Adarsh Ganeshram | Searcharxiv

arXiv2026

Stability Framework for the Singularity of the Euler Equations on $\mathbb{R}^3$

In a companion numerical study, we found a high-precision singular profile for the Euler equations on the unbounded domain $\mathbb{R}^3$. The present manuscript complements that study by establishing in detail a framework for proving (nonlinear) stability of the approximate self-similar profile, reducing the analysis to a large but finite collection of explicit estimates and computable constants. Conditional on rigorous certification of the estimates and constants appearing in the argument, the framework closes the stability proof and, crucially, allows the stable rescaled profile to be reconstructed as an admissible solution in the original variables that becomes singular in finite physical time. With the stability and reconstruction mechanisms established, the remaining work is largely quantitative: certifying the explicit constants and, where necessary, sharpening selected analytic estimates to close the stability proof.

math.AP↗

arXiv2026

Stable Singularity of the Euler Equations on $\mathbb{R}^3$

We provide evidence of a stable finite-time singularity in the 3D Euler equations on the unbounded domain. Using a physics-informed neural network (PINN) with a self-similar ansatz, we find an approximate singular profile for the Euler system at the critical blowup rate of $0.5$ and certify it using a spline representation. The transport field associated with the obtained profile suggests that linear damping can be established throughout the domain, providing strong evidence for the overall stability of the candidate profile. We also establish a complete framework for proving nonlinear stability of the approximate self-similar profile, reducing the analysis to a large but finite collection of explicit estimates and computable constants.

math.AP↗

arXiv2022

FC-PINO: High Precision Physics-Informed Neural Operators via Fourier Continuation

The physics-informed neural operator (PINO) is a machine learning paradigm that has demonstrated promising results for learning solutions to partial differential equations (PDEs). It leverages the Fourier Neural Operator to learn solution operators in function spaces and leverages physics losses during training to penalize deviations from known physics laws. Spectral differentiation provides an efficient way to compute derivatives for the physics losses, but it inherently assumes periodicity. When applied to non-periodic functions, this assumption can lead to significant errors, including Gibbs phenomena near domain boundaries which degrade the accuracy of both function representations and derivative computations. To overcome this limitation, we introduce the FC-PINO (Fourier-Continuation-based Physics-Informed Neural Operator) architecture which extends the accuracy and efficiency of PINO and spectral differentiation to non-periodic and non-smooth PDEs. In FC-PINO, we propose integrating Fourier continuation into the PINO framework, and test two different continuation approaches: FC-Legendre and FC-Gram. By transforming non-periodic signals into periodic functions on extended domains in a well-conditioned manner, Fourier continuation enables fast and accurate derivative computations. This approach avoids the discretization sensitivity of finite differences and the memory overhead of automatic differentiation. We demonstrate that standard PINO fails (without padding) or struggles (even with padding) to solve non-periodic and non-smooth PDEs with high precision, across challenging benchmarks. In contrast, the proposed FC-PINO provides accurate, robust, and scalable solutions, substantially outperforming PINO alternatives, and demonstrating that Fourier continuation is critical for extending PINO to a wider range of PDE problems when high-precision solutions are needed.

cs.LG↗