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

TL;DR

This thesis combines numerical, theoretical, and machine learning methods to analyze singularity formation in PDEs, addressing complex equations like Navier-Stokes and Keller-Segel.

Contribution

It introduces a new analytical framework for singular PDEs, enhances numerical approaches with machine learning, and proposes the novel Kolmogorov-Arnold Network architecture.

Findings

Validated the new analytical approach on NLH and CGL equations.

Demonstrated machine learning improves identification of blowup solutions.

Introduced the Kolmogorov-Arnold Network with interpretability and scalability.

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, we introduce a robust analytical framework to simplify and systematize pen-and-paper proofs for simpler 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. 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 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.