Koopman Operator Theory

The universe is governed by rules of change, but these rules are rarely simple, straight lines. From the turbulent flow of a river to the firing of neurons in the brain, nonlinear dynamics shape our world, making prediction and control notoriously difficult. How can we find order in this apparent chaos? What if there was a way to look at these complex systems through a different lens—one that transforms their tangled, nonlinear behavior into the elegant, predictable language of linear algebra? This is the revolutionary promise of Koopman operator theory. This article demystifies this powerful framework, addressing the fundamental challenge of analyzing systems whose future is not simply proportional to their present. It guides the reader through the foundational principles of the theory, its computational implementation, and its profound impact across science and engineering. In the first chapter, "Principles and Mechanisms," we will explore how the theory shifts focus from system states to observables, revealing the system's secrets through eigenvalues and eigenfunctions. Following this, the "Applications and Interdisciplinary Connections" chapter will demonstrate how this abstract idea becomes a practical tool for everything from designing stable robots to discovering physical laws from data.

Imagine you are trying to understand the weather. You could try to follow the path of a single air molecule as it gets tossed and turned in the turbulent atmosphere—a dizzying, impossibly complicated task. Or, you could take a different view. You could stand still and watch how the temperature, pressure, and humidity change at your location. You're no longer tracking the particle; you're tracking the values of measurements (or "observables") as the system flows past. This shift in perspective is the heart of Koopman operator theory. It’s a trick, a beautiful piece of mathematical jujitsu that allows us to use the simple, powerful tools of linear algebra to understand the chaotic and complex dance of nonlinear dynamics.

Most of the universe is nonlinear. From the orbits of planets to the firing of neurons and the crashing of waves, the rules of change are rarely simple straight lines. A small push doesn't always lead to a small effect; sometimes it can lead to a gigantic one. This makes predicting the future of such systems notoriously difficult. The governing equations, like $\dot{x}=f(x)$, describe how a state $x$ (perhaps the position and velocity of our air molecule) changes in time. The function $f$ is the nonlinear rulebook, and following it can be a wild ride.

The Koopman operator, proposed by Bernard Koopman in the 1930s, offers a startlingly different approach. Instead of focusing on the state $x$, we focus on an observable function, let's call it $g(x)$. This could be any measurement you can imagine: the temperature at position $x$, the kinetic energy of a particle, or even just one of its coordinates. The Koopman operator, which we'll call $\mathcal{K}^t$, doesn't evolve the state $x$. It evolves the observable function $g$.

How? It simply asks: "If the system starts at state $x$ and evolves for a time $t$ to a new state $\Phi^t(x)$, what is the value of our original observable $g$ at this new location?" Mathematically, we write this as:

$(\mathcal{K}^t g)(x) = g(\Phi^t(x))$

Look closely at this definition. On the right-hand side, the nonlinearity is all bundled up inside the flow map $\Phi^t(x)$. But the operator $\mathcal{K}^t$ itself, in how it acts on functions, is perfectly ​​linear​​. If you take a combination of two observables, say $g_1$ and $g_2$, the operator acts on each one separately: $\mathcal{K}^t(c_1 g_1 + c_2 g_2) = c_1 (\mathcal{K}^t g_1) + c_2 (\mathcal{K}^t g_2)$. This is its magic trick. It trades the finite-dimensional, nonlinear problem of evolving states for an infinite-dimensional, but linear, problem of evolving functions. And with linearity comes a treasure trove of powerful tools, most notably the concepts of eigenvalues and eigenfunctions.

Because the Koopman operator is linear, we can ask a question familiar to anyone who has studied linear algebra or quantum mechanics: are there any special functions that, when the operator acts on them, don't change their shape, but are just scaled by a constant factor? These are the ​​eigenfunctions​​ of the operator, and the scaling factors are their ​​eigenvalues​​.

An eigenfunction $\phi(x)$ of $\mathcal{K}^t$ with eigenvalue $\lambda^t$ satisfies:

$(\mathcal{K}^t \phi)(x) = \phi(\Phi^t(x)) = \lambda^t \phi(x)$

These are not just mathematical curiosities; they are the fundamental building blocks of the dynamics. An eigenfunction represents a pattern or a coordinated mode of behavior in the system. The corresponding eigenvalue tells you how that pattern evolves in time.

• If the eigenvalue $\lambda$ has a magnitude $|\lambda| > 1$, the pattern described by $\phi(x)$ will grow exponentially over time. This might correspond to an instability.
• If $|\lambda| < 1$, the pattern will decay and vanish. This could represent a transient behavior settling down.
• If $|\lambda| = 1$, the pattern persists forever, neither growing nor shrinking. Its value may oscillate, and the frequency of that oscillation is encoded in the complex phase of $\lambda$.

