Nikolaos Pallikarakis, PhD

Nikolaos Pallikarakis

Applied Mathematics and Computation

from differential equations to machine learning and engineering applications

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About

I am an applied mathematician working in differential equations, inverse problems, deep/machine learning, numerical methods, and scientific computing, with applications to acoustic and electromagnetic scattering and semiconductor metrology. I received my PhD in Applied/Computational Mathematics from the National Technical University of Athens (NTUA), and I have over 13 years of experience in academia and industry.

I currently work as a Senior Application Engineer and Data Scientist Consultant, while continuing my postdoctoral research. I am particularly interested in problems where mathematical analysis and computational methods can be brought together in practice.

Research

Postdoctoral research at NTUA. Differential equations, inverse problems, eigenvalue problems, direct and inverse scattering, deep learning.

Teaching

Teaching assistant and tutor in differential equations, numerical methods and mathematical analysis.

Industry

Senior Application Engineer at n&k Technology, Inc.: thin-film and semiconductor metrology, computational methods and machine learning for optical measurements.

Research Overview

Research Interests

Direct & Inverse Spectral Problems

Eigenvalue problems in inhomogeneous and discontinuous media. Transmission eigenvalues, Sturm-Liouville theory, spectral analysis of differential operators and uniqueness results. Applications to material characterization.

Deep/Machine Learning

Neural network architectures (CNNs, MLPs, PINNs) and machine/statistical learning for solving direct and inverse problems. Applications to acoustic and electromagnetic scattering.

Numerical & Computational Methods

Computational methods for direct and inverse problems and differential equations. Applications to scientific computing and data-driven analysis.

Publications

11 Published Articles in peer-reviewed journals, 3 Published Articles in international conferences.

Articles in peer-reviewed journals

11. Inverse Electromagnetic Scattering for Doubly-Connected Cylinders using Convolutional Neural Networks

L. Mindrinos, N. Pallikarakis and N. L. Tsitsas

Inverse Problems, 42, 075008, 2026 (corresponding author)

DOI: 10.1088/1361-6420/ae84f6 | Code →

10. Neumann series of Bessel functions in direct and inverse spherically symmetric transmission eigenvalue problems

V. V. Kravchenko, L. E. Murcia-Lozano and N. Pallikarakis

Applied Mathematics and Computation, 531, 130188, 2026 (corresponding author)

DOI: 10.1016/j.amc.2026.130188

9. Estimating properties of a homogeneous bounded soil using machine learning models

K. Kalimeris, L. Mindrinos and N. Pallikarakis

Communications on Analysis and Computation, 6, 62–93, 2025

DOI: 10.3934/cac.2025019

8. A review on the direct and inverse transmission eigenvalue problem for the spherically symmetric refractive index

N. Pallikarakis

Boletín de la Sociedad Matemática Mexicana, 30, 92, 2024 (invited review article)

DOI: 10.1007/s40590-024-00661-0

7. Exploring the inverse line-source scattering problem in dielectric cylinders with deep neural networks

N. Pallikarakis, A. Kalogeropoulos and N. L. Tsitsas

Physica Scripta, 99, 116013, 2024 (corresponding author)

DOI: 10.1088/1402-4896/ad852c | Code →

6. Application of machine learning regression models to inverse eigenvalue problems

N. Pallikarakis and A. Ntargaras

Computers and Mathematics with Applications, 154, 162–174, 2024 (corresponding author)

DOI: 10.1016/j.camwa.2023.11.038 | Code →

5. Analysis of the transmission eigenvalue problem with two conductivity parameters

R. Ceja Ayala, I. Harris, A. Kleefeld and N. Pallikarakis

Applicable Analysis, 103, 211–239, 2023

DOI: 10.1080/00036811.2023.2181167

4. Uniqueness of a spherically symmetric refractive index from modified transmission eigenvalues

D. Gintides, N. Pallikarakis and K. Stratouras

Inverse Problems, 38, 085010, 2022 (corresponding author)

