Deep Learning for Dynamical Systems

Predicting spatiotemporal activity of complex system is challenging as the dynamics can span multiple scales. Machine learning allows building models that can predict the complex dynamics with high fidelity. I have developed several machine learning pipelines that can achieve this task.

Machine Learning for predicting complex systems

Kuramoto–Sivashinsky Chaotic Model

Fitz-Hugh Nagumo Model

Adaptive time-sampling based on hierarchical deep learning

Multiscale systems are ubiquitous in science and engineering. Resolving all temporal scales for such systems often incurs a cost that prevents experimentation. In this study, we use techniques from deep learning to learn a hierarchy of different time-scale behaviour that characterize the dynamics of multiscale systems.

AHiTS method

Surrogate modeling via Deep learning

Deep learning is revolutionizing every domain of science and engineering ranging from predictive modeling to new drug discoveries. Since large-scale models naturally produce tremendous volumes of data, this can be leveraged by deep-learning technologies to obtain computationally tractable models that facilitate control and optimization tasks in a variety of engineering applications. Towards this direction, I am actively working on developing novel deep-learning frameworks that can learn lower dimensional manifolds that encode the high-fidelity map.

Projection using nonlinear manifolds

IEEE 118 system results

A. Hamid, D. Rafiq, S. A. Nahvi, M. A. Bazaz, Deep Learning Assisted Surrogate Modelling of Large-Scale Power Grids, Sustainable Energy, Grids & Networks, 2023. 

Nonlinear Model Order Reduction

Nonlinear model order reduction using system theoretic measures

Models emerging from the time-dependent nonlinear partial differential equations are highly large-scale and nonlinear resulting in huge computational resources. My work builds upon the idea of the existing generalization of moment-matching to nonlinear systems based on steady-state considerations. Towards this direction, I have proposed various efficient reduction methods for nonlinear state-space models based on approximate moment-matching and techniques from Koopman theory.

Graphical highlight of nonlinear MOR techniques

D. Rafiq, Advanced Techniques for Dimensionality Reduction of Nonlinear and Parametric Dynamical Systems using System-Theoretic Measures, Ph.D. Thesis, 2022.

Collection of large-scale models for nonlinear model reduction

We provide a publicly available collection of sixteen large-scale benchmark nonlinear state-space models coded in MATLAB. The models are scalable in spatiotemporal degrees of freedom. The aim is to provide the active research community with a suite of high-dimensional nonlinear models to test the state-of-the-art nonlinear model reduction strategies.

Chafee Infante Equation

2D Burgers' Model

1D Burgers' Model

Nonlinear Circuit Model

D. Rafiq, M. A. Bazaz, A Collection of Large Scale Benchmark Models for Nonlinear Model Order Reduction, Archives of Computational Methods in Engineering, July 2022. 

Data-driven Power Grid Parameter Estimation

Data-driven modeling of power systems

Modern power grids consist of thousands of loads, interconnects, and sources. With the increasing integration of decentralized renewable generation, it is getting even more complex. The de-facto computational barrier limits our ability to analyze the power systems thoroughly. My work focuses on using modern Koopman analysis to obtain low-dimensional models that are amenable to linear control theory. Also, I have worked on discovering models from power system data that are parsimonious; thus promoting sparsity and generalizability.

Suggested reading:

A. Hamid, D. Rafiq, S. A. Nahvi and M. A. Bazaz "Discovering low-rank representations of large-scale power-grid models using Koopman theory", IEEE TEECCON 2022.

A. Hamid, D. Rafiq, S. A. Nahvi, and M. A. Bazaz, "Power Grid parameter estimation
using Sparse Identification of Nonlinear Dynamics", IEEE ICICCSP 2022.