Blog Posts

18 posts

The pressure field adjusting itself to satisfy the incompressibility constraint

navier–stokes 04/05

EducationalOctober 15th, 2026

Putting the Navier–Stokes Equations Together: The Pressure Poisson Equation

Newton's law joins a parcel's acceleration and the forces on it, and the no-pile-up rule stands beside them. Stepping the system forward needs the pressure, which has no equation of its own. This post derives the one hidden in the constraint: the pressure Poisson equation.

Pressure, viscous friction and gravity acting on a small parcel of fluid

navier–stokes 03/05

EducationalOctober 7th, 2026

The Forces on a Fluid Parcel: Pressure, Viscosity and Gravity

Pressure pushes onto a fluid parcel's faces, viscous friction drags along them, and gravity pulls on every kilogram inside. This post derives each force and turns it into its share of the parcel's acceleration.

Fluid flowing through a box fixed in space, entering and leaving through its faces

navier–stokes 02/05

EducationalOctober 1st, 2026

Conservation of Mass: The Continuity Equation

Mass can't appear out of nowhere or simply disappear. This post turns that simple idea into the continuity equation, explains what the divergence of a flow measures, and shows why water obeys the constraint ∇·u = 0.

A velocity field with a fluid parcel drifting along its path

navier–stokes 01/05

EducationalSeptember 29th, 2026

The Material Derivative: How a Fluid Parcel Accelerates

Newton's F = ma follows one fluid parcel, but the velocity field is only known at fixed points. This post bridges the gap between Eulerian and Lagrangian viewpoints, deriving the material derivative.

Diagram of the Fourier Neural Operator architecture with lifting, Fourier layers and projection

physics simulation 09/14

EducationalSeptember 7th, 2026

Fourier Neural Operators (FNOs)

A step by step guide through the architecture of Fourier Neural Operators, their applications, and how they efficiently process global information through the Fourier transform.

A 2D Fourier mode shown as a striped pattern on a grid

physics simulation 08/14

EducationalAugust 17th, 2026

The Fourier Transform, explained for Neural Operators

A semi-deep dive into the Fourier transform and its applications in signal processing and neural networks.

DeepONet branch and trunk networks mixing basis functions

physics simulation 07/14

EducationalAugust 10th, 2026

Neural Operators and DeepONet

An introduction to neural operators, DeepONet, and how they can be used to learn mappings between function spaces.

Stable and unstable numerical wave solutions on either side of the CFL limit

physics simulation 06/14

EducationalJuly 31st, 2026

The CFL Condition of the Wave Equation

A deep dive into the Von Neumann stability analysis of the wave equation and its connection to the CFL condition.

Numerical error of a finite difference scheme growing over time

physics simulation 05/14

EducationalJuly 30th, 2026

Von Neumann Stability Analysis

A deep dive into the von Neumann stability analysis, its connection to the CFL condition, and how it determines the stability of numerical schemes.

Neural network prediction of a propagating wave compared to the numerical solution

physics simulation 04/14

EducationalJuly 2nd, 2026

Neural Networks as Wave Equation Approximators

A deep dive into the wave equation, its reformulation for neural network approximation, and the challenges of learning dynamics with finite speed.

Forces acting on a small segment of a vibrating string

physics simulation 03/14

EducationalJune 10th, 2026

Deriving the Wave Equation

A step-by-step derivation of the wave equation from first principles, exploring how tension and mass density lead to wave propagation with finite speed.

Heat diffusing across a 2D grid, solved with a finite difference scheme

physics simulation 02/14

EducationalJune 3rd, 2026

Partial Differential Equations and Neural PDEs

From the mathematics of change to neural networks that learn dynamics — understanding PDEs, numerical solvers, and how modern generative models are PDEs in disguise.

Trajectory of an ODE solution integrated step by step

physics simulation 01/14

EducationalMay 21st, 2026

Ordinary Differential Equations and Neural ODEs

From the mathematics of change to neural networks that learn dynamics — understanding ODEs, numerical solvers, and how modern generative models are ODEs in disguise.

Noise being transformed into structured data along a probability path

EducationalJune 21st, 2025

The Mathematics Behind Flow and Diffusion Models

A deep dive into the mathematical foundations of flow-based models and diffusion processes in deep learning.

Breakdown of context-reliance subtypes in the RAG faithfulness study

Research PaperMay 30th, 2026

Context-Parametric Inversion

A controlled faithfulness study on how QASPER fine-tuning affects context reliance across a RAG pipeline for NLP research papers.

Project RecapApril 28th, 2025

Dual-MIDI Transformer

A recap of my work on the Dual-Midi Transformer project, which aims to generate music from MIDI data.

Audio waveform used as input for a music generation model

Literature ReviewMay 2nd, 2025

An Introduction to Artificial Music Generation

A comprehensive overview of the current state of music generation techniques

Gradient descent stepping down a loss surface

EducationalApril 6th, 2025

Introduction to Gradient Descent and Backpropagation

An introduction into Neural Networks, Gradient Descent and Backpropagation.