How DeepMind’s AI Uncovered New Unstable Singularities in Fluid Dynamics

Problem Context

Fluid‑dynamics equations such as the three‑dimensional Euler, Navier‑Stokes, CCF, IPM and Boussinesq equations can develop singularities (blow‑ups) where the solution gradient becomes infinite. Unstable singularities require infinitely precise initial data; any tiny perturbation diverts the trajectory, making them extremely hard to capture with conventional numerical methods.

Two‑Stage Research Workflow

Stage 1 – Solution Discovery

Researchers used physics‑informed neural networks (PINNs) to search for self‑similar candidate solutions with scaling exponent λ. The workflow iteratively refined the PINN architecture and training process, producing high‑precision candidate solutions that guided both mathematical modeling and network design.

Stage 2 – Solution Analysis

For each discovered solution of the CCF, IPM and Boussinesq equations, the surrounding PDE was linearized. The linear stability analysis revealed a set of unstable modes sharing the same symmetry assumptions, indicating that the discovered families are complete within the considered λ range. An empirical asymptotic formula linking the blow‑up rate to the instability order was derived.

Key Technical Innovations

Gauss‑Newton optimizer : Replaced standard gradient optimizers (Adam, L‑BFGS). Achieved residuals as low as 1e‑8 within ~50 000 iterations, providing faster convergence and higher accuracy.

Multi‑stage training : First train a network to obtain an approximate solution; then train a second network to correct the residual errors of the first, effectively reaching double‑precision accuracy.

Embedding mathematical structure : Symmetry, periodicity and infinite‑domain handling were encoded directly into the network architecture, turning PINNs into a tool that respects the underlying physics.

Experimental Results

The combined approach reduced maximum residuals by five orders of magnitude for both stable and first‑unstable solutions of the CCF and IPM equations, reaching the limits of double‑precision floating‑point arithmetic (errors only due to GPU rounding). Visualizations of three‑dimensional flow fields and two‑dimensional vorticity maps showed that the model’s error corresponds to predicting the Earth’s diameter within a few centimeters.

Resources

Pre‑print on arXiv:

https://go.hyper.ai/iGh6t