The collection of all these eigenvalues is the ​​Koopman spectrum​​. It is a fingerprint of the dynamical system, a hidden bar code that reveals its deepest secrets—its frequencies, its growth rates, its conserved quantities, and even its long-term behavior.

What if an eigenvalue is exactly equal to 1? Then we have $\mathcal{K}^t \phi = \phi$, which means $\phi(\Phi^t(x)) = \phi(x)$ for all time $t$. This tells us that the value of the observable $\phi$ is constant along any trajectory of the system. In physics, we call such a quantity a ​​conserved quantity​​ or an integral of motion.

For any dynamical system, there's always one obvious conserved quantity: a constant function. If you have an observable $g(x) = c$, then no matter where the system moves, the value of the observable remains $c$. This means that constant functions are always eigenfunctions of the Koopman operator with eigenvalue $\lambda=1$. This might seem trivial, but it forms a crucial baseline. The truly interesting question is: are there any other, non-constant eigenfunctions with eigenvalue 1? The answer to this question tells us whether the system is "stuck" in some way, or if it explores its entire domain. This leads us to the profound concept of ergodicity.

In everyday language, "ergodic" might mean a system that, given enough time, explores all the states it possibly can. Think of a gas molecule in a box; eventually, it will have visited every nook and cranny. A more precise definition relates two kinds of averages. The ​​time average​​ of an observable is what you get by following a single trajectory for a very long time and averaging the values you see. The ​​space average​​ is what you get by "freezing" the system at one moment and averaging the observable's value over the entire state space.

A system is ​​ergodic​​ if, for almost any starting point, the time average equals the space average. This is an incredibly powerful property. It means that a single, long simulation can tell you about the average properties of the entire system.

The Koopman operator provides a stunningly elegant criterion for ergodicity. A system is ergodic if and only if the only eigenfunctions with eigenvalue 1 are the constant functions. If a non-constant eigenfunction with eigenvalue 1 exists, it means there's a conserved quantity that partitions the state space. A trajectory starting in a region where this function has value $c_1$ can never cross into a region where its value is $c_2$. The system is "stuck," and it cannot explore the whole space.

Let's see this in action with a beautiful, simple example: a point moving around a circle.

• ​​Rational Rotation:​​ Consider the map $T(x) = x + p/q \pmod 1$, where $p/q$ is a fraction. If you start at any point, you will visit only $q$ distinct spots before returning to the beginning. The trajectory is periodic and clearly doesn't cover the whole circle. The system is not ergodic. And behold, we can easily find a non-constant function that is invariant under this map: $f(x) = \exp(2\pi i q x)$. This function has a period of $1/q$, and it is an eigenfunction of the corresponding Koopman operator with eigenvalue 1. The existence of this non-constant invariant is the spectral proof of non-ergodicity.
• ​​Irrational Rotation:​​ Now, consider the map $T(x) = x + \alpha \pmod 1$, where $\alpha$ is irrational. The trajectory of any point will never exactly repeat; it will eventually come arbitrarily close to every point on the circle, densely filling it. This system is ergodic. And if we check its Koopman spectrum, we find that the only functions satisfying $f(x+\alpha) = f(x)$ are the constant functions. The eigenspace for eigenvalue 1 is one-dimensional, confirming ergodicity.

This principle is general and can be used to prove ergodicity in far more complicated systems, like certain dynamics on a torus. The reward for establishing ergodicity is the ​​Mean Ergodic Theorem​​. It guarantees that the long-time average of an observable converges to its space average. For a chaotic system like the Baker's Map, known to be ergodic, this allows us to calculate the long-term average behavior of any quantity simply by integrating it over the unit square—a task that is often much easier than simulating a long trajectory.

Ergodicity is a statement about where a system goes in the long run. A stronger property is ​​mixing​​. Imagine stirring a drop of cream into coffee. Ergodicity says that eventually, every part of the coffee will have some cream in it. Mixing says that the cream will become smoothly and indistinguishably blended throughout the entire cup. Any initial concentration of cream will spread out until its density is uniform everywhere.

In the language of observables, mixing means that for any two observables $f$ and $g$, the correlation between the initial state of $f$ and the future state of $g$ eventually fades to nothing. Mathematically, $\langle f, \mathcal{K}^n g \rangle \to \langle f, 1 \rangle \langle 1, g \rangle$ as $n \to \infty$. The system loses all memory of its initial details. This is a hallmark of chaos.