DOI: 10.1088/1361-6420/ac7b3f

3. On the modified transmission eigenvalue problem with an artificial metamaterial background

D. Gintides, N. Pallikarakis and K. Stratouras

Research in the Mathematical Sciences, 8, 2021

DOI: 10.1007/s40687-021-00278-z

2. The inverse transmission eigenvalue problem for a discontinuous refractive index

D. Gintides and N. Pallikarakis

Inverse Problems, 33, 055006, 2017

DOI: 10.1088/1361-6420/aa5bf0

1. A computational method for the inverse transmission eigenvalue problem

D. Gintides and N. Pallikarakis

Inverse Problems, 29, 104010, 2013

DOI: 10.1088/0266-5611/29/10/104010

Articles in international conferences with proceedings

3. DUV-Vis-IR Optical Metrology Enhancements with Machine Learning

J. Roberts, A. Ntargaras, N. Pallikarakis, K. Florios and M. Colli

36th Annual SEMI Advanced Semiconductor Manufacturing Conference (ASMC), Albany, NY, USA, 2025, pp. 1-5

DOI: 10.1109/ASMC64512.2025.11010784

2. Predicting the number of line sources inside a cylinder using classification neural networks

N. Pallikarakis, A. Kalogeropoulos and N. L. Tsitsas

2024 IEEE International Symposium on Antennas and Propagation and INC/USNC-URSI Radio Science Meeting (AP-S/INC-USNC-URSI), Florence, Italy, 2024, pp. 675-676

DOI: 10.1109/AP-S/INC-USNC-URSI52054.2024.10687110

1. A computational approach for the inverse problem of reconstructing a spherically symmetric refractive index using modified transmission eigenvalues

D. Gintides, N. Pallikarakis and K. Stratouras

34th General Assembly and Scientific Symp. Int. Union of Radio Science (URSI GASS), 2021

DOI: 10.23919/URSIGASS51995.2021.9560389

Presentations & Talks

19th Panhellenic Conference in Mathematical Analysis

Athens, Greece | December 18-20, 2025

Invited Talk: "Transmission Eigenvalues in Spherically Symmetric Media: From Theory to NSBF Computation" (with V. Kravchenko and L.E. Murcia - Lozano)

Conference Programm

2025 36th SEMI Advanced Semiconductor Manufacturing Conference (ASMC)

Albany, NY, USA | May 5-8, 2025

Talk: "DUV-Vis-IR Optical Metrology Enhancements with Machine Learning" (with J. Roberts, A. Ntargaras, K. Florios, M. Colli)

Conference Presentation

Seminar on Analysis, Differential Equations and Mathematical Physics

Online | March 6, 2025

Invited Talk: "Direct and Inverse Spherically Symmetric Transmission Eigenvalue Problems: Past, Present, and Future"

Seminar Series

2024 IEEE International Symposium on Antennas and Propagation (AP-S/INC-USNC-URSI)

Florence, Italy | July 14-19, 2024

Talk: "Predicting the number of line sources inside a cylinder using classification neural networks" (with A. Kalogeropoulos, N. L. Tsitsas)

DOI: 10.1109/AP-S/INC-USNC-URSI52054.2024.10687110 | Session

SEMICON West

San Fransisco, California, USA | July 9-11, 2024

Poster: "DUV-Vis-NIR OCD Metrology utilizing Machine Learning for BCD Semiconductor Manufacturing Yield Enhancements" (with J. Roberts, J. Lam, K. Florios, M. Colli, M. Lombardo, M. Ravasio)

Poster Details

International Summer School "Mathematics and Machine Learning for Image Analysis"

University of Bologna, Italy | June 4-12, 2024

Poster: "Inverse line-source scattering problems in dielectric cylinders: A Divide-and-Conquer DNN approach"

Summer School

Numerical Analysis Seminars, Department of Mathematics

University of Bologna, Italy | May 20, 2024

Invited Talk: "Numerical solution of inverse eigenvalue and inverse source problems"

Seminar Talk

International Conference on Frontiers of Characterization and Metrology for Nanoelectronics (FCMN)

