My research is driven by the goal of advancing robust and efficient numerical frameworks for solving complex real-world problems modeled by fractional-order partial differential and integro-partial differential equations (PDEs), particularly those involving weak singularities. These types of problems frequently arise in diverse fields such as viscoelasticity, anomalous diffusion, and biological transport, where classical integer-order models fail to capture essential memory and hereditary effects.
A central focus of my work is the development and analysis of high-accuracy numerical schemes, including finite difference and finite element methods, for fractional models with singularities. I am particularly interested in combining these classical techniques with modern approaches such as wavelet-based methods for improved spatial adaptivity and computational efficiency.
Building upon this foundation, I aim to integrate wavelet theory and physics-informed neural networks (PINNs) to design hybrid algorithms capable of handling data-driven problems, inverse modeling, and uncertainty quantification in fractional-order systems. Wavelets offer multiscale representation and sparsity, making them ideal for enhancing PINNs in the presence of singularities or localized features.
Looking forward, my vision is to establish a comprehensive and scalable computational framework that unifies traditional numerical analysis with data-driven machine learning techniques to solve high-dimensional, nonlinear, and nonlocal problems encountered in scientific and engineering applications. This vision includes:
- Developing adaptive, stable, and convergent numerical solvers for fractional PDEs with singular kernels and multi-scale behaviors.
- Leveraging wavelets to improve resolution and compression in both deterministic and data-driven models.
- Designing novel fPINNs frameworks for weakly singular and integro-differential operators with provable convergence properties.
- Exploring real-time simulation and control of complex systems using hybrid PDE-PINN models.
Ultimately, I aspire to contribute to a deeper theoretical understanding and practical computation of fractional-order models, with applications spanning medical science, materials engineering, and computational finance.
