A parametric framework for kernel-based dynamic mode decomposition using deep learning

TL;DR

The paper addresses efficient parametric prediction of dynamical systems by extending kernel-based DMD with the LANDO algorithm into a two-stage offline-online framework. It constructs parameter-specific LANDOs offline and then learns a parametric map to states at a target time using a deep neural network, with optional POD-based reduction for high-dimensional problems. Key contributions include a practical parametric LANDOscheme, explicit ALD-based sparse dictionary construction, and demonstration on Lotka-Volterra, heat diffusion, and Allen-Cahn dynamics, showing accuracy and extrapolation capabilities. This approach enables rapid many-query evaluations in uncertainty quantification and design optimization while managing computational cost via model reduction.

Abstract

Surrogate modelling is widely applied in computational science and engineering to mitigate computational efficiency issues for the real-time simulations of complex and large-scale computational models or for many-query scenarios, such as uncertainty quantification and design optimisation. In this work, we propose a parametric framework for kernel-based dynamic mode decomposition method based on the linear and nonlinear disambiguation optimization (LANDO) algorithm. The proposed parametric framework consists of two stages, offline and online. The offline stage prepares the essential component for prediction, namely a series of LANDO models that emulate the dynamics of the system with particular parameters from a training dataset. The online stage leverages those LANDO models to generate new data at a desired time instant, and approximate the mapping between parameters and the state with the data using deep learning techniques. Moreover, dimensionality reduction technique is applied to high-dimensional dynamical systems to reduce the computational cost of training. Three numerical examples including Lotka-Volterra model, heat equation and reaction-diffusion equation are presented to demonstrate the efficiency and effectiveness of the proposed framework.

Figures (11)

• Figure 1: A schematic diagram for proposed parametric LANDO framework. At the offline stage, a series of LANDO models are prepared and used for data generation at particular time instant $t^*$ during online stage. A DNN surrogate model $\tilde{\mathcal{M}}^{t^*}_{\bm{\theta}}(\bm{\mu})$ is subsequently applied to learn the mapping $\mathcal{M}^{t^*}(\bm{\mu})$ between parameters and states based on those data. For high-dimensional dynamical systems, dimensionality reduction technique based on POD is employed to the state data before DNN learning and recover to the full state after DNN prediction (demonstrated in blue boxes).
• Figure 2: A comparison between the parametric LANDO predictions and the ground truth of the Lotka-Volterra model with varying parameter $\alpha$ at time instants 100, 300, 450 and 500. The dynamical system is initiated from $\mathbf{x}_0=[80, 20]^{\top}$.
• Figure 3: A comparison between the parametric LANDO predictions and the ground truth of the Lotka-Volterra model varying parameter $\alpha$ at time instants 100, 300, 450 and 500. The dynamical system is initiated from $\mathbf{x}_0=[70, 20]^{\top}$.
• Figure 4: (a) Mean and standard deviation of the relative $L_2$ error of parametric LANDO prediction for time instants from 50 to 600. (b) Mean and standard deviation of the relative $L_2$ error of parametric LANDO prediction with different size of the training dataset at time instants 50, 300 and 600.
• Figure 5: The relative $L_2$ error of parametric LANDO prediction for the Lotka-Volterra model at time instants 100 and 500. Both $\alpha$ and $\beta$ varied simultaneously. (a) with initial condition $x_0=[70,20]^{\top}$, (b) with initial condition $x_0=[80,20]^{\top}$