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

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.