Speaker : Mr. Abhishek Ajayakumar
S.R. Number : 06-18-01-10-12-18-1-16176
Title : “Sparsification of Reaction-Diffusion Dynamical Systems in Complex Networks.”
Research Supervisor: Prof. Soumyendu Raha
Date & Time : July 31, 2023 (Monday) at 11:30 AM
Venue : The Thesis Colloquium will be held on HYBRID Mode # 102 CDS Seminar Hall /MICROSOFT TEAMS.
Please click on the following link to join the Thesis Colloquium:
MS Teams link:
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Abstract
Graph sparsification is an area of interest in computer science and applied mathematics. Sparsification of a graph, in general, aims to reduce the number of edges in the network while preserving specific properties of the graph, like cuts and subgraph counts. Modern deep learning frameworks, which utilize recurrent neural network decoders and convolutional neural networks, are characterized by a significant number of parameters. Pruning redundant edges in such networks and rescaling the weights can be useful. Computing the sparsest cuts of a graph is known to be NP-hard, and sparsification routines exist for generating linear sized sparsifiers in almost quadratic running time. The complexity of this task varies, and it is closely linked to the level of sparsity we desire to achieve. In our study, we extend the concept of sparsification to the realm of reaction-diffusion complex systems. We aim to address the challenge of reducing the number of edges in the network while preserving the underlying flow dynamics.
Sparsification of such complex networks is approached as an inverse problem guided by data representing flows in the network, where we adopt a relaxed approach considering only a subset of trajectories. We map the network sparsification problem to a data assimilation problem on a Reduced Order Model (ROM) space with constraints targeted at preserving the eigenmodes of the Laplacian matrix under perturbations. The Laplacian matrix is the difference between the diagonal matrix of degrees and the graph’s adjacency matrix. We propose approximations to the eigenvalues and eigenvectors of the Laplacian matrix subject to perturbations for computational feasibility and include a custom function based on these approximations as a constraint on the data assimilation framework.
In the latter phase of the study, we developed a framework to enhance POD-based model reduction techniques inreaction-diffusion complex systems. This framework incorporates techniques from stochastic filtering theory and pattern recognition.
Getting optimal state estimates from a noisy model and noisy measurements forms the core of the filtering problem. By integrating the particle filtering technique, we generate the reaction-diffusion state vector at various time steps, utilizing noisy measurements obtained from ROM. To ensure the framework’s effectiveness, we make intermittent updates to the system variables during the particle filtering step, employing the carefully crafted sparse graph. The framework is utilized for experimentation, and results are presented on random graphs, considering the diffusion equation on the graph and the chemical Brusselator model as the dynamical systems embedded in the graph. We discuss the method’s limitations, and the proposed framework is evaluated by comparing its performance against the Neural ODE-based approach, which serves as a compelling reference due to its demonstrated robustness in specific applications.
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