Research Survey ⁴

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.