TY - JOUR

T1 - U p -Net

T2 - a generic deep learning-based time stepper for parameterized spatio-temporal dynamics

AU - Stender, Merten

AU - Ohlsen, Jakob

AU - Geisler, Hendrik

AU - Chabchoub, Amin

AU - Hoffmann, Norbert

AU - Schlaefer, Alexander

N1 - Funding Information: J. Ohlsen was supported by the Hamburg University of Technology initiative (funding ID T-LP-E01- WTM-1801-02).

PY - 2023/6

Y1 - 2023/6

N2 - In the age of big data availability, data-driven techniques have been proposed recently to compute the time evolution of spatiotemporal dynamics. Depending on the required a priori knowledge about the underlying processes, a spectrum of black-box end-to-end learning approaches, physics-informed neural networks, and data-informed discrepancy modeling approaches can be identified. In this work, we propose a purely data-driven approach that uses fully convolutional neural networks to learn spatio-temporal dynamics directly from parameterized datasets of linear spatio-temporal processes. The parameterization allows for data fusion of field quantities, domain shapes, and boundary conditions in the proposed Up-Net architecture. Multi-domain Up-Net models, therefore, can generalize to different scenes, initial conditions, domain shapes, and domain sizes without requiring re-training or physical priors. Numerical experiments conducted on a universal and two-dimensional wave equation and the transient heat equation for validation purposes show that the proposed Up-Net outperforms classical U-Net and conventional encoder–decoder architectures of the same complexity. Owing to the scene parameterization, the UpNet models learn to predict refraction and reflections arising from domain inhomogeneities and boundaries. Generalization properties of the model outside the physical training parameter distributions and for unseen domain shapes are analyzed. The deep learning flow map models are employed for long-term predictions in a recursive time-stepping scheme, indicating the potential for data-driven forecasting tasks. This work is accompanied by an open-sourced code.

AB - In the age of big data availability, data-driven techniques have been proposed recently to compute the time evolution of spatiotemporal dynamics. Depending on the required a priori knowledge about the underlying processes, a spectrum of black-box end-to-end learning approaches, physics-informed neural networks, and data-informed discrepancy modeling approaches can be identified. In this work, we propose a purely data-driven approach that uses fully convolutional neural networks to learn spatio-temporal dynamics directly from parameterized datasets of linear spatio-temporal processes. The parameterization allows for data fusion of field quantities, domain shapes, and boundary conditions in the proposed Up-Net architecture. Multi-domain Up-Net models, therefore, can generalize to different scenes, initial conditions, domain shapes, and domain sizes without requiring re-training or physical priors. Numerical experiments conducted on a universal and two-dimensional wave equation and the transient heat equation for validation purposes show that the proposed Up-Net outperforms classical U-Net and conventional encoder–decoder architectures of the same complexity. Owing to the scene parameterization, the UpNet models learn to predict refraction and reflections arising from domain inhomogeneities and boundaries. Generalization properties of the model outside the physical training parameter distributions and for unseen domain shapes are analyzed. The deep learning flow map models are employed for long-term predictions in a recursive time-stepping scheme, indicating the potential for data-driven forecasting tasks. This work is accompanied by an open-sourced code.

KW - Machine learning

KW - Partial differential equations

KW - Representation learning

KW - Sensor data fusion

KW - Time integration

KW - Wave propagation

UR - http://www.scopus.com/inward/record.url?scp=85150650970&partnerID=8YFLogxK

U2 - 10.1007/s00466-023-02295-x

DO - 10.1007/s00466-023-02295-x

M3 - Article

VL - 71

SP - 1227

EP - 1249

JO - Computational mechanics

JF - Computational mechanics

SN - 0178-7675

IS - 6

ER -