Yongho Kim

Yongho Kim, Jan Heiland

Simulations of large-scale dynamical systems require expensive computations. Low-dimensional parametrization of high-dimensional states such as Proper Orthogonal Decomposition (POD) can be a solution to lessen the burdens by providing a certain compromise between accuracy and model complexity. However, for really low-dimensional parametrizations (for example for controller design) linear methods like the POD come to their natural limits so that nonlinear approaches will be the methods of choice. In this work we propose a convolutional autoencoder (CAE) consisting of a nonlinear encoder and an affine linear decoder and consider combinations with k-means clustering for improved encoding performance. The proposed set of methods is compared to the standard POD approach in two cylinder-wake scenarios modeled by the incompressible Navier-Stokes equations.

Yongho Kim, Yongho Choi

Physics-informed neural networks have been widely applied to partial differential equations with great success because the physics-informed loss essentially requires no observations or discretization. However, it is difficult to optimize model parameters, and these parameters must be trained for each distinct initial condition. To overcome these challenges in second-order reaction-diffusion type equations, a possible way is to use five-point stencil convolutional neural networks (FCNNs). FCNNs are trained using two consecutive snapshots, where the time step corresponds to the step size of the given snapshots. Thus, the time evolution of FCNNs depends on the time step, and the time step must satisfy its CFL condition to avoid blow-up solutions. In this work, we propose deep FCNNs that have large receptive fields to predict time evolutions with a time step larger than the threshold of the CFL condition. To evaluate our models, we consider the heat, Fisher's, and Allen-Cahn equations with diverse initial conditions. We demonstrate that deep FCNNs retain certain accuracies, in contrast to FDMs that blow up.

Jan Heiland, Yongho Kim, Steffen W. R. Werner

Polytopic autoencoders provide low-di\-men\-sion\-al parametrizations of states in a polytope. For nonlinear PDEs, this is readily applied to low-dimensional linear parameter-varying (LPV) approximations as they have been exploited for efficient nonlinear controller design via series expansions of the solution to the state-dependent Riccati equation. In this work, we develop a polytopic autoencoder for control applications and show how it improves on standard linear approaches in view of LPV approximations of nonlinear systems. We discuss how the particular architecture enables exact representation of target states and higher order series expansions of the nonlinear feedback law at little extra computational effort in the online phase and how the linear though high-dimensional and nonstandard Lyapunov equations are efficiently computed during the offline phase. In a numerical study, we illustrate the procedure and how this approach can reliably outperform the standard linear-quadratic regulator design.

Jan Heiland, Yongho Kim

With the advancement of neural networks, there has been a notable increase, both in terms of quantity and variety, in research publications concerning the application of autoencoders to reduced-order models. We propose a polytopic autoencoder architecture that includes a lightweight nonlinear encoder, a convex combination decoder, and a smooth clustering network. Supported by several proofs, the model architecture ensures that all reconstructed states lie within a polytope, accompanied by a metric indicating the quality of the constructed polytopes, referred to as polytope error. Additionally, it offers a minimal number of convex coordinates for polytopic linear-parameter varying systems while achieving acceptable reconstruction errors compared to proper orthogonal decomposition (POD). To validate our proposed model, we conduct simulations involving two flow scenarios with the incompressible Navier-Stokes equation. Numerical results demonstrate the guaranteed properties of the model, low reconstruction errors compared to POD, and the improvement in error using a clustering network.

Yongho Kim, Yongho Choi

In recent years, Physics-informed neural networks (PINNs) have been widely used to solve partial differential equations alongside numerical methods because PINNs can be trained without observations and deal with continuous-time problems directly. In contrast, optimizing the parameters of such models is difficult, and individual training sessions must be performed to predict the evolutions of each different initial condition. To alleviate the first problem, observed data can be injected directly into the loss function part. To solve the second problem, a network architecture can be built as a framework to learn a finite difference method. In view of the two motivations, we propose Five-point stencil CNNs (FCNNs) containing a five-point stencil kernel and a trainable approximation function for reaction-diffusion type equations including the heat, Fisher's, Allen-Cahn, and other reaction-diffusion equations with trigonometric function terms. We show that FCNNs can learn finite difference schemes using few data and achieve the low relative errors of diverse reaction-diffusion evolutions with unseen initial conditions. Furthermore, we demonstrate that FCNNs can still be trained well even with using noisy data.