State-space reconstruction from partial observables using an invertible neural network with structure-preserving properties for nonlinear structural dynamics

Data-driven machine learning models are useful for modeling complex-typically nonlinear-structures based on empirical observations, bypassing the need to generate a physical model in cases where the physics is not well known or easily modeled. 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. 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. A challenge with this hybrid data-driven, physics-constrained approach is the problem of limited observability, i.e., not being able to measure structural response in a complete coordinate system that is compatible with the physical constraints being enforced. To overcome this issue, we propose combining an invertible neural network autoencoder architecture with knowledge from embedding theory to enrich the limited observable data with time-delay embeddings. From Taken’s theorem, we know that a sufficient time-delay embedding is diffeomorphically equivalent to the underlying state space of the system. We use this information to find time-delays of the original data and build the diffeomorphic mapping with a neural network encoder. The approach is demonstrated on computational and experimental examples.