Posts ¹⁷

Recent Advances in KAN-Based Numerical PDE Solvers

Kolmogorov-Arnold Networks (KANs), introduced in 2024, have rapidly become one of the most active frontiers in scientific machine learning for solving partial differential equations (PDEs) (Liu et al., 2024). Unlike Multi-Layer Perceptrons (MLPs), which apply fixed activation functions at nodes, KANs place learnable univariate activation functions on edges, grounded in the Kolmogorov-Arnold representation theorem: every continuous multivariate function can be expressed as a composition of univariate functions and summations. This structural difference gives KANs two key properties relevant to PDE numerics — higher interpretability and parameter efficiency — making them an appealing successor to MLP-based Physics-Informed Neural Networks (PINNs).

Recent Advances in Numerical PDEs

Numerical methods for partial differential equations (PDEs) have entered a period of rapid transformation, driven by two converging forces: deep learning’s maturation as a tool for high-dimensional function approximation, and the resurgence of classical methods augmented by machine learning. The field broadly divides into physics-informed machine learning, neural operator learning, foundation models for PDEs, and the continuing evolution of classical high-order, structure-preserving, and data-driven discovery methods. Quantum computing and laser-based hardware solvers are also beginning to enter the landscape. This survey organises the most active research fronts, highlights landmark and recent key papers, and identifies open problems as of early 2026.

Recent Advances in Steady States of Navier-Stokes Equations

The study of steady-state and self-similar solutions of the incompressible Navier-Stokes equations (NSE) has undergone remarkable progress in the 2020s. This post surveys landmark results from 2024–2026 touching on existence, uniqueness, classification, and stability of such solutions. The stationary (steady) NSE in $\mathbb{R}^3$ reads: $$-\nu \Delta u + (u \cdot \nabla) u + \nabla p = 0, \quad \operatorname{div} u = 0.$$ A central object of the self-similar theory is the class of $(-1)$-homogeneous (scale-invariant) solutions: a function $u$ is $(-1)$-homogeneous if $u(\lambda x) = \lambda^{-1} u(x)$ for all $\lambda > 0$. These are precisely the profiles of forward self-similar solutions $u(x,t) = t^{-1/2} U(x/\sqrt{t})$ of the time-dependent NSE.

Recent Research Directions in Analysis of PDEs 2021–2026

The arXiv section of Analysis of Partial Differential Equations is one of the most prolific areas of pure mathematics, producing over 400 preprints per month as of early 2026. The period 2021–2026 has witnessed landmark breakthroughs — including a computer-assisted proof of finite-time singularity in the 3D Euler equations, the resolution of Hilbert’s Sixth Problem via kinetic theory, and the emergence of probabilistic and nonlocal operator methods as dominant paradigms. This survey identifies, categorises, and profiles the key research directions and landmark papers in math.AP during this era.

Paper Reading - Optimization problems for elliptic PDEs (2601.01591)

This paper is a panoramic tour of three families of optimal control problems for elliptic PDEs: where the control is the coefficient, the potential, or the source term, unifying and sharpening results from the authors’ previous works. Three ways to control an elliptic PDE # The authors always consider a Dirichlet problem on a bounded domain $\Omega \subset \mathbb{R}^d$, with the solution $u$ as the state and a function (or measure) as the control. They study three settings:

Paper Reading - Optimal coefficients for elliptic PDEs (2512.08431)

This paper gives a clear, fairly complete picture of how to optimally choose the coefficient $a(x)$ (think “material quality”) in an elliptic PDE, with compliance as the main model and then a general optimal control formulation. Problem Setup # Considering the boundary value problem: $$ -{\rm div}(a(x)\nabla u) = f \quad\text{in } \Omega,\qquad u=0 \text{ on } \partial\Omega, $$ where $\Omega$ is a bounded domain, $f$ is a given load, and $a(x)$ is the design variable.

Paper Reading - Optimal sources for elliptic PDEs (2509.01521)

Introduction # The authors study how to “best choose” a source term $f$ in a Poisson-type equation $$ -\Delta u = f \quad\quad\text{in }\Omega,\quad u = 0\text{ on }\partial\Omega, $$ so that a given performance measure (a cost functional) is optimized. The twist is that the source itself is the control, and it can be subject to various constraints (size, bounds, sign, etc.). This makes the problem sit at the intersection of optimal control, shape optimization, and regularity theory.