Monterey, California, USA | April 15-18, 2024

Poster: "DUV-Vis-NIR OCD Metrology for BCD Semiconductor Manufacturing Yield Enhancements" (with J. Roberts, J. Lam, K. Florios, M. Colli, M. Lombardo, M. Ravasio)

Conference Website

Seminars on Inverse Problems Theory and Applications

Online | March 26, 2024

Invited Talk: "Exploring Inverse Eigenvalue Problems through Machine Learning"

Seminar Talk

Seminars of the Department Mathematics NTUA

Athens, Greece | March 22, 2024

Invited Talk: "Solving Inverse Problems with Supervised Machine Learning Models"

Seminar Talk

11th Applied Inverse Problems (AIP) Conference

Göttingen, Germany | September 4-8, 2023

Talk: "The inverse spectral problem for a spherically symmetric refractive index using modified transmission eigenvalues" (with D. Gintides, K. Stratouras)

Conference Website

Seminars of Industrial Mathematics - Centro de Investigación y Tecnología Matemática de Galicia

University of A Coruña, Spain | January 25, 2023

Invited Talk: "Inverse eigenvalue problems for inhomogeneous media: theoretical and numerical advances"

Seminar Talk

Seminars on Inverse Problems Theory and Applications

Online | October 4, 2022

Invited Talk: "Inverse Spectral Problems for classic and modified transmission eigenvalues"

Seminar Talk

Scientific Symposium of the International Union of Radio Science (URSI GASS)

Rome, Italy | August 28 - September 4, 2021

Talk: "A computational approach for the inverse modified transmission eigenvalue problem" (with D. Gintides, K. Stratouras)

DOI: 10.23919/URSIGASS51995.2021.9560389 | Proceedings

10th Applied Inverse Problems (AIP) Conference

Grenoble, France | July 8-12, 2019

Talk: "The Inverse ITE Problem for Discontinuous Refractive Index" (with D. Gintides)

Conference Website

Researcher's Night NTUA

Athens, Greece | September 30, 2016

Poster: "The reconstruction of a refractive index from transmission eigenvalues"

Event Page

International Conference: Modern Mathematical Methods in Science and Technology (M3ST)

Kalamata, Greece | August 30 - September 1, 2015

Talk: "Uniqueness theorems for the inverse transmission eigenvalue problem for a discontinuous refractive index" (with D. Gintides)

Conference Information

7th International Conference Inverse Problems: Modeling and Simulation (IPMS)

Fethiye, Turkey | May 26-31, 2014

Talk: "The inverse transmission eigenvalue problem for a discontinuous refractive index" (with D. Gintides)

Conference Series

Franco-German Summer School: Inverse Problems and PDEs

Bremen, Germany | October 7-11, 2013

Poster: "The Inverse Transmission Eigenvalue Problem" (with D. Gintides)

Summer School

Research Grants

High-Performance Computing (HPC) Resource Allocation
Project: "Deep Learning for Applied Inverse Problems (Deep-AIP)"
 HPC ARIS
19th Call		 02/2026–02/2027 Principal Investigator
High-Performance Computing (HPC) Resource Allocation
Project: "Application of Machine Learning to Inverse Problems (AMALIP)"
 HPC ARIS
16th Call		 06/2024–06/2025 Principal Investigator
Researchers' support with emphasis on young researchers (2nd Cycle)
Project: "Direct and Inverse Spectral Problems in Scattering Theory"		 Greece & EU
European Social Fund (ESF)		 06/2020–08/2021 Post-Doctoral Researcher
PEBE Program
Project: "Inverse Scattering Theory in Linear Elasticity"		 NTUA
Basic Research		 09/2011–11/2012 Research Assistant

Editorial Service

Guest Editor

  • 2026: Special issue on Asymptotics, optimization and machine learning
    Boletín de la Sociedad Matemática Mexicana (Springer)
    Dedicated to the rich interplay between asymptotic analysis, optimization, and modern machine learning.
    Covering spectral asymptotics, large-scale optimization, deep learning theory, stochastic optimization, and numerical PDEs.
    🔗 Call for papers

