Category Archives: Blog

Problem: State the Poisson-Boltzmann mean-field equation for the electrostatic potential \(\phi(\mathbf x)\) at temperature \(T\) in a dielectric of isotropic permittivity \(\varepsilon\) containing mobile and immobile charge carriers. Explain what it means to apply Debye-Huckel linearization to the Poisson-Boltzmann mean-field … Continue reading →

Problem: State the Schwarzschild metric solution to the Einstein field equations and the assumptions underlying it. Solution: The Schwarzschild metric is given by: \[ds^2=\left(1-\frac{r_s}{r}\right)d(ct)^2-\frac{dr^2}{1-\frac{r_s}{r}}-r^2d\Omega^2\] (where \(r_s:=2GM/c^2\) is the Schwarzschild radius) and is (by Birkhoff’s theorem) the unique isotropic solution of … Continue reading →

Problem: Explain why for a random walk in \(\mathbf R^d\), the probability distribution of the random vector sum \(\mathbf x:=\mathbf x_1+…+\mathbf x_N\) of \(N\) i.i.d. random vectors \(\mathbf x_i\) each with identical mean \(\boldsymbol{\mu}:=\langle\mathbf x_i\rangle\) and identical covariance matrix \(\sigma^2:=\langle\mathbf … Continue reading →

Problem: Define the \(2\) words in the phrase “ideal paramagnet“. Show that a classical ideal paramagnet of \(N\) spins each with the same fixed magnetic dipole moment \(\mu:=|\boldsymbol{\mu}|\) placed in a uniform external magnetic field \(B:=|\mathbf B|\) will develop a … Continue reading →

Problem: Train a physics-informed neural network (PINN) on both the van der Pol oscillator and the drift-free Fokker-Planck diffusion equation. Solution: report Spectral Bias of Physics-Informed Neural Networks¶ 1. Introduction and Background¶ The physics-informed neural network (PINN) paradigm is a … Continue reading →

Problem: State and prove Tweedie’s formula. Solution: Tweedie’s formula asserts that if \(p(\mathbf x|\boldsymbol{\mu},\sigma)=\frac{1}{\det(\sqrt{2\pi}\sigma)}e^{-(\mathbf x-\boldsymbol{\mu})^T\sigma^{-2}(\mathbf x-\boldsymbol{\mu})/2}\) is normally distributed, then without needing to know anything about the prior \(p(\boldsymbol{\mu}|\sigma)\) on the mean random vector \(\boldsymbol{\mu}\), one has the following Bayesian … Continue reading →

Problem: Let \(\boldsymbol{\Theta}\) be a smooth statistical manifold, and let \(D:\boldsymbol{\Theta}^2\to [0,\infty)\) be a smooth function. What does it mean for \((\boldsymbol{\Theta},D)\) to be a “divergence manifold“? Solution: The notion of a divergence manifold relaxes the axioms of a metric … Continue reading →

VAE $\textbf{Problem}$: Apply the PRAC-DTDT-ID workflow to a (vanilla) $\textit{autoencoder}$. $\textbf{Solution}$: Problem: to learn a lower-dimensional latent manifold representation of the input manifold (this is justified by the $\textit{submanifold hypothesis}$, namely that the data-generating distribution $p(\mathbf x)$ is essentially supported … Continue reading →

Problem: Give a broad sketch of the current state of the field of research in graph neural networks. Solution: Problem: Okay, so now explain what a graph neural network (GNN) actually is. Solution: A GNN is basically any neural network … Continue reading →

Problem: Consider a Landau-Ginzburg statistical field theory involving a single real scalar field \(\phi(\mathbf x)\) for \(\mathbf x\in\mathbf R^d\) governed by the canonically normalized free energy density: \[\mathcal F(\phi,\partial\phi/\partial\mathbf x,…)=\frac{1}{2}\biggr|\frac{\partial\phi}{\partial\mathbf x}\biggr|^2+\frac{\phi^2}{2\xi^2}+…\] Explain what the \(+…\) means, explain which terms have … Continue reading →