The spectral signature of mixing is even more stringent than that of ergodicity. While an ergodic system can still have a rich spectrum of discrete eigenvalues with magnitude 1 (as in the irrational circle rotation), a mixing system has a purely continuous spectrum (aside from the simple eigenvalue at 1). For a truly chaotic system like the dyadic map $T(x) = 2x \pmod 1$, correlations decay, and the long-time averaged correlation between two functions simply becomes the product of their individual averages.

This is all wonderful, but there's a catch. The Koopman operator acts on an infinite-dimensional space of functions. How can we ever compute its spectrum in practice? This is where modern developments have transformed the theory from an elegant abstraction into a powerful computational tool.

One path is to be clever. For some special nonlinear systems, like the Stuart-Landau equation which models the onset of turbulence, one can identify a special subspace of observables that is "invariant" under the Koopman operator. That is, the operator maps any function in this family to another function within the same family. By restricting the operator to this simpler, smaller world, we can sometimes solve for its eigenvalues exactly, revealing the precise growth rates and frequencies of the underlying nonlinear modes.

A more general and revolutionary approach is to use data. This is the idea behind ​​Dynamic Mode Decomposition (DMD)​​. We don't even need to know the governing equations! Suppose we have a series of "snapshots" of our system—measurements of the state $x_k$ at regular time intervals $\Delta t$.

1. We choose a dictionary of observable functions, $\mathbf{g}(x) = [g_1(x), g_2(x), \dots, g_m(x)]^T$. These could be simple polynomials, Fourier modes, or even the raw state variables themselves.
2. We collect data, forming two big matrices: one matrix $G_X$ with columns $\mathbf{g}(x_k)$ for our snapshots, and another matrix $G_Y$ with columns $\mathbf{g}(x_{k+1})$.
3. Since we know that $(\mathcal{K}^{\Delta t}g)(x_k) = g(x_{k+1})$, we are looking for a linear operator that maps our observables at one time step to the next. DMD finds the best possible finite-dimensional matrix, let's call it $A$, that does this job in a least-squares sense: $G_Y \approx A G_X$.
4. This matrix $A$ is our prize. It is a finite-dimensional, computable approximation of the infinite-dimensional Koopman operator $\mathcal{K}^{\Delta t}$. The eigenvalues of $A$ give us approximations of the true Koopman eigenvalues, and its eigenvectors (the DMD modes) approximate the Koopman eigenfunctions.

This data-driven approach bridges the gap between abstract theory and messy reality. It allows us to take complex data—from video footage of a turbulent fluid, financial time series, or brain activity scans—and decompose it into its fundamental, dynamically relevant patterns, each associated with a specific frequency and growth or decay rate. It is the fulfillment of Koopman's original vision: a linear framework for making sense of a nonlinear world.

Having journeyed through the principles and mechanisms of the Koopman operator, you might be left with a sense of mathematical elegance, but perhaps also a question: What is this beautiful abstraction truly for? It is one thing to know that we can view a nonlinear world through a linear lens; it is another entirely to see what this new perspective allows us to do. As it turns out, this shift in viewpoint is not merely a theoretical curiosity. It is a master key, unlocking profound insights and powerful new tools across a breathtaking range of scientific and engineering disciplines. It transforms the art of analysis into a science of construction, and the task of prediction into a path toward discovery.

Let us now explore this landscape, to see how the ghost of linearity that Koopman theory summons can be put to work in the tangible world.

Imagine you are an engineer tasked with designing a control system for a robot, an autonomous vehicle, or a complex power grid. Your primary concern is stability. You need to ensure that if the system is nudged slightly from its desired operating state, it will gracefully return, rather than spiraling out of control. For over a century, the cornerstone of stability analysis has been the method of Aleksandr Lyapunov. The idea is wonderfully intuitive: if you can find a function, let's call it $V(\mathbf{x})$, that is always positive (except at the equilibrium point) and that always decreases as the system evolves, then the system must be stable. This function acts like an "energy" bowl; the system state is like a marble that will always roll down to the bottom.

The great difficulty, however, has always been in finding this Lyapunov function $V(\mathbf{x})$. Its discovery has often been considered more of an art than a science, relying on clever guesses and painstaking trial and error.

