Speaker
Description
Model-order Reduction (MOR) is a technique used to reduce the complexity of high-dimensional dynamical systems while preserving their essential properties. It has been been successfully used in many scientific and engineering fields, such as Fluid Dynamics and Control Theory by methods like Proper Orthogonal Decomposition (POD), Autoencoder, Dynamic Mode decomposition (DMD) and so on. However, most MOR techniques assume that the full state is available which may not be the case, e.g. in molecular dynamics simulations and climate simulations. Hence techniques with partially observed states have been developed, since common MOR procedure cannot be applied.
Partial of the state space/observation is available of dynamical systems, unlike traditional MOR techniques mentioned above, there is not enough accessible information to build a ROM according to FOM. That is, we should propagate dynamics through the partially observed state space. The Mori-Zwanzig formalism (MZ) provides a powerful and systematic way to model the system by partially observed information by introducing a projection operator and memory kernel. The projection operator decomposes observed variables and observed variables to construct a model that only involves observed variables. We have implemented MZ in a large-scale linear system with partial observation, for which the matrix $A$ of the dynamics $\dot{x} = Ax$ is partitioned into $A_{11}$, $A_{12}$, $A_{21}$ and $A_{22}$ corresponding to observed states, unobserved states and their interactions respectively. Our work demonstrates that the solution of the Mori-Zwanzig (MZ) formalism is quite accurate compared to the RK45 ODE solver in a large-scale linear system when a partial dimension (e.g., 5 out of 10) is observed. However, there is no free lunch; as a consequence of missing information about the unobserved dimensions, an integro-differential equation that includes a non-Markovian memory kernel must be solved. Consequently, the MZ approach suffers form high computational costs, especially when the integration time step is long. We plan to implement Neural Networks (NN) to approximate the memory kernel from data, thereby avoiding expensive on-the-fly computations. Subsequently, we intend to apply the MZ formalism to nonlinear large-scale systems to make it more widely applicable.
