Manifold meta-learning for reduced-complexity neural system identification

System identification has greatly benefited from deep learning techniques, particularly for modeling complex, nonlinear dynamical systems with partially unknown physics where traditional approaches may not be feasible. However, deep learning models often require large datasets and significant computational resources for both training and inference due to their high-dimensional parameterizations. To address this challenge, we propose a meta-learning framework that discovers a low-dimensional manifold within the parameter space of an over-parameterized neural network architecture. This manifold is learned from an abundant meta-dataset of input-output sequences generated by a simulator of the class of systems of interest, and thereafter enables efficient model training while preserving the network's expressive power for the considered system class. Unlike bilevel meta-learning approaches, our method employs an auxiliary neural network to map datasets directly onto the learned manifold, eliminating the need for costly second-order gradient computations during meta-training and generally reducing the number of first-order updates required in inference, which could be expensive for large models. Finally, we derive a probabilistic bound that quantifies the performance gap between training on the learned low-dimensional manifold and the full parameterization. We validate our approach on a family of Bouc-Wen oscillators, which is a well-studied nonlinear system identification benchmark. We demonstrate that we are able to learn accurate models even in small-data scenarios.