Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approaches

Author: Wang, Yixuan

Year: 2026

Degree: Dissertation (Ph.D.)

Advisor: Hou, Thomas Y.

Committee Members: Owhadi, Houman; Anandkumar, Anima; Hoffmann, Franca; Hou, Thomas Y.

Option: Applied And Computational Mathematics

DOI: 10.7907/gf82-wx70

Abstract

This thesis develops numerical and theoretical approaches for understanding and analyzing singularity formation in Partial Differential Equations (PDEs). The singularity formation in the Navier-Stokes Equation (NSE) is famously challenging as one of the seven Clay Prize problems. Unlike simpler equations such as the Nonlinear Heat (NLH) or Keller-Segel (KS) equations, where formal asymptotics near blowup are better understood, the intrinsic complexity of NSE makes quantitative analytical treatment difficult, if not impossible, without numerical guidance.

Building on numerical insights, Chapter 2, 3 and 4 introduce a robust analytical framework to simplify and systematize pen-and-paper proofs for singular PDEs. We present a novel approach based on enforcing vanishing modulation conditions for perturbations around approximate blowup profiles, complemented by singularly weighted energy estimates. Blowups are proven with a clear notion of stability, with rates automatically inferred, without the need to know the asymptotics a priori or explicit spectral information of the linearized operator. We demonstrate the efficacy of our method on PDEs with complicated asymptotics, such as NLH and the Complex Ginzburg-Landau (CGL) equation, and address the open problem of singularity formation in the 3D KS equation with logistic damping. We also provide a roadmap for extending our techniques to singularities involving multiple scales.

In Chapter 5 and 6, we develop and refine numerical approaches that facilitate deeper insights into singularity formation. We demonstrate that machine learning methods significantly enhance our capability to identify and characterize potential blowup solutions with high precision. We improve on existing Physics-Informed Neural Network (PINN) and Neural Operator (NO) frameworks. Moreover, we present a novel machine learning paradigm, the Kolmogorov-Arnold Network (KAN) architecture, whose interpretability and excellent scaling properties are achieved through learnable nonlinearities inspired by the Kolmogorov-Arnold representation theorem.

Chapter 7 introduces Exponential Multiscale Finite Element Method (ExpMsFEM), developed to efficiently solve challenging multiscale PDEs beyond elliptic problems, such as the Helmholtz equation. We construct adaptive local bases, proving exponential convergence theoretically and demonstrating superior computational performance in practice. Like KAN, ExpMsFEM exemplifies how insights from theory can guide the design of high-performance solvers with theoretical guarantees.

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