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

2026-04-18

AI Summary

This work addresses the analytical and numerical challenges in singularity formation for partial differential equations such as the Navier–Stokes and Keller–Segel systems. It proposes a unified framework integrating theoretical analysis with machine learning: by introducing vanishing modulation conditions and singularly weighted energy estimates, it streamlines and generalizes analytical proofs of blow-up solutions; furthermore, it pioneers the incorporation of Kolmogorov–Arnold Networks (KANs)—featuring learnable nonlinearities and strong interpretability—into enhanced physics-informed neural networks (PINNs) and neural operators to accurately resolve singularity structures. The approach is successfully applied to the nonlinear heat equation, the complex Ginzburg–Landau equation, and the three-dimensional Keller–Segel equation with logistic damping, resolving the long-standing open problem of singularity formation in the latter and offering new theoretical insights and efficient computational tools for investigating potential singularities in the Navier–Stokes equations.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• Singularity Formation
• Navier-Stokes Equation
• Partial Differential Equations
• Blowup
• Keller-Segel Equation

Innovation

• singularity formation
• vanishing modulation conditions
• singularly weighted energy estimates
• Kolmogorov-Arnold Network
• Physics-Informed Neural Networks