Dynamic mode decomposition (DMD)-guided deep surrogates for parametrized dynamical systems

Parameterized dynamical systems arise in many scientific and engineering areas. Developing surrogate models for parameterized problems are important in applications demanding rapid, long-term predictions across a large parameter space. In this work, we introduce efficient deep learning-based surrogate models for the simulation of parametrized dynamical systems with the help of Dynamic Mode Decomposition (DMD) method. We first propose a vanilla approach for systems where standard DMD is effective. In this approach, a neural network is trained to map parameters directly to their corresponding DMD modes and eigenvalues, using a hybrid loss that ensures accuracy in both the components and the reconstructed dynamics. For more complex systems requiring a large number of modes, we present an order-reduced approach that integrates an autoencoder to compress the high-dimensional state into some hidden features. This allows the dynamics to be captured by fewer modes in latent space, significantly reducing memory and optimization challenges. The optimization of the autoencoder and the parameter-to-latent-dynamics network performed in an iterative manner. The effectiveness and accuracy of proposed methods for long-term prediction are demonstrated through several parametric problems.