Scientists now have a new AI tool that turns chaotic, real-world motion into simple, testable rules — without needing physics equations. Duke University’s framework learns compact linear models from messy data, enabling faster discovery in climate science, neuroscience, and engineering.
Duke University engineers have built an artificial intelligence system capable of transforming chaotic, real-world motion into simple, human-readable rules — something scientists have long chased but never achieved at scale. Led by Boyuan Chen, director of Duke’s General Robotics Lab, and PhD candidate Sam Moore, the team’s work appears in npj Complexity. Their innovation doesn’t just predict future behavior; it reveals underlying structure — offering a powerful new lens for understanding complex systems.
Their framework analyzes time-series data — measurements taken over time — then generates compact mathematical equations that describe how systems evolve. This applies across domains: weather patterns, electrical circuits, mechanical devices, biological signals, and more. The goal isn’t merely prediction, but insight — turning raw observations into actionable scientific knowledge.
“Scientific discovery has always depended on finding simplified representations of complicated processes,” said Chen, Dickinson Family Assistant Professor of Mechanical Engineering and Materials Science at Duke. “We increasingly have the raw data needed to understand complex systems, but not the tools to turn that information into the kinds of simplified rules scientists rely on. Bridging that gap is essential.”
Why Complex Systems Resist Simple Explanations
Since Isaac Newton’s 1687 Principia, science has relied on dynamical systems theory to track state variables as they evolve — describing everything from planetary orbits to cellular activity. Yet many real-world systems resist simplification. You can measure their behavior, but identifying the rules driving them remains elusive.
Nonlinear behavior compounds the problem. Small changes can trigger vastly different outcomes — a hallmark of chaos theory. High dimensionality adds another layer: when systems involve dozens or hundreds of interacting states, interpretation becomes nearly impossible, and existing analysis tools often fail.
Even seemingly simple motions illustrate this tradeoff. A cannonball’s trajectory depends on speed, angle, drag, wind, and temperature — yet physicists still use simple equations based on just speed and angle to get close enough approximations. Science advances precisely by finding those “good enough” reductions.
A 1930s Mathematical Idea, Upgraded with Modern AI
The Duke team builds on a concept proposed by mathematician Bernard Koopman in 1931: nonlinear systems can sometimes be represented through linear models — if described using the right coordinates. Linear models offer advantages because they enable global analysis and spectral decomposition — tools that reveal modes and stability.
But Koopman’s approach faced a critical limitation: scale. Representing nonlinear systems often required embedding them into huge, even infinite-dimensional spaces — making models unwieldy and prone to redundancy, false modes, and overfitting. Past methods like Dynamic Mode Decomposition or Extended DMD struggled with this inflation. Even deep learning approaches landed on latent spaces far larger than the original systems.
“The catch is scale,” Chen explained. “Koopman-style modeling often pushes you into a very large, even infinite, space of variables. That reality has fed a long-running problem in the field.”
The team tested their method against famous benchmarks. For example, the two-dimensional Duffing system was previously represented using embeddings reaching 100 dimensions — sometimes much more. Similar inflation occurred with the Van der Pol oscillator. While these large embeddings worked, they were inefficient and risky.
How the New Framework Shrinks the Problem
The Duke framework prioritizes minimalism while preserving predictive power. It takes experimental time series data and uses deep learning combined with physics-inspired constraints to discover a reduced set of hidden variables — called latent dimensions — that still capture the system’s core behavior.
In practice, the model learns a latent space labeled ψ, where dynamics behave like a linear system. It leverages time-delay embedding — feeding past states into the model to infer future behavior — and employs a mutual-information method to select optimal time-delay lengths since this choice heavily influences prediction accuracy.
Training focuses on long-horizon accuracy. The researchers use a discounted loss function over future steps, gradually adjusting the discount rate over time — similar to a curriculum — to help the model generalize beyond its training window. They also explore multiple latent dimensions and select the smallest one that doesn’t meaningfully harm performance.
“What stands out is not just the accuracy, but the interpretability,” Chen emphasized. “When a linear model is compact, the scientific discovery process can be naturally connected to existing theories and methods that human scientists have developed over millennia. It’s like connecting AI scientists with human scientists.”
Nine Testbeds, From Pendulums to Neural Circuits to Weather Models
To validate their method, the team created nine datasets spanning simulated and experimental nonlinear systems — ranging from simple to highly complex. This breadth matters: a method only working on textbook motion offers little practical value.
The simplest case: a single pendulum with two measured variables and a stable resting state. Next came the Van der Pol oscillator — featuring a repeating limit cycle — followed by the Hodgkin-Huxley model, which describes neuron firing with four variables and strong nonlinearity. Then came the Lorenz-96 system — used in weather predictability research — introducing high dimensionality with periodic and chaotic behaviors.
The study also focused on multistability — systems that can settle into multiple long-term patterns — exemplified by the Duffing oscillator. Other testbeds included interacting magnetic systems, nested limit cycles, an experimental magnetic pendulum, and a double pendulum exhibiting chaotic behavior.
Across most systems, the framework found reduced models more than 10 times smaller than previous machine-learning approaches — while maintaining reliable long-term forecasts. For instance, the Van der Pol oscillator required only three dimensions, and the Duffing oscillator six. In a higher-dimensional case — a limit-cycle Lorenz-96 system — the model reduced 40 states down to just 14 latent dimensions without sacrificing performance.
Finding the “Landmarks” That Explain Stability and Change
Prediction is valuable — but so is interpretation. The framework also aims to reveal structures dynamicists care about, including attractors — stable states or patterns a system tends toward — helping scientists judge whether a system behaves normally, drifts, or edges toward instability.
“For a dynamicist, finding these structures is like finding the landmarks of a new landscape,” said Moore. “Once you know where the stable points are, the rest of the system starts to make sense.”
The method supports spectral analysis of learned linear systems — extracting eigenvalues and eigenfunctions that describe modes, frequencies, and decay rates. It also produces neural Lyapunov functions — learned stability tools built from decaying modes — providing a practical way to assess global stability, an area where nonlinear systems often force researchers to settle for local answers.
Researchers penalized unstable growth during training by discouraging eigenvalues with positive real parts — ensuring learned dynamics remain physically realistic rather than exploding in ways that fit training data but fail in reality.
“This is not about replacing physics,” Moore clarified. “It’s about extending our ability to reason using data when the physics is unknown, hidden, or too cumbersome to write down.”
Practical Implications for Science and Industry
This work represents a paradigm shift: AI tools that do more than spot patterns. If a model can uncover compact, interpretable rules from messy measurements, researchers can test hypotheses faster, design better experiments, and accelerate discovery — especially in fields where governing equations are incomplete or too hard to derive.
Applications span climate science, neuroscience, and complex engineering systems. The framework could also improve early warning and control systems — identifying when systems drift toward instability — crucial for electrical grids, aircraft dynamics, and biological rhythms.
Moreover, the method can guide what data to collect next — reducing cost in expensive experiments. Over time, this approach could support “machine scientists” — AI collaborators that help human researchers move from raw measurements to clear, testable rules.
Research findings are available online in npj Complexity.
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