DeepMind Uses AI to Uncover New Unstable Singularities in Three Fluid Equations

Discovery and Analysis Two‑Stage Structure

Fluid dynamics has long posed fundamental challenges, with singularities or blow‑ups remaining unresolved for equations such as the three‑dimensional Euler and Navier‑Stokes equations. An unstable singularity occurs when a solution that starts from smooth initial data evolves to infinite gradient, requiring infinitely precise initial conditions because any tiny perturbation diverts the solution from the blow‑up trajectory.

To address this, the researchers adopted a two‑stage approach—discovery and analysis—to achieve high‑precision identification of unstable singularities. First, they searched for candidate solutions exhibiting self‑similar scaling rate λ using a Burgers‑equation example, iteratively refining the machine‑learning pipeline and solution accuracy. The empirical results guided both mathematical modeling and neural‑network architecture design, while the modeling informed inductive biases such as input coordinate transformations and output field shapes.

In the analysis stage, each discovered unstable solution for the CCF, IPM and Boussinesq equations was linearized to assess stability. For the n ‑th unstable solution, the team identified n unstable modes sharing the same symmetry assumptions, indicating that the discovered solution family is complete within the admissible λ range.

Mathematical Insight + Neural Networks Make PINN a New Tool

The study enhanced physics‑informed neural networks (PINNs) by embedding mathematical insights directly into the network architecture, steering optimization toward mathematically relevant solutions. Constraints derived from the governing equations—such as symmetry, periodicity, and handling of infinite domains—were enforced through architectural design, creating a strong training prior.

Iterative feedback between numerical experiments and mathematical analysis refined the network, explicitly factorizing residual equations and markedly improving stability.

Improved High‑Precision Training

To meet the extreme precision required for unstable singularities, the researchers introduced a Gauss‑Newton optimizer and a multi‑stage training strategy. Standard gradient optimizers (e.g., Adam, L‑BFGS) failed to produce high‑quality solutions, whereas the Gauss‑Newton method reduced residuals to 10⁻⁸ within roughly 50,000 iterations, outperforming standard optimizers in speed and convergence.

Multi‑stage training first obtained an approximate solution with one network, then trained a second network to correct residual errors. Combining the two outputs pushed the solution to higher accuracy. In experiments on stable and first‑order unstable solutions for the CCF and IPM equations, this approach improved the maximum residual by five orders of magnitude, reaching precision sufficient for rigorous mathematical verification.

The resulting models achieved unprecedented accuracy, visualized through three‑dimensional representations and two‑dimensional vorticity fields, with maximum errors comparable to predicting the Earth’s diameter within a few centimeters.

Authors and References

First author Yongji Wang is a postdoctoral researcher at NYU’s Courant Institute and a visiting scholar at Stanford, focusing on continuous media mechanics, geophysics, and scientific machine learning. The work is documented in the arXiv preprint titled “Discovery of Unstable Singularities” and references DeepMind’s blog and related news articles.