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 submitted to arXiv:math.NA (2026)

[P1] A. Yadav — "Energy-Preserving POD-Galerkin Reduced-Order Models with Neural Closure Corrections for Parameterized Navier–Stokes Equations" — arXiv:26XX.XXXXX, 2026

• Proved discrete energy inequality for ROM with neural closure; derived computable a posteriori error bounds for the coupled system
• Demonstrated long-time stability on lid-driven cavity (Re = 100–500) and flow past cylinder benchmarks

[P2] A. Yadav — "Residual-Based A Posteriori Error Bounds for PINN Solutions of Elliptic PDEs in Sobolev Norms" — arXiv:26XX.XXXXX, 2026

• Derived rigorous H¹ error bounds with explicit constants via Riesz representer approach
• Achieved near-optimal effectivity indices η ≈ 1.1–1.4 on Poisson, variable-coefficient, L-shaped domain, and convection-dominated benchmarks

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 · ArXiv Paper #1 · FEniCS · PyTorch · POD/Galerkin

Constructed POD-Galerkin ROM for parameterized incompressible Navier–Stokes with energy-constrained neural closure enforcing discrete energy inequality dE/dt ≤ −ν‖∇uᵣ‖². Proved long-time boundedness theorem; derived a posteriori error bounds decomposing POD truncation and closure approximation errors. Validated on 2D benchmarks where standard POD-ROM diverges at T → ∞.

2026 · ArXiv Paper #2 · FEniCS · PyTorch · Functional Analysis

Derived computable H¹ error bounds for PINN solutions of elliptic PDEs: ‖u − û‖₁ ≤ Cₚ‖r‖₋₁ with explicit Poincaré and coercivity constants. Implemented residual dual-norm evaluation via auxiliary FEM solve. Demonstrated that training loss is unreliable: PINN with low training loss had true error 10× larger than estimated bound predicted.

2026 · emcee · PyMC · Bayesian UQ

Formulated Bayesian inverse problem for spatially varying diffusion coefficient recovery from noisy observations. Implemented MCMC posterior sampling with PINN forward surrogate vs. FEM forward model. Quantified surrogate-induced posterior error via Wasserstein-2 distance.

2025–2026 · NumPy · SciPy · FEM from scratch

Implemented P1 and P2 Lagrange FEM from scratch on unstructured triangular meshes. Verified optimal convergence rates O(h), O(h²) for P1 and O(h²), O(h³) for P2, matching Brenner–Scott theory. Performed condition number analysis and mesh refinement studies.

2026 · DeepONet · FNO · NumPy/SciPy · Non-trivial geometry

Benchmarked DeepONet-RFF and FNO-POD on parametric elliptic PDEs on L-shaped, annular, and perforated domains. Identified systematic failure at re-entrant corner singularities (u ~ r²/³): DeepONet shows 7.4× error amplification near the corner. Demonstrated FNO achieves 0.03–4.4% in-distribution L² error but degrades 7–17× OOD vs. DeepONet's 1.2–2.5×. Proposed domain decomposition preprocessing as mitigation.

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. The platform is 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."