations of complex systems by learning their effective dynamics. Nat Mach Intell 4, 359–366 (2022).

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