When it comes to science and medicine, knowing where you are and where you want to go is only part of the problem. That’s because no matter how strictly you try to control your system, chaos theory dictates that there will be perturbations which, if not corrected for, will lead you far astray. Trying to stay on course in a dynamic environment has been a problem since humanity began traveling out of sight of its cave or campfire. Ancient humans navigated by the sun and stars by establishing landmarks and making course corrections. The best example of a type of ancient calculator for this purpose is the Antikythera mechanism, a super slide rule.
Another possible ancient tool of navigation was the sunstone used by the Vikings, now believed to have been Icelandic spar, a seemingly magic crystal made of calcium carbonate that, when held to the sky on a cloudy day, is able to show the location of the hidden sun by polarizing the light coming through the clouds or over the horizon. Along with that, we discovered magnetism and created the compass. Now, we have satellite navigation systems that we all take for granted, but that did not exist when I had to learn old-school navigation tactics in the Marines. But these devices only tell you where you are and point to where you are going. It takes more to get there.
That’s because no instrument is perfect, and tiny errors in azimuth and velocity can cause big problems later, as these errors are amplified and become cumulative without correction. That is where control algorithms come into play. In 1892, a mathematical genius named Aleksandr Lyapunov wrote The General Problem of Stability of Motion, which provided a systematic way to study the stability of dynamical systems, allowing for the analysis of both linear and nonlinear systems. In mathematics and engineering, the Lyapunov function stands as a monumental tool, instrumental in the analysis of dynamical systems. These functions are employed to study the stability of equilibrium points in various systems, ranging from simple mechanical systems to complex, multi-faceted networks.
A linear system is like a weight hanging on a spring. This system’s oscillations will follow the principle of superposition, meaning the output is always directly proportional to the input. In this case, the spring will follow Hooke’s Law (F= -kx), where F can be directly and exactly determined by the function. A nonlinear system, like a large swinging pendulum, does not follow simple harmonic motion. Think of the weird things that playground swings do after they have started swinging. Chemical reactions are also nonlinear, as are climate systems.
In brief, linear systems are simple and predictable, and nonlinear systems are complex and, over sufficient time, unpredictable. For this reason, complicated things like the actions of medications in the human body or the reactions of human beings to medications are not perfectly or even approximately predictable for any individual human being. That is because the complexity of the system extinguishes absolute predictability, creating a chaotic system that can be analyzed to some degree of accuracy as a whole but not specifically for any individual component.
The integration of artificial intelligence (AI) in health care has the potential to revolutionize patient care, diagnostics, treatment planning, and operational efficiency. However, ensuring the stability, safety, and robustness of AI systems in this critical sector is paramount. Lyapunov functions, a mathematical tool traditionally used in stability analysis of dynamical systems, can play a crucial role in achieving these goals. This article explores how the U.S. health care system can harness Lyapunov functions to enhance the reliability and effectiveness of AI in health care. AI systems in health care are employed in various applications, including diagnostic imaging, predictive analytics, personalized medicine, and robotic surgery. The performance and reliability of these systems directly impact patient outcomes and safety. Ensuring that these AI systems operate within safe and stable parameters is critical.
This is where Lyapunov functions and mathematical savants come into play. They can look at the chaos of a nonlinear system and devise formulas and techniques for determining when random perturbations are causing you to deviate from your goal. In the 1950s, Lyapunov’s functions were being used to design and analyze control systems for electrical circuits, chemical processes, and even complicated mechanical devices like inertial navigation systems. Inertial navigation systems were developed during World War II. One system used a spinning gyroscope on the V2 rocket to help keep it on course. On the Allied side, the Norden Bombsight used a gyrostabilized platform to maintain orientation during bombing runs. This is critical for flight, as it is easy to lose your bearings, so to speak, and even, in curved flight, to think up is down. That is what killed John Kennedy Jr.
By the 1960s, Lyapunov’s functions were used to correct errors in Inertial Navigation Systems (INS), like those developed at MIT for the Apollo program. In the 1960s, when humanity was racing to put a man on the moon where the stakes were astronomically high, NASA built a computer for the Apollo command module using the same core memory as an old cell phone. Why? Because of the physics of space travel, we couldn’t carry enough propellant for the astronauts to slow down and re-enter Earth’s orbit on their return. Every pound sent to space had to be lifted off Earth first, and we just couldn’t haul that much fuel. Hitting the re-entry corridor was like threading a needle from a mile away. Too shallow, and you’d skip off into space; too steep, and you’d burn up. No human could handle that precision, not even with the best sights in the window. We needed computers to rapidly and accurately calculate the trajectory and make necessary thrust corrections. This precision was tested on Apollo 13 when, after losing fuel cells, the astronauts had to rely on manual calculations for a critical burn—a hair-raising reminder of human limitations in a crisis.
Fast forward to today. Our computers are light-years ahead in terms of processing speed and storage capacity, yet the core principles remain unchanged. Computers are still deterministic machines—they calculate precise outcomes based on absolute facts. But intelligence? That’s a different beast. Intelligence isn’t just crunching numbers; it’s making sense of incomplete data, inferring from the unknown, and understanding context. Apollo missions relied on precise calculations, and that same need for stability and precision is crucial in health care AI. Here’s where Lyapunov functions come in. They’re mathematical tools that help ensure stability in dynamic systems, and they could be game changers for health care AI. The Apollo spacecraft had to navigate farther than any other human-crewed craft had before and, to this day, is still the farthest humanity has gone from its home. The Apollo Guidance Computer took the input from the INS and added Kalman filtering for further correction of state estimation and control system functions. But these systems must be tested for accuracy, and that requires Lyapunov functions, which can evaluate the answers you are being given to see how near the target they are.
