Akhilesh Yadav

Applied Mathematician · Numerical PDEs · Scientific Machine Learning

M.Sc. Applied Mathematics — IIEST Shibpur

Applied mathematician working at the intersection of numerical analysis, reduced-order modelling, and scientific machine learning. Research focuses on structure-preserving model reduction for parametric PDEs, a posteriori error certification for neural PDE solvers, and Bayesian inverse problems with surrogate-accelerated uncertainty quantification. Experienced in finite element methods, physics-informed neural networks, and computational fluid dynamics. Seeking a fully funded PhD to develop mathematically rigorous, ML-enhanced numerical methods for complex physical systems.

Two independent research preprints, hosted on Zenodo (arXiv math.NA endorsement pending)

[P1] A. Yadav — "Dissipative Neural Closure for Energy-Stabilized POD–Galerkin Models of Incompressible Flows"

• Proved discrete energy inequality for ROM with neural closure via skew-symmetric + negative-semidefinite decomposition; derived computable a posteriori error bounds via a Gronwall-type argument
• Validated on the DFG 2D-2 cylinder benchmark at Re = 100

[P2] A. Yadav — "Residual-Based A Posteriori Error Bounds for Physics-Informed Neural Networks in Energy Seminorm"

• Derived the bound $|\nabla(u-\hat u)|{L^2} \le \frac{1}{\alpha}|R|{H^{-1}} + |\nabla w|_{L^2}$ with harmonic lifting for boundary mismatch
• Achieved near-optimal effectivity indices $\eta \in [1.05, 1.31]$

Structure-preserving reduced-order models for parametric PDEs
A posteriori error estimation and certification for neural PDE solvers
Bayesian inverse problems with learned surrogates and certified UQ
Hybrid FEM–neural operator methods with convergence guarantees
Neural operators on complex geometries with geometric priors (FEEC)
Computational fluid dynamics

2026 · energy-stable-rom · Zenodo DOI · FEniCS · PyTorch

Structure-preserving POD-Galerkin ROM for incompressible flows with a neural closure decomposed into skew-symmetric and negative-semidefinite parts, guaranteeing dissipativity via a Gronwall-type bound. Validated on the DFG 2D-2 cylinder benchmark at Re = 100.

2026 · pinn-aposteriori-bounds · Zenodo DOI · FEniCS · PyTorch

Computable $H^1$-seminorm error bound for PINN solutions of elliptic PDEs using a harmonic-lifting correction for boundary mismatch. Effectivity indices $\eta \in [1.05, 1.31]$ across benchmark problems, demonstrating that training loss alone is an unreliable proxy for true error.

2026 · bayesian-pinn-inversion · emcee · Bayesian UQ

Bayesian inverse problem for spatially varying diffusion coefficient recovery from noisy observations, using a PINN forward surrogate. Posterior sampled via MCMC; surrogate-induced posterior error quantified via Wasserstein-2 distance against the FEM-forward-model posterior.

2025–2026 · fem-solver-suite · NumPy · SciPy

$P_1$/$P_2$ Lagrange FEM implemented from scratch on unstructured triangular meshes for Poisson, heat, and convection–diffusion equations. Verified optimal convergence rates matching Brenner–Scott theory, with condition number analysis and mesh refinement studies.

2026 · neural-operator-benchmarks · DeepONet · FNO

Benchmarked DeepONet-RFF and FNO-POD on parametric elliptic PDEs over L-shaped, annular, and perforated domains. Identified systematic failure at re-entrant corner singularities ($u \sim r^{2/3}$), with DeepONet showing 7.4× error amplification near the corner; FNO achieves lower in-distribution error but degrades more sharply out-of-distribution.

Finite Element Methods (FEniCS/Firedrake) · Finite Differences · Finite Volumes · POD/Galerkin ROM · Spectral Methods

PINNs · DeepONet · FNO · Neural Closure Models · PyTorch · Automatic Differentiation · Loss Landscape Analysis

MCMC (emcee, PyMC) · Polynomial Chaos Expansions · Gaussian Processes · Bayesian Optimization

Python (NumPy, SciPy, Matplotlib) · MATLAB · LaTeX · Git/GitHub · Jupyter

Sobolev Spaces · Weak Formulations · A Priori/A Posteriori Error Analysis · Functional Analysis · Measure Theory

M.Sc. Applied Mathematics · IIEST Shibpur · 2022–2024 · CGPA 8.15/10 Specialization: Fluid Mechanics & Numerical Methods Coursework: Advanced Fluid Mechanics · PDEs · Numerical Analysis · Linear Algebra · Probability & Statistics · Functional Analysis

B.Sc. Mathematics & Physics · VBSPU Jaunpur · 2018–2021

I am building MathLumen — an academic platform bridging rigorous mathematical foundations with modern scientific machine learning, PINNs, and physics-informed modeling, aimed at researchers and graduate students working at the intersection of numerical analysis and deep learning.

I am open to academic collaboration in:

• Scientific Machine Learning & operator learning
• A posteriori error estimation for neural PDE solvers
• Structure-preserving reduced-order models
• Bayesian inverse problems and certified UQ
• Hybrid FEM–neural operator methods

📧 akhileshyadav.maths@gmail.com

"The goal of numerical analysis is not merely to compute answers, but to understand when and why those answers are correct."