4. Ensemble of Numerics-Informed Neural Networks with Embedded Hamiltonian Constraints for Modeling Nonlinear 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 readily otherwise model-able. 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 model architecture via the model topology or numerical constraints in the formulation. For large-scale systems, relevant physics, 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 (e.g., accelerations) 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 all the relevant equations, including computing the generalized inertia matrix by forming the Hessian of the kinetic energy with respect to the generalized coordinates. Moreover, a Bayesian approach is used to estimate model-form error to predict domain shifts in the data and enable model correction. The network incorporates a numerics-informed loss function via the residual of a multistep integration term, allowing the ensemble of networks to be time-integrated with new initial conditions and an arbitrary external force after it has been trained. The approach is demonstrated on simple exemplars, such as a two degree-of-freedom (DOF) damped oscillator with cubic nonlinearities.