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

arXiv 2604.16842 ↗First posted 18 Apr 2026math.NA cs.LG cs.NAcited by 0

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Abstract (arXiv)

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.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 66 of its 418 references that are in the map. First posted on arXiv 18 Apr 2026.

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References · 66

  1. 2020Scaling Laws for Neural Language Modelscited by 6,586
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  3. 2023Sparse Autoencoders Find Highly Interpretable Features in Language Modelscited by 1,420
  4. 2022Toy Models of Superpositioncited by 971
  5. 2022In-context Learning and Induction Headscited by 964
  6. 2017Deep Learning Scaling is Predictable, Empiricallycited by 582
  7. 2020Scaling Laws for Autoregressive Generative Modelingcited by 522
  8. 2019Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networkscited by 368
  9. 2021Explaining Neural Scaling Lawscited by 363
  10. 2017A Conceptual Introduction to Hamiltonian Monte Carlocited by 209
  11. 2024U-KAN Makes Strong Backbone for Medical Image Segmentation and Generationcited by 179
  12. 2017Searching for Activation Functionscited by 169
38 more
  1. 2024Kolmogorov-Arnold Networks are Radial Basis Function Networkscited by 128
  2. 2024Wav-KAN: Wavelet Kolmogorov-Arnold Networkscited by 122
  3. 2024Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximationcited by 118
  4. 2024Challenges in Training PINNs: A Loss Landscape Perspectivecited by 113
  5. 2016Extrapolation and learning equationscited by 110
  6. 2024A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networkscited by 104
  7. 2024Kolmogorov-Arnold Networks (KANs) for Time Series Analysiscited by 98
  8. 2024DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for Mechanics Problemscited by 84
  9. 2024Convolutional Kolmogorov-Arnold Networkscited by 77
  10. 2024TKAN: Temporal Kolmogorov-Arnold Networkscited by 71
  11. 2024Kolmogorov-Arnold Networks for Time Series: Bridging Predictive Power and Interpretabilitycited by 70
  12. 2024Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networkscited by 67
  13. 2024fKAN: Fractional Kolmogorov-Arnold Networks with trainable Jacobi basis functionscited by 64
  14. 2025Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspectivecited by 56
  15. 2023Interpretable Machine Learning for Science with PySR and SymbolicRegression.jlcited by 54
  16. 2024GKAN: Graph Kolmogorov-Arnold Networkscited by 53
  17. 2020A Neural Scaling Law from the Dimension of the Data Manifoldcited by 52
  18. 2024A Temporal Kolmogorov-Arnold Transformer for Time Series Forecastingcited by 52
  19. 2024KAGNNs: Kolmogorov-Arnold Networks meet Graph Learningcited by 49
  20. 2024Kolmogorov-Arnold Network for Satellite Image Classification in Remote Sensingcited by 47
  21. 2024BSRBF-KAN: A combination of B-splines and Radial Basis Functions in Kolmogorov-Arnold Networkscited by 46
  22. 2024Enhancing Graph Collaborative Filtering with FourierKAN Feature Transformationcited by 42
  23. 2025Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networkscited by 42
  24. 2024Demonstrating the Efficacy of Kolmogorov-Arnold Networks in Vision Taskscited by 40
  25. 2024Adaptive Training of Grid-Dependent Physics-Informed Kolmogorov-Arnold Networkscited by 38
  26. 2024KANQAS: Kolmogorov-Arnold Network for Quantum Architecture Searchcited by 37
  27. 2024Exploring the Potential of Polynomial Basis Functions in Kolmogorov-Arnold Networks: A Comparative Study of Different Groups of Polynomialscited by 35
  28. 2024rKAN: Rational Kolmogorov-Arnold Networkscited by 35
  29. 2019Learning Activation Functions: A new paradigm for understanding Neural Networkscited by 34
  30. 2024Unveiling the Power of Wavelets: A Wavelet-based Kolmogorov-Arnold Network for Hyperspectral Image Classificationcited by 34
  31. 2024Suitability of KANs for Computer Vision: A preliminary investigationcited by 34
  32. 2024GraphKAN: Enhancing Feature Extraction with Graph Kolmogorov Arnold Networkscited by 32
  33. 2024Kolmogorov-Arnold Graph Neural Networkscited by 32
  34. 2022On the Activation Function Dependence of the Spectral Bias of Neural Networkscited by 28
  35. 2024Gauss-Newton Natural Gradient Descent for Physics-Informed Computational Fluid Dynamicscited by 28
  36. 2019Multi-scale Deep Neural Networks for Solving High Dimensional PDEscited by 25
  37. 2024Predictive Modeling of Flexible EHD Pumps using Kolmogorov-Arnold Networkscited by 24
  38. 2024Tuning the Frequencies: Robust Training for Sinusoidal Neural Networkscited by 19

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