Peer Review Activity

Reviewer for Journals

  • 2026 IEEE Transactions on Antennas and Propagation, Journal of Computational Physics (Elsevier), Lobachevskii Journal of Mathematics (Springer), Applicable Analysis (Taylor & Francis)
  • 2025 Inverse Problems (IOP), Journal of Inverse and Ill-posed Problems (De Gruyter), Mathematical Reviews/MathSciNet
  • 2024 Inverse Problems (IOP), Mathematical Reviews/MathSciNet
  • 2023 Journal of Scientific Computing (Springer)

Work Experience

Senior Application Engineer and Data Scientist Consultant

n&k Technology, Inc., San Jose, California, USA
Nov 2018 – Present
  • Analyze raw reflectance/transmittance data and develop metrology models using proprietary analysis software (GUI/API)
  • Apply RCWA (Rigorous Coupled-Wave Analysis) and computational electromagnetism models combined with thin film material characterization for optical semiconductor metrology
  • Program in C++ and Python for data processing, automation, and model deployment
  • R&D projects for machine learning enhanced metrology
  • Work on non-destructive testing of semiconductor devices; bridge theory and practical manufacturing needs

Post-Doctoral Collaborations

National Technical University of Athens, Department of Mathematics, Greece
Jun 2020 – Present
  • Direct and inverse scattering theory and computational methods for eigenvalue problems in nonhomogeneous domains
  • Machine/Deep learning methods with applications in direct and inverse problems

Tutor in Mathematics

Avakas, Athens, Greece
Apr 2017 – Dec 2018
  • Tutor to undergraduate and postgraduate students in: Differential Equations, Mathematical Analysis, Mathematical Programming with MATLAB/Octave, Complex Analysis, Computational Methods, Numerical Analysis, Linear Algebra

Teaching Assistant

National Technical University of Athens, School of Applied Mathematical and Physical Sciences, Greece
Oct 2012 – Apr 2017
  • Teaching assistant in undergraduate Ordinary and Partial Differential Equations courses (exercises and laboratory)
  • Courses: ODE (SEFME, School of Civil Engineering), PDE (SEMFE, Chemical Engineering School), PDE II (SEMFE)

Research Assistant

National Technical University of Athens, School of Applied Mathematical and Physical Sciences, Greece
Sep 2011 – Nov 2012
  • Research assistant for the Project: "The inverse scattering theory in linear elasticity using indicator functions"
  • Numerical implementation of algorithms using MATLAB software for inverse problems

Education

PhD in Applied Mathematics/Differential Equations

National Technical University of Athens
Department of Mathematics
Sep 2012 – Apr 2017

Thesis: "The inverse spectral problem for the reconstruction of the refractive index from the interior transmission problem"

Available in: English & Greek

Supervisor: Prof. Drossos Gintides

MSc in Applied Mathematical Sciences

National Technical University of Athens
School of Applied Mathematical and Physical Sciences
Sep 2010 – May 2012

Grade: 9.3/10 (Excellent)

Direction: Mathematical Analysis, Differential Equations and Computational Mathematics

Core Subjects: Numerical Methods for Differential Equations, Numerical Analysis, Non Linear Differential Equations, Functional Analysis, Integral Equations and Inverse Problems

Thesis: "The interior transmission problem in inverse scattering and applications"

Available in: Greek

Supervisor: Prof. Drossos Gintides

Diploma (MEng) - Integrated Master

National Technical University of Athens
School of Applied Mathematical and Physical Sciences
Sep 2005 – June 2010

Grade: 7.9/10 (Very Good)

Direction: Applied Mathematics

Core Subjects: Linear Algebra, Mathematical Analysis, Complex Analysis, Probability Theory, Statistics, Mathematical Software, Applied Physics, Applied Mechanics, Object-Oriented Programming, Algorithms

Specialization Streams:

