L^2 based stability for general type I blowup, see [L^2-based Stability of Blowup with Log Correction for semilinear Heat Equation].
Characterization of blowup for reduced models of Euler equations, see [Blowup Analysis for a Quasi-exact 1D Model of 3D Euler and Navier-Stokes].
Numerical investigation of blowup profiles using dynamic rescaling equations and neural networks, see [Fourier Continuation for Exact Derivative Computation in Physics-Informed Neural Operators].
Numerical profile construction and computer-assisted proof for singularity.
New paradigm for neural network architecture, see [KAN: Kolmogorov-Arnold Networks].
Improved methods for operator learning, see [Fourier Continuation for Exact Derivative Computation in Physics-Informed Neural Operators].
Model reduction and homogenization techinques in multiscale equation, see [Exponential Convergence for Multiscale Linear Elliptic PDEs via Adaptive Edge Basis Functions], [Exponentially Convergent Multiscale Methods for 2D High Frequency Heterogeneous Helmholtz Equations].
See also the review paper [Exponentially Convergent Multiscale Finite Element Method].
Hermite spectral method as equilibrium-preserving techniques for kinetic equations, see [Approximation to Singular Quadratic Collision Model in Fokker-Planck-Landau Equation].
Convergence and stability of nonlinear numerical schemes for kinetic equations.
Ensemble sampling method for inverse problems based on underdamped Langevin equation, see [Second Order Ensemble Langevin Method for Sampling and Inverse Problems].
Reduced approximation model for principal-component analysis based on operator approximation via the Schrödinger equation, see [Schrödinger principal-component analysis: On the duality between principal-component analysis and the Schrödinger equation].