Lyapunov functions are mathematical constructs used to prove the stability of equilibrium points in dynamical systems. The efficacy of Lyapunov functions lies in their key properties, where they must be positive definite around an equilibrium point, and their time derivative along the system trajectories must be negative definite or negative semi-definite, which ensures the system’s energy decreases, leading to stability. If a Lyapunov function can be identified for a system, it provides a measure of the system’s tendency to remain stable under perturbations. In the context of AI, Lyapunov functions can help ensure that learning algorithms and AI-driven control systems behave predictably and safely. In health care, AI algorithms are often used to analyze vast amounts of data and provide decision support. By applying Lyapunov functions, we can ensure that these algorithms converge to optimal solutions without diverging or exhibiting unstable behavior. This is particularly important in adaptive learning systems, where parameters evolve over time based on new data.
Medical devices, such as insulin pumps, pacemakers, and robotic surgical systems, rely on precise control mechanisms. Lyapunov-based stability analysis can be used to design and verify control algorithms that ensure these devices operate safely and effectively under varying conditions and patient-specific parameters. AI-driven predictive analytics can identify potential health crises before they occur, allowing for early intervention. The primary use of Lyapunov functions is in stability analysis. Through Lyapunov’s Direct Method, researchers construct these functions to prove the stability of an equilibrium point in a dynamical system. This method not only demonstrates stability but also shows asymptotic stability, where the system’s state converges to an equilibrium point over time. Moreover, if a Lyapunov function can be extended over the entire state space, it proves global stability, a powerful testament to the system’s robustness. Stability analysis using Lyapunov functions ensures that these predictive models remain reliable and do not produce false alarms or miss critical warnings due to unstable behavior.
In optimization, Lyapunov functions prove the convergence and stability of algorithms, especially in non-convex optimization problems. In machine learning, they demonstrate the stability and convergence properties of training algorithms like gradient descent, ensuring reliable model training. In control theory, Lyapunov functions are indispensable. They aid in designing controllers that ensure the stability of a closed-loop system. This is crucial in adaptive control, where system parameters change over time. Lyapunov functions help design adaptive control laws to maintain stability despite these changes. In robust control, they ensure controllers can handle uncertainties and disturbances, maintaining system performance. For nonlinear systems, where traditional linearization methods fall short, Lyapunov functions come to the rescue. They analyze the stability of such systems and identify invariant sets and regions of attraction within the state space. This capability is particularly useful in systems exhibiting complex, unpredictable behavior.
In ecological and biological systems, Lyapunov functions play a critical role. They help study the stability of populations and interactions in population dynamics and analyze the spread and control of diseases in epidemiology. This application is vital for understanding and managing biological systems and public health. Hybrid systems, combining continuous and discrete dynamics, pose unique stability challenges. Lyapunov functions are used to analyze stability across different modes in these systems. Similarly, for switched systems that transition between different subsystems, these functions are vital in ensuring overall stability.
While Lyapunov functions are not directly used in determining the weights of AI models, they intersect with AI to ensure the stability and convergence of learning algorithms. In adaptive learning, robust training, and reinforcement learning, they provide a framework for stable adaptation and robust learning rules. In control and robotics, they ensure learned control laws stabilize systems. For recurrent neural networks (RNNs) and other feedback systems, they analyze and ensure the stability of network dynamics. In hybrid systems combining AI with traditional control methods, Lyapunov functions help analyze overall system stability. To effectively leverage Lyapunov functions in health care AI, the U.S. health care system should consider the following strategies immediately:
Funding and support for research. Provide funding through agencies like the National Institutes of Health (NIH) and the National Science Foundation (NSF) for research into the application of Lyapunov functions in health care AI. Encourage interdisciplinary collaboration between mathematicians, engineers, and health care professionals.
Development of standards and guidelines. Task organizations such as the Food and Drug Administration (FDA) and the Department of Health and Human Services (HHS) with developing standards and guidelines for the use of Lyapunov-based stability analysis in health care AI systems. This will ensure consistency and reliability across the industry.
Public-private partnerships. Foster partnerships between government agencies, academic institutions, and private sector companies to accelerate the development and implementation of stable and safe AI systems in health care. Collaborative efforts can lead to the creation of robust frameworks and best practices.
Training and education. Invest in training programs for health care professionals, AI developers, and regulatory personnel on the principles and applications of Lyapunov functions. Enhancing the knowledge base will facilitate the adoption of these techniques in practical settings.
Pilot programs and case studies. Launch pilot programs to test and validate the use of Lyapunov functions in real-world health care scenarios. Document and disseminate case studies that demonstrate the benefits and challenges of this approach, providing valuable insights for wider adoption.
Lyapunov functions are fundamental in both theoretical research and practical applications and provide a systematic way to study the stability of a wide range of systems. From ensuring the stability of autonomous drones and spacecraft to analyzing the spread of diseases and the stability of financial systems, these functions are indispensable. Their use across various fields highlights their versatility and importance in maintaining the stability and robustness of complex systems. The potential of artificial intelligence in health care is immense, but realizing this potential safely and effectively requires rigorous stability analysis. By leveraging Lyapunov functions, U.S. health care systems can ensure that AI systems in health care are not only innovative but also reliable and safe. Through research funding, standard development, public-private partnerships, education, and pilot programs, the government can lead the way in integrating this powerful mathematical tool into the future of health care.
L. Joseph Parker is a research physician. Neil Anand is an anesthesiologist.
