At the turn of the millennium, the Clay Mathematics Institute announced the Millenium Prize. The Institute’s Scientific Discovery panel, consisting of four leading mathematicians, selected seven problems that they deemed the be the most important open problems in mathematics. The Clay Institute announced that it would award a $1,000,000 prize to any mathematician or group of mathematicians who discovered a solution to any of the problems. Apart from their centrality to research mathematics, the seven problems have substantial implications for other disciplines. A solution to one of the problems, the ‘P vs NP’ problem, for example, would determine whether modern public-key cryptography is secure. Earlier this month, to the astonishment of the mathematicians, a research group at Open AI published an AI generated solution to one of the millennium problems, the ‘Navier-Stokes Smoothness and Existence’ problem. Experts, including the Clay Institute, are still verifying the result. However, based on a preliminary examination, most experts think the solution is legitimate. This startling breakthrough is one of many recent mathematical and scientific discoveries facilitated by AI models. These discoveries span fields as disparate as biology, chemistry and number theory. The pace of AI progress in science and mathematics is remarkable, it remains to be seen where the frontier lies.
Navier-Stokes
In the 18th century a mathematician named Leonhard Euler developed a series of equations known as the Euler Equations, that described the dynamics of an ideal fluid, that is a fluid without viscosity (thickness). Given a set of initial conditions or starting conditions for an ideal fluid, such as the fluid’s starting velocity, pressure, and density, the Euler Equations can accurately predict the fluid’s future behaviour. While a remarkable mathematical discovery, the Euler Equations are ineffective when applied to practical situations. Real fluids are viscous and their viscosity plays an important role in determining their behaviour.
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In the 19th century, two mathematicians Claude-Louis Navier and George Stokes working independently, discovered techniques to extend Euler’s results to viscous fluids. Over the years, mathematicians and physicists expanded upon and refined the work of Navier and Stokes to produce a series of equations known as the Navier-Stokes Equations. These equations serve as a powerful tool. In a manner similar to the Euler Equations, given the initial conditions of a viscous fluid, the Navier-Stokes Equations can accurately predict many important characteristics of a fluid’s behaviour. For example, they can determine the future turbulence of a fluid or the exact lift and drag experienced by an object moving through the fluid. The Navier-Stokes Equations are used widely by physicists and engineers. Aeronautical engineers, for example, use the Navier-Stokes equations to model the airflow around an aircraft. While, chemical engineers use the equations to model the flow, mixing rate and heat transfer of fluids in industrial processes.
The Navier-Stokes equations produce highly accurate predictions for a broad range of initial conditions, however for centuries, mathematicians have been unsure about whether Navier-Stokes is a ‘complete theory’, that is, would the equations always produce realistic predictions. The Clay Institute codified this problem as the ‘Navier-Stokes Smoothness and Existence’ problem. The goal is to determine whether for all possible initial conditions the Navier-Stokes equations produce finite and non-jagged predictions for the turbulence and dispersion of a fluid.
Historically, mathematicians have been unable to approach this problem directly. Instead, they tried to start with an easier problem, they tried to identify initial conditions for which Euler Equations led to a ‘finite-time blow ups’. In other words, they tried to find reasonable initial conditions for the Euler Equations that led to impossible predictions, like the fluid moving at an infinite velocity or the entire volume of the fluid converging to a single point. They hoped that these examples could be used as a starting point to identify cases for which the Navier-Stokes Equations resulted in unrealistic predictions. Over the last decade, mathematicians have been able to find such examples for the Euler Equations but were unable to extend these results to Navier-Stokes.
Open AI’s model took the same approach, it started by looking for initial conditions that led to ‘finite-time blow ups’ for the Euler Equations and then attempted to extend these results to Navier-Stokes. Astonishingly, the model succeeded where mathematicians had failed. The key insight was to choose an initial condition, where the fluid experienced a smooth external force that could overcome the smoothing effect of viscosity and allow ‘instability’ to develop. The model was able to show mathematically that when the Navier-Stokes Equations were applied to those initial conditions; they would predict that the fluid would become increasingly concentrated resulting in it reaching an infinite velocity in a finite amount of time. Mathematicians are still carefully verifying this result, however, if legitimate, this example will provide a key insight into the limitations on Navier-Stokes and allow us to develop a more accurate theory of fluids.
Drug Discovery
Another scientific domain in which AI has made remarkable contributions is drug discovery. Historically, drug discovery has been a tedious and lengthy process. Traditionally, researchers would need to review the existing literature, isolate biological targets, and screen thousands of possibly viable compounds, before they could initiate clinical trials. Deepmind launched the Co-scientist project with the aim of addressing this problem. Their system consists of a network of AI agents that collaboratively synthesize vast amounts of scientific literature, propose biological targets, and debate their viability in real time. Through this process, the system is able to rapidly isolate the most promising drug candidates, drastically reducing the time and effort the preclinical phase would typically require.
One of the Co-Scientist project’s most significant accomplishments was finding an effective drug for Acute Myeloid Leukemia (AML), a form of cancer that affects the patient’s blood and bone marrow by producing abnormalities in the white blood cells of those organs. Researchers at DeepMind tasked the model with reviewing the existing scientific literature and suggesting ways in which existing drugs could be repurposed to treat AML. The system’s network of agents collaboratively generated thousands of hypotheses, computationally simulated the relevant molecular interactions, and then rigorously debated the viability of different compounds. Through this process, the model successfully pinpointed an overlooked, existing drug known as KIRA6 that could theoretically target the relevant vulnerabilities in the white blood cells. Subsequent laboratory experiments confirmed that the identified drug could kill white blood cells infected with leukemia, in the lab at realistic dosage levels that would be safe for patients. The drug is now undergoing clinical trials.
The Hardest Problem: Formulating New Problems
Solving the Navier-Stokes Millenium problem and finding an effective drug for Myeloid Leukemia are by no means the only major accomplishments of AI models. In the recent past, AI models have proposed novel configurations for nuclear fusion reactors, solved the protein-folding problem, and significantly improved upon traditional weather forecasts.
However, while those breakthroughs are extraordinarily accomplishments, the problems had already been defined by mathematicians or scientists. The research community had identified a question worth answering, developed a framework for investigating it, and decided what constitutes a meaningful result. The real test for AI will be whether it can go one step further: identify gaps in our understanding, formulate entirely new questions, and independently investigate these solutions to those problems.
The article has been written by Aman Adukoorie, Quantitative Strategist at a Leading Bank
