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A structure-preserving neural differential operator with embedded Hamiltonian constraints for modeling structural dynamics

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Abstract

Data-driven machine learning models are useful for modeling complex structures based on empirical observations, bypassing the need to generate a physical model where the physics is not well known or otherwise difficult to model. One disadvantage of purely data-driven approaches is that they tend to perform poorly in regions outside the original training domain. To mitigate this limitation, physical knowledge about the structure can be embedded in the neural network architecture. For large-scale systems, relevant physical properties such as the system state matrices may be expensive to compute. One way around this problem is to use scalar functionals, such as energy, to constrain the network to operate within physical bounds. We propose a neural network framework based on Hamiltonian mechanics to enforce a physics-informed structure to the model. The Hamiltonian framework allows us to relate the energy of the system to the measured quantities through the Euler–Lagrange equations of motion. In this work, the potential, kinetic energy, and Rayleigh damping terms are each modeled with a multilayer perceptron. Auto-differentiation is used to compute partial derivatives and assemble the relevant equations. The network incorporates a numerics-informed loss function via the residual of a multi-step integration term for deployment as a neural differential operator. Our approach incorporates a physics-constrained autoencoder to perform coordinate transformation between measured and generalized coordinates. This approach results in a physics-informed, structure-preserving model of the structure that can form the basis of a digital twin for many applications. The technique is demonstrated on computational examples.

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Acknowledgements

This work was funded by Sandia National Laboratories, Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525. The University of California San Diego acknowledges Sub-Contract Agreement 2169310 from Sandia National Laboratories for its particpation in this work.

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Authors and Affiliations

  1. Department of Structural Engineering, University of California San Diego, 9500 Gilman Dr, La Jolla, CA, 92093, USA

    David A. Najera-Flores & Michael D. Todd

  2. ATA Engineering, Inc., 13290 Evening Creek Dr S, San Diego, CA, 92128, USA

    David A. Najera-Flores

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Correspondence to Michael D. Todd.

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Appendix A: Autoencoder hyperparameters

Appendix A: Autoencoder hyperparameters

The autoencoder hyperparameters were obtained by running a grid search using the platform Weights and Biases [33]. The validation loss plotted as a function of epochs is shown in Fig. 17. For all of these runs, the nonlinear activation function was the Swish function [34]. Table 2 summarizes the results for all the iterations considered. The names of the iterations were automatically generated by Weights and Biases. Based on these results, the autoencoder used had 4 layers and 14 nodes per layer.

Fig. 17
figure 17

Validation loss as a function of epochs during training

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Table 2 Summary of hyperparameter grid search results
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Najera-Flores, D.A., Todd, M.D. A structure-preserving neural differential operator with embedded Hamiltonian constraints for modeling structural dynamics. Comput Mech 72, 241–252 (2023). https://doi.org/10.1007/s00466-023-02288-w

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