Restriction and extension

Considering a smooth compact hyper-surface $\mathcal{S}$ in $\mathbb{R}^d$ with surface measure $d\sigma$. Given $f \in L^1(\mathbb{R}^d)$, the Fourier transform defined as follow: $$ \begin{equation} \hat{f}(x) = \int_{\mathbb{R}^d}e^{-2\pi i x \xi}f(x)dx \end{equation} $$ which by Riemann-Lebesgue is a bounded, continuous function vanishing at infinity. Since $\hat{f}$ is continuous on $\mathbb{R}^d$, by the Rimann-Lesbegue lemma its restriction to the compact hyper-surface $S \subset \mathbb{R}^d$ is is well-defined pointwise. Specifically, the restriction $\hat{f}\mid_{S}: S \rightarrow \mathbb{C}$ is the continuous function given by $$ \begin{equation} \hat{f}\mid_{S}(\sigma) = \hat{f}(\sigma) = \int_{\mathbb{R}^d}e^{-2\pi i x \xi}f(x)dx \end{equation} $$ for each $\sigma \in S$. This is bounded (as $\hat{f}$ is bounded) and can be integrated against the surface measure $d\sigma$ on $S$.

Proof of Theorem of solution of wave equation in the case $n = 1$

Solution of Brezis Problem 8.24 (1) and (2)

Solution of Evans PDE Problem 13

A lemma of J. L. Lions

This post explores J. L. Lions’ lemma about Banach spaces with compact injection, including applications to functional analysis. Lemma statement: Let $X$, $Y$, and $Z$ be three Banach spaces with norms $|| \cdot ||_X$, $|| \cdot ||_Y$, and $|| \cdot ||_Z$. Assume that $X \subset Y$ with compact injection and that $Y \subset Z$ with continuous injection. Prove that $$ \forall \varepsilon > 0, \exists C_\varepsilon > 0 \text{ satisfying } || u ||_Y \leq \varepsilon || u ||_X + C _{\varepsilon}|| u ||_Z,\quad \forall u \in X $$

Complex Hahn-Banach Theorem

Let $X$ be a complex vector space, $X_0$ one of its subspaces, $p: X \to \mathbb{R}_+$ such that $$ p(\lambda x) = |\lambda| p(x), \quad \forall \lambda \in \mathbb{C}, x \in X \text{ and } p(x + y) \leq p(x) + p(y), \quad \forall x, y \in X, $$ satisfying $|f(x)| \leq p(x)$, $\forall x \in X_0$, where $f: X_0 \to \mathbb{C}$ is linear. Under these conditions, there exists a linear functional $F: X \to \mathbb{C}$ such that $F|_{X_0} = f$ and

Real Hahn-Banach Theorem

Suppose $X$ is a vector space over $\mathbb{R}$, $p: X \to \mathbb{R}$ has the following properties: $p(X) = \lambda p(x)$, $\forall x \in X$, $\lambda \in \mathbb{R}_+$ and $p(x + y) \leq p(x) + p(y)$, $\forall x, y \in X$. Let $X_0$ be a subspace of $X$ and $u: X_0 \to \mathbb{R}$ a linear functional such that $u(x) \leq p(x)$, $\forall x \in X_0$. Then we can find $f: X \to \mathbb{R}$ a linear functional such that $f|_{X_0} = u$ and $f(x) \leq u(x)$, $\forall x \in X$.

Riesz Representation Theorem

1. Riesz Representation Theorem # Let $H$ be a Hilbert space over $\mathbb{R}$ or $\mathbb{C}$, and $T$ be a bounded linear functional on $H$ (a bounded operator from $H$ to the field $\mathbb{R}$ or $\mathbb{C}$, where $H$ is defined over that field). The following is known as the Riesz Representation Theorem: Theorem 1: If $T$ is a bounded linear functional on the Hilbert space $H$, then there exists $g \in H$ such that for every $f \in H$, we have: $$ T(f) = \langle f, g \rangle. $$

The application of Hahn-Banach Theorem 01

Suppose $X$ is a normed space and $X_0$ is a closed subspace of $X$ and $x_0 \in X \setminus X_0$. Then we can find $f \in X’$ such that $f(x_0) = 1$ and $f(x) = 0$, $\forall x \in X_0$. Proof: Since $x_0 \notin X_0$, we can find $\delta > 0$ such that $|x_0 - x| \geq \delta$, $\forall x \in X_0$, which is equivalent to $1 \leq \dfrac{|x_0 - x|}{\delta}$, $\forall x \in X_0$.

The application of Hahn-Banach Theorem 02

$X'$ = $\{ f: X \to \mathbb{K} \}$ where $f$ is is linear and continuous and $X$ is a Banach space over $\mathbb{K}$. Prove that $X' \neq {0}$, in fact, for every $x \neq 0 \in X$, we can find $f \in X’$ such that $f(x) = |x|$ and $|f| = 1$. Proof: Pick $x_0 \in X$. Define $X_0 = x_0 \cdot \mathbb{K}$, a subspace of $X$, and $g: X_0 \to \mathbb{K}$, $g(x) = x$, which is linear. Since $g$ and $|\cdot|$ satisfy the conditions of the Hahn-Banach theorem, we can find $f: X \to \mathbb{K}$ such that $f|_{X_0} = g$, $f$ is linear and $f(x) \leq |x|$, $\forall x \in X$. Therefore $f(x_0) = g(x_0) = |x_0|$ and $|f| \leq 1$. The equality $f(x_0) = |x_0|$ guarantees that $|f| = 1$.