Research

Approximation theory & numerical analysis on ODEs/PDEs
I have grown interests towards the theory of convergence of orthogonal expansions crucial from the persepective of developing efficient numerical/computational schemes for solving DEs. In particular, I have an inclination towards numerical analysis concerning spectral Fourier-Chebyshev methods.

Spectral method aided deep learning for ODEs/PDEs
Work on solving PDEs through neural networks was recently made popular in the form of Physics-informed neural networks (PINNs) and neural operators. There has been a lot of progress in accelerating neural PDE solvers involving variants of PINNs, particularly using transfer learning. However, these methods rely on backpropagation and thus necessitate computing the derivatives of the given PDE for each forward pass, rendering them computationally intensive, even for simple PDEs, and often resulting in limited approximation accuracy. Spectral methods for solving PDEs promise fast convergence with a small number of basis functions. Recent progresses in research promises deep learning based spectral method in which hierarchical spatial basis functions are extracted from a trained Deep Operator Nets (DeepONets) and employed in a spectral method to solve the PDE.

Optimisation in continuous domains
Motivated from the exposure during my MITACS project and my formal training to earn a minor, I have started to explore the field of optimisation over continuous domains which are crucial from the perspective of deep learning. Particularly, I have dedicated my attention towards optimisation over manifolds and their applications in developing efficient derivative algorithms of gradient descent optimisers.