Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks (2201.06780v3)

Abstract: Whether there exist finite time blow-up solutions for the 2-D Boussinesq and the 3-D Euler equations are of fundamental importance to the field of fluid mechanics. We develop a new numerical framework, employing physics-informed neural networks (PINNs), that discover, for the first time, a smooth self-similar blow-up profile for both equations. The solution itself could form the basis of a future computer-assisted proof of blow-up for both equations. In addition, we demonstrate PINNs could be successfully applied to find unstable self-similar solutions to fluid equations by constructing the first example of an unstable self-similar solution to the C\'ordoba-C\'ordoba-Fontelos equation. We show that our numerical framework is both robust and adaptable to various other equations.

Asymptotic Self-Similar Blow-Up Profiles in Axisymmetric Euler Equations via Neural Networks

The paper "Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks" addresses the longstanding open question in fluid dynamics about the existence of finite-time singularities in the Euler equations. Finite-time blow-up solutions from smooth initial data have been elusive, especially in three-dimensional (3D) flows, a problem that parallels the famous Navier-Stokes Millennium Prize challenge. This research introduces a novel numerical framework employing Physics-Informed Neural Networks (PINNs) to discover, for the first time, smooth self-similar blow-up profiles for both the 3D Euler and 2D Boussinesq equations.

Summary of Research Findings

1. Self-Similar Blow-Up Discovery:
   • The approach via PINNs enables the identification of a smooth asymptotic self-similar blow-up profile, specifically for the scenario proposed by Luo and Hou in their paper of axisymmetric 3D Euler equations. This paper presents a critical link between the formation of finite-time singularities in the presence of cylindrical boundaries and self-similarity.
2. Neutral and Unstable Self-Similar Solutions:
   • The framework demonstrated its robustness in discovering both stable and unstable self-similar solutions, including an unstable self-similar solution to the Córdoba-Fontelos equation. This is significant as such solutions are challenging to obtain through traditional numerical methods due to inherent instabilities.
3. Numerical Strategy:
   • PINNs leveraged here stand out by incorporating the intrinsic symmetries and expected qualitative behaviors of the solutions. This method allowed for the determination of not just the self-similar profile but also the associated self-similarity exponent $\lambda$.
4. Implications for Future Research:
   • The solutions and methodologies presented suggest pathways towards crafting computer-assisted proofs of blow-up for both the 3D Euler and 2D Boussinesq equations. The method extends naturally to other PDEs exhibiting self-similarity and shows promise in addressing the open question of blow-up in the Navier-Stokes equations.

Implications and Future Directions

The utilization of PINNs for identifying self-similar blow-up profiles represents a significant step in understanding singularities in fluid mechanics. The framework’s adaptability to various equations opens up new avenues for computational fluid dynamics research. Moreover, the potential for PINNs to handle unstable solution spaces implies possible extensions into more complex systems and under-researched parameter regimes, including higher-dimensional simulations.

The prospect of using these results to facilitate computer-assisted proofs points towards a transformative approach to longstanding problems in mathematical physics. As such, this work could serve as a prototype for using machine learning-driven methods to tackle other complex dynamical systems beyond fluid dynamics.

Conclusion

This paper demonstrates the efficacy of PINNs in discovering and analyzing complex self-similar blow-up phenomena in fluid dynamic equations. The neural network-based approach not only confirms prior numerical and theoretical findings concerning singularity formation but also provides a robust new toolset for exploring unresolved problems in PDE analysis. Moreover, the methodological innovations introduced could have broad applications across numerous fields where self-similarity and singularities play key roles.