Understanding how systems change over time has been a central pursuit of science for centuries. From the motion of planets to the behavior of electrical circuits and biological networks, scientists have sought simplified rules that explain complex dynamics. Recently, researchers at Duke University have taken a significant step toward this goal by developing a novel artificial intelligence (AI) framework capable of uncovering clear, interpretable equations from highly complex, nonlinear systems. Their work, published online on December 17 in npj Complexity, represents a powerful fusion of modern machine learning and classical dynamical systems theory.
The new AI system is inspired by the tradition of great “dynamicists,” scientists who study systems that evolve over time. Isaac Newton, often regarded as the first dynamicist, transformed science by expressing motion and force through concise mathematical equations. Similarly, the Duke framework analyzes time-series data describing how complex systems evolve and then produces equations that accurately capture this behavior. What makes this approach remarkable is its ability to operate at a level of complexity far beyond human analytical capacity, reducing systems with hundreds or thousands of interacting variables into simplified, low-dimensional representations.
The Challenge of Complexity in Modern Science
Scientific discovery has long depended on simplification. Natural systems are influenced by countless factors, yet useful theories often rely on only a few key variables. A classic example is projectile motion. Although the trajectory of a cannonball depends on numerous influences—air resistance, wind, temperature, and more—it can be closely approximated using a simple linear equation involving just launch speed and angle. This ability to reduce complexity without losing essential behavior lies at the heart of scientific modeling.
In modern science, however, researchers increasingly face systems whose complexity defies traditional analytical approaches. Climate dynamics, neural activity, power grids, and advanced mechanical systems generate massive quantities of data, but converting that data into meaningful, interpretable rules remains difficult. According to Boyuan Chen, director of Duke’s General Robotics Lab and assistant professor of mechanical engineering and materials science, the challenge is no longer data availability, but the lack of tools to transform data into the simplified representations scientists rely on. Bridging this gap is essential for progress.
Revisiting Koopman’s Vision with AI
The Duke framework builds upon a mathematical idea introduced in the 1930s by Bernard Koopman. Koopman demonstrated that nonlinear dynamical systems could be represented using linear operators—an insight that opened new ways of analyzing complex behavior. In theory, this means that even highly nonlinear systems can be described using linear equations, which are far easier to analyze and interpret.
The difficulty, however, lies in implementation. Representing a complex system in a linear framework often requires constructing hundreds or thousands of equations tied to different variables. For human researchers, managing and interpreting such massive representations is impractical. This is precisely where artificial intelligence becomes invaluable.
The new AI framework uses deep learning to analyze time-series data from experiments or simulations. By incorporating constraints inspired by physics and dynamical systems theory, the model identifies the most meaningful patterns governing system evolution. Rather than preserving every variable, it discovers a compact set of hidden variables that capture the system’s essential dynamics. The result is a simplified, linear-like model that remains faithful to real-world behavior while being far more interpretable than traditional black-box machine learning methods.
Demonstrating Versatility Across Systems
To validate their approach, the researchers applied the framework to a wide range of systems with vastly different characteristics. These included the simple oscillatory motion of a pendulum, nonlinear electrical circuits, climate-related models, and neural circuit dynamics. Despite the diversity of these systems, the AI consistently identified a small number of governing variables that explained their behavior.
Notably, the resulting models were often more than ten times smaller than those produced by earlier machine-learning techniques, yet they still provided reliable long-term predictions. This reduction in size is not merely a technical improvement; it directly enhances scientific understanding. Compact linear models can be analyzed using well-established mathematical tools, allowing researchers to connect AI-derived insights with centuries of theoretical knowledge.
Chen emphasizes that interpretability is a key strength of the framework. When models are compact and linear, scientists can naturally integrate them with existing theories and analytical methods. In this sense, the AI does not replace human scientists but acts as a bridge, connecting modern data-driven approaches with classical scientific reasoning.
Discovering Stability and Early Warning Signals
Beyond prediction, the AI framework can also identify stable states, known as attractors, toward which systems naturally evolve. In dynamical systems theory, attractors play a crucial role in understanding long-term behavior, stability, and transitions to instability. Recognizing these structures allows scientists to determine whether a system is operating normally, drifting gradually, or approaching a critical tipping point.
Sam Moore, the study’s lead author and a PhD candidate in Chen’s lab, compares this process to mapping a new landscape. Once the stable landmarks are identified, the overall structure of the system becomes much easier to understand. This capability is particularly valuable for monitoring complex systems where early warning signs of failure or instability are difficult to detect.
Importantly, the researchers stress that their approach is not intended to replace physics-based modeling. Instead, it extends scientific reasoning into domains where traditional equations are unavailable, incomplete, or prohibitively complex. By learning directly from data while respecting physical structure, the AI provides a complementary tool for discovery.
Toward the Era of Machine Scientists
Looking ahead, the Duke team envisions broader applications for their framework. One promising direction is guiding experimental design, where the AI actively selects which data to collect in order to reveal a system’s structure more efficiently. This could significantly reduce experimental costs and accelerate discovery. The researchers also plan to extend the method to richer data types, including video, audio, and signals from complex biological systems.
Ultimately, this work supports a long-term vision of developing “machine scientists”—AI systems that assist humans in uncovering fundamental laws of nature. By combining modern artificial intelligence with the mathematical language of dynamical systems, the Duke framework points toward a future in which AI does more than recognize patterns. It becomes an active partner in scientific discovery, helping to reveal the hidden rules that govern both the physical world and living systems.
In this sense, the new AI framework represents not just a technological advance, but a philosophical shift in how science may be conducted in the data-rich era—reviving the spirit of classical dynamicists while extending their reach through artificial intelligence.
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