We propose a new system identification method Violina (various-of-trajectories identification of linear time-invariant non-Markovian dynamics). In the Violina framework, we optimize the coefficient matrices of state-space model and memory kernel in the given space using a projected gradient descent method so that its model prediction matches the set of multiple observed data. Using Violina we can identify a linear non-Markovian dynamical system with constraints corresponding to a priori knowledge on the model parameters and memory effects. Using synthetic data, we numerically demonstrate that the Markovian and non-Markovian state-space models identified by the proposed method have considerably better generalization performances compared to the models identified by an existing dynamic decomposition-based method.

Modeling and controlling complex spatiotemporal dynamical systems driven by partial differential equations (PDEs) often necessitate dimensionality reduction techniques to construct lower-order models for computational efficiency. This paper explores a deep autoencoding learning method for reduced-order modeling and control of dynamical systems governed by spatiotemporal PDEs. We first analytically show that an optimization objective for learning a linear autoencoding reduced-order model can be formulated to yield a solution closely resembling the result obtained through the dynamic mode decomposition with control algorithm. We then extend this linear autoencoding architecture to a deep autoencoding framework, enabling the development of a nonlinear reduced-order model. Furthermore, we leverage the learned reduced-order model to design controllers using stability-constrained deep neural networks. Numerical experiments are presented to validate the efficacy of our approach in both modeling and control using the example of a reaction-diffusion system.