The laws of fluid dynamics are the invisible architecture of our world. For centuries, elegant equations have described the fundamental physics governing everything from the swirling vortex of a hurricane to the delicate airflow lifting an airplane’s wing. These mathematical models are incredibly powerful, yet they harbor a deep and persistent mystery that has challenged mathematicians and physicists for over a century. Under certain theoretical conditions, the equations break down. This breakdown occurs at points called “singularities” or “blow ups,” where the mathematics predict physically impossible outcomes. In these moments, quantities like fluid velocity or pressure could theoretically become infinite, signaling a fundamental limit in the predictive power of the equations themselves. Understanding these singularities isn’t just an academic exercise; it helps scientists identify the absolute boundaries of our physical theories and improve our understanding of how the world truly functions. For generations, this has remained one of the most enduring challenges in mathematics, with the famous Navier-Stokes problem standing as one of the unsolved Millennium Prize Problems. Now, a breakthrough has arrived not just from a new answer, but from an entirely new way of asking the question, one that forges a powerful new partnership between human mathematical intuition and the brute-force precision of AI. So how exactly was this century-old problem cracked? To Solve the Big Problem, Scientists Hunt for Fragile, “Unstable” Solutions The key to tackling this grand challenge lies in a counter-intuitive approach: instead of looking for robust, easily formed singularities, researchers focused on finding ones that are exceptionally fragile and elusive. A “stable” singularity is a robust outcome, forming from a wide range of initial conditions. In contrast, an “unstable” singularity is a mathematical phantom; it requires initial conditions tuned with infinite precision, and any infinitesimal perturbation will cause the system to veer away, preventing the blow up from ever forming. The crucial and surprising insight is that for the biggest unsolved problems in fluid dynamics including the boundary-free 3D Euler and Navier-Stokes equations. mathematicians hypothesize that only these hyper-sensitive, unstable singularities are expected to exist. This makes them extraordinarily difficult to find. Traditional numerical simulations, which step forward through time, struggle to stay on the knife’s edge required to track such a solution; the slightest rounding error can create an exponential deviation, causing the entire simulation to diverge uselessly. This is why finding a way to systematically discover these elusive solutions is such a critical first step. It provides the first concrete evidence for phenomena that were previously only theoretical. By developing a method to capture these fragile mathematical structures, researchers have opened a new pathway toward confronting one of the most profound open questions in science. AI Is Now a Discovery Tool for Pure Mathematics This research pioneers a new role for AI, not just as a data processor, but as a genuine discovery tool for pure mathematics. The method is built on a specific type of AI called Physics-Informed Neural Networks (PINNs). Unlike conventional neural networks that learn patterns from vast datasets, PINNs are trained to directly satisfy the fundamental equations of physics. The network’s output is constantly checked against what the physical equations demand, and it learns by minimizing the “residual”; a measure of how badly its current guess violates the laws of physics. This research The true innovation, however, was not just applying PINNs but fundamentally transforming them into a specialist tool. Researchers embedded deep mathematical insights such as known symmetries, physical constraints, and asymptotic behaviors directly into the AI’s architecture. This created a powerful feedback loop: initial AI experiments revealed hidden mathematical structures; for example, that solutions vanished faster near the origin than expected which the researchers then incorporated back into the network’s architecture as new constraints, further refining and accelerating the search. As Yongji Wang, the study’s first author, explains, this synthesis of mathematical knowledge and computational power was the key to the breakthrough. By embedding mathematical insights and achieving extreme precision, we transformed PINNs into a discovery tool that finds elusive singularities. The AI Achieved Almost Unimaginable Precision To find solutions that require infinite precision, the researchers had to develop a method that achieved an almost unimaginable level of accuracy. A clever trick was to first reframe the problem. Instead of simulating the fluid’s evolution over time, the researchers used a mathematical lens called “self-similar coordinates” to transform the explosive blow-up into a search for a single, stable spatial profile, turning a chaotic moving target into a stationary one. Even then, extreme precision was required. To put the scale of this achievement in perspective, the team notes that their “largest errors addressed are equivalent to predicting the diameter of the Earth to within a few centimeters.” This extreme accuracy is not merely for show; it is an essential requirement for the ultimate goal of mathematical proof. Such precision meets the incredibly high standard required for creating “rigorous computer-assisted proofs,” which are essential for turning a numerical discovery into a mathematically proven fact. While this peak precision was achieved for the CCF model solutions, the method delivered accuracy multiple orders of magnitude better than previous work across all the systems studied. This level of performance was made possible by moving beyond standard AI training techniques. The team employed advanced methods, including a “full-matrix Gauss-Newton optimizer” and “multi-stage training,” where a second neural network is trained to correct the tiny, high-frequency errors of the first. These sophisticated approaches significantly outperformed standard optimizers, enabling the discovery of solutions with unprecedented accuracy. A Hidden Pattern Emerged from the Chaos Armed with a method capable of this unprecedented precision, the researchers could do more than just find a single, isolated solution. They were able to systematically discover whole families of new unstable singularities for three canonical fluid systems: the Incompressible Porous Media (IPM), the Boussinesq, and a simplified 1D model known as the CCF equations. As they mapped out these new mathematical objects, a surprising and elegant pattern emerged from the complexity. For two major, long-studied systems (the IPM and Boussinesq equations) a simple, clear linear relationship appeared. This pattern connects a key parameter of the singularity (its blow-up speed, λ), where a smaller λ corresponds to a faster, more intense blow-up capable of overcoming dissipative effects to its order of instability, which is the number of unique ways the solution can deviate and fail to form the blow up. As the solutions became more and more unstable, their corresponding blow-up speeds aligned along a predictable straight line. The significance of this discovery is profound. It suggests that a deeper, underlying structure governs these chaotic phenomena, transforming what seemed like a random collection of isolated solutions into an organized hierarchy. This empirical formula now provides a predictive guide for future research, offering candidate values for even higher-order unstable solutions and making the search for them faster and more reliable. A New Playbook for Science’s Hardest Problems This work is more than just a breakthrough in fluid dynamics; it establishes a “new playbook for exploring the complex landscape of nonlinear partial differential equations (PDEs).” It demonstrates a powerful new way of conducting mathematical research, proving that when deep, domain-specific human knowledge is combined with cutting-edge AI computation, problems that have resisted purely analytical or purely numerical methods for decades can finally be tackled. The approach of embedding mathematical structure directly into a neural network’s architecture, and then refining it to a level of precision sufficient for computer-assisted proof, represents a new paradigm. It transforms AI from a general-purpose solver into a specialized discovery engine, capable of navigating vast and complex mathematical landscapes to find the specific, elusive solutions that matter most. The success of this interdisciplinary method opens the door to a new era of computer-assisted mathematics. It leaves us with a compelling question: What other long-standing scientific mysteries, from fundamental physics to materials science, could finally be unraveled by combining human mathematical insight with the precision of AI?

The is an opinion piece based on the author’s POV and does not necessarily reflect the views of HackerNoon.

This story contains AI-generated text. The author has used AI either for research, to generate outlines, or write the text itself. 

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