Dr. Anshima Singh

I am a Postdoctoral Research Associate in the Department of Mathematics at the University of Manchester, based on the UK EPSRC Prob_AI Hub, working with Prof. David Silvester and Prof. Catherine Powell.

Before moving to Manchester, I spent an amazing year as a Project Associate at the Indian Institute of Science, Bangalore, where I worked with Dr. Ratikanta Behera on designing physics-informed neural networks (PINNs) and their applications.

I completed my PhD in Applied Mathematics at the Indian Institute of Technology (BHU), Varanasi, supervised by Dr. Sunil Kumar. My PhD thesis is titled “Analysis of high-order methods for time-fractional partial differential equations with smooth and non-smooth solutions.” Prior to my PhD, I received my Bachelor and Master in Mathematics from the University of Lucknow.

---

My research lies at the intersection of numerical analysis and scientific machine learning, with a focus on developing efficient and accurate computational methods for differential equations. I am particularly interested in:

• Applications of flow-based deep generative models, as an adaptive sampling tool for problems such as bifurcation detection in parametrised fluid flow, and for improving PINN training and accuracy
• Physics-informed neural networks, including wavelet-based approaches for problems with steep gradients, rapid oscillations, and other singular behaviour that standard PINNs struggle with
• High-order numerical schemes for fractional partial differential equations, with rigorous stability and convergence analysis, especially for solutions with weak singularities

Publications

Preprints and articles under review

1. Adaptive sampling for hydrodynamic stability
   Singh, A., & Silvester, D.J. · Under review, 2025 · arXiv:2512.13532
2. DAS-PINNs for high-dimensional partial differential equations: extending deep adaptive sampling to spacetime domains
   Singh, A., & Silvester, D.J. · Under review, 2026 · arXiv:2606.06314

Journal articles

1. High-order hybrid computational method with adaptive mesh strategies for time-fractional nonlinear reaction–diffusion problems featuring weak singularity
   Singh, A. · Mathematics and Computers in Simulation, 244, 90–113, 2026 · DOI
2. An efficient wavelet-based physics-informed neural network for multiscale problems
   Pandey, H., Singh, A., & Behera, R. · Neural Networks, 108860, 2026 · DOI
3. A modified cubic B-spline collocation based numerical approximation of a fractional sub-diffusion equation
   Singh, A., Kumar, S., & Ramos, H. · International Journal of Applied and Computational Mathematics, 11(4), 157, 2025 · DOI
4. Error analysis of a fast high order spline approximation method for time-fractional nonlinear reaction-diffusion problems exhibiting singular behavior
   Priyanka, Singh, A., & Kumar, S. · Journal of Computational and Nonlinear Dynamics, 20(11), 111008, 2025 · DOI
5. A fast high-order nonpolynomial spline method for nonlinear time-fractional telegraph model for neutron transport in a nuclear reactor
   Priyanka, Singh, A., Kumar, S., & Vigo-Aguiar, J. · Mathematical Methods in the Applied Sciences, 48(9), 9751–9769, 2025 · DOI
6. Error analysis of a high-order fully discrete method for two-dimensional time-fractional convection-diffusion equations exhibiting weak initial singularity
   Singh, A., & Kumar, S. · Numerical Algorithms, 99(1), 251–284, 2025 · DOI
7. A novel high-order approximation method for higher-dimensional time-fractional reaction-diffusion problems with weak initial singularity
   Singh, R., Singh, A., Kumar, S., & Vigo-Aguiar, J. · Computational Methods for Differential Equations, 1–28, 2025 · DOI
8. Higher order numerical approximations for non-linear time-fractional reaction–diffusion equations exhibiting weak initial singularity
   Singh, A., & Kumar, S. · Communications in Nonlinear Science and Numerical Simulation, 139, 108317, 2024 · DOI
9. On new approximations of Caputo–Prabhakar fractional derivative and their application to reaction-diffusion problems with variable coefficients
   Singh, A., Kumar, S., & Vigo-Aguiar, J. · Mathematical Methods in the Applied Sciences, 47(1), 268–296, 2024 · DOI
10. An efficient numerical method based on exponential B-splines for time-fractional Black–Scholes equation governing European options
    Singh, A., & Kumar, S. · Computational Economics, 64(4), 1965–2002, 2024 · DOI
11. An efficient computational method based on exponential B-splines for a class of fractional sub-diffusion equations
    Singh, A., Kumar, S., & Ramos, H. · Computational Methods for Differential Equations, 12(4), 719–740, 2024 · DOI
12. A fully discrete scheme based on cubic splines and its analysis for time-fractional reaction–diffusion equations exhibiting weak initial singularity
    Singh, A., Kumar, S., & Vigo-Aguiar, J. · Journal of Computational and Applied Mathematics, 434, 115338, 2023 · DOI
13. High-order schemes and their error analysis for generalized variable coefficients fractional reaction–diffusion equations
    Singh, A., Kumar, S., & Vigo-Aguiar, J. · Mathematical Methods in the Applied Sciences, 46(16), 16521–16541, 2023 · DOI
14. A convergent exponential B-spline collocation method for a time-fractional telegraph equation
    Singh, A., & Kumar, S. · Computational and Applied Mathematics, 42(2), 79, 2023 · DOI

Full list on Google Scholar.

Talks

Invited talks

Retreats, workshops, seminars and winter schools

Conference presentations

Teaching and service

Teaching assistant, IIT (BHU) Varanasi, 2019–2023

• CSM-421: Numerical Solutions of PDEs
• MA-201: Numerical Techniques
• MA-203: Mathematical Method
• MA-101: Engineering Mathematics I

Reviewer for

• Applied Mathematical Modelling
• Neural Networks
• Communications in Nonlinear Science and Numerical Simulation
• Mathematics and Computers in Simulation
• Journal of Computational and Applied Mathematics
• Journal of Computational and Nonlinear Dynamics
• SN Computer Science
• Computational Methods for Differential Equations

Repositories

Code I have published. More on my GitHub profile.

CV

Download my CV (PDF)