  • Applied Mathematical Analysis: Differential Equations, Numerical Analysis, Optimization, Non Linear Systems, Real and Functional Analysis, Measure Theory, Operator Theory, Stochastic Processes, Mathematical Modeling, Dynamical Systems
  • Applied Mechanics / Computational Simulation: Computational Mechanics, Fluid Mechanics, Continuum Mechanics, Elasticity Theory, Non Destructive Testing, Asymptotic Methods, Composite Materials

Thesis: "Integral equations and representations in linear elasticity"

Supervisor: Prof. Drossos Gintides

Awards & Scholarships

  • ELKE NTUA Scholarship (2013–2017): 4-year PhD scholarship
  • Thomaidio Award (2017, 2014): For publications in SCI-indexed journals (awarded twice)

Skills & Expertise

Programming Languages

  • Python (NumPy, SciPy, Pandas)
  • C++ (Data processing & optimization)
  • MATLAB/Octave (Numerical computation)
  • Mathematica (Symbolic computation)
  • LaTeX (Academic writing)
  • JavaScript/HTML (Web development)

Machine Learning & Deep Learning

  • TensorFlow/Keras
  • Scikit-learn (supervised & unsupervised learning algorithms)
  • Neural Networks (MLP, CNN, PINNs)
  • Regression & Classification
  • Model Optimization & Regularization
  • Dimension Reduction (PCA, UMAP)

Scientific & Mathematical Computing

  • Numerical Methods & PDE Solvers
  • Signal Processing & Spectral Analysis
  • Inverse Problems & Reconstruction
  • Acoustic and Electromagnetic Scattering Problems
  • Eigenvalue & Eigenfunction Problems

Domain Expertise

  • Semiconductor Metrology
  • RCWA (Rigorous Coupled-Wave Analysis)
  • Optical Measurement & Characterization
  • Non-destructive Testing
  • Thin Film and Trench (OCD) Metrology

Tools & Platforms

  • Google Colab & HPC Computing
  • GitHub & Version Control
  • Overleaf (LaTeX)
  • Linux/Unix & Command Line
  • Windows
  • Microsoft Office Suite

Languages

  • Greek (Native)
  • English (B2 - FCE Cambridge, ECCE Michigan)
  • Spanish (B2 - Universidade de Vigo)
  • Italian (B1 - Istituto Italiano di Cultura)
  • German (B1 - Zertifikat Deutsch)

Soft Skills

  • Self-motivated & work efficiently unsupervised
  • Team player & team organizer
  • Technical presentations & science communication
  • Ability to explain complex concepts to diverse audiences
  • Mentoring & academic teaching
  • Collaborative research & interdisciplinary communication

Lifelong Learning & Professional Development

Summer Schools & Workshops

Online Courses & Certifications

  • Machine Learning with Python – EdX & IBM (August 2020)
  • Introduction to Python for Data Science – EdX (May 2017)
  • Introduction to R for Data Science – EdX (May 2017)

Blog & Articles

Read my articles for insights into mathematics and practical applications.

Solving the Heat Equation with Physics-Informed Neural Networks: PyTorch vs TensorFlow

PyTorch implementation of the heat equation PINN with a detailed comparison between PyTorch and TensorFlow concepts.

Medium · 2026

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Solving the Heat Equation with Physics-Informed Neural Networks: A Practical Guide

Step-by-step tutorial on building PINNs to solve PDEs without labeled data, using TensorFlow and automatic differentiation

Medium · 2026

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Direct and Inverse Eigenvalue Problems — a gentle introduction

An introduction to eigenvalue theory, direct problems, and inverse reconstruction methods

Medium · 2024

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Numerical solution of the Sturm-Liouville eigenvalue problem — matslise (MATLAB) and pyslise (Python) solvers

Practical guide to solving Sturm-Liouville problems numerically using specialized libraries and tools

Medium · 2024

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Follow me on Medium: @pallikarakis.n for more articles on inverse problems, deep learning, and computational mathematics.

Get In Touch

I'm interested in collaborations, and discussions on inverse problems, deep learning, and scientific computing.

Email (NTUA) Email (Personal)