This study focuses on the intersection of machine learning and scientific computing, specifically on solving partial differential equations (PDEs) using neural network-based approaches. PDEs play a fundamental role in modeling physical systems across disciplines, but they are challenging to solve analytically and numerically, especially for large-scale systems. Many PDEs are NP-hard, and traditional numerical solvers typically rely on iterative schemes applied to discretized sub-problems, which demand significant computational resources and high-performance computing (HPC) infrastructure.

To address these challenges, Physics-Informed Neural Networks (PINNs), a class of machine learning models that incorporate physical laws directly into the learning process are explored. However, training these models is non-trivial due to issues like optimization difficulty, poor convergence, spectral bias (where neural networks preferentially learn low-frequency solution components before high-frequency ones), and many more. To target these bottlenecks, following novel solutions are developed:

• Spectral Bias Mitigation: The role of trainable activation functions (that can capture local features) in overcoming spectral bias is investigated.

• Architectures for Fluid–Structure Interaction (FSI): A hybrid neural network architecture capable of capturing the coupled dynamics of fluid and solid domains is designed. This system extends traditional PINN models to handle multiphysics problems where accurate interface modeling and force coupling are crucial.

• Quantum Neural Networks for PDEs: Research is further extended to quantum machine learning through the development of a quantum neural network architecture. By encoding neural network complexity into quantum circuits, quantum networks offer a promising approach to solving large-scale or computationally PDEs with improved scalability.