Here, Koopman theory offers a stunningly direct and constructive approach. Recall that Koopman eigenfunctions, $\phi_j(\mathbf{x})$, are the special observables that transform simply under the dynamics: their time derivative is just a number, the eigenvalue $\lambda_j$, times the function itself. Now, suppose we find a set of eigenfunctions whose corresponding eigenvalues all have negative real parts, i.e., $\text{Re}(\lambda_j) 0$. This means each of these observables decays exponentially to zero along any trajectory. What happens if we construct a candidate Lyapunov function as a simple sum of squares of these eigenfunctions, say $V(\mathbf{x}) = |\phi_1(\mathbf{x})|^2 + |\phi_2(\mathbf{x})|^2$?

Let's look at its time derivative:

Using the eigenfunction property $\dot{\phi}_j = \lambda_j \phi_j$, this becomes:

Since we chose the eigenvalues to have negative real parts, $\dot{V}$ is guaranteed to be negative! We have systematically constructed a valid Lyapunov function, turning the art of guesswork into a clear-cut procedure. By finding the special "coordinates" in which the system decays linearly, we can certify the stability of the entire nonlinear system. This provides a powerful, practical recipe for engineers to analyze and guarantee the safety of complex dynamical systems.

The engineer's problem is often one of analyzing a system whose governing equations are, at least approximately, known. But what if we are explorers in an unknown land? What if we have no equations, only data? This is the frontier of modern science, where we seek to understand complex phenomena—from the intricate dance of proteins to the turbulence of the climate—by observing them.

Imagine a nanoscientist using an Atomic Force Microscope (AFM) to probe the surface of a material. A tiny cantilever tip taps against the surface, and its motion reveals information about the nanoscale forces at play. These forces—arising from quantum mechanics and electrostatics—are notoriously complex and nonlinear. Writing down an exact, simple equation for the tip's motion is often impossible. What we can do is measure the tip's position and velocity thousands of times per second.

This is where Koopman theory, in its computational form known as Dynamic Mode Decomposition (DMD) and its powerful extension (EDMD), provides a kind of "crystal ball." The core idea is to not limit ourselves to the raw data of position $z$ and velocity $v$. Instead, we tell our computer to also watch a whole dictionary of related observables: $z^2$, $v^2$, $zv$, $z^3$, and so on. We are "lifting" our view into a much larger, abstract space. The magic is that, within this vast space of possibilities, we can find a combination of observables that does evolve linearly. The computer sifts through the data to find a finite-dimensional linear model in this lifted space that best predicts the future. This allows us to build an incredibly accurate predictive model from data alone, capturing the essence of the nonlinear tip-sample interactions without ever writing down the messy force law itself. Of course, this magic requires good data; we must "excite" the system enough to see all its characteristic behaviors, a principle well-understood through this framework.

We can even raise our ambition from mere prediction to outright discovery. Can we teach a machine to be a physicist—to deduce the fundamental laws of nature from data? Consider a system of interacting particles from a molecular dynamics simulation. Its entire evolution is governed by a single, sacred quantity: the Hamiltonian, or total energy. If we knew the formula for the Hamiltonian $H$, we would know everything. The problem is, we don't. But we have trajectory data.

A modern machine learning approach, inspired by Koopman theory, tackles this head-on. A neural network is designed to learn a set of special observables $\phi(\mathbf{z})$ from the raw state data $\mathbf{z}$. The training is guided by two competing goals. First, the predicted dynamics must obey the known structure of physics (in this case, Hamilton's equations) based on a guessed Hamiltonian. Second, the learned observables themselves must evolve as linearly as possible, just as the Koopman operator demands. By minimizing the error in these two goals simultaneously, the model is forced to find a set of observables and a corresponding energy function that are mutually consistent. Incredibly, this approach allows the machine to learn the underlying Hamiltonian—the physical law itself—directly from the data of the system's motion.

The quest to find a simple representation of complex dynamics is not unique to physics and engineering; it is at the very heart of modern artificial intelligence. Many of the most challenging problems in AI, from financial forecasting to weather prediction and natural language processing, involve making sense of complex sequences of data.

In recent years, a new class of deep learning architectures, known as State-Space Models (SSMs), has achieved remarkable success on these tasks. On the surface, they are complex "black boxes" with millions of parameters. Yet, if we look under the hood with our Koopman lens, we find something astonishing. The success of these models can be understood as an implicit discovery of the Koopman operator.

The internal "latent state" of an SSM acts as a learned set of observables. The model's architecture forces this latent state to evolve according to a simple linear rule. In essence, when we train an SSM on a complex time series, the neural network learns to perform the Koopman program automatically: it finds a transformation of the raw data into a special feature space where the dynamics are linear and thus easy to predict. The theory tells us the conditions under which this is possible: the underlying system must possess some regularity (like ergodicity), and the dominant, slow dynamics must be separable from the faster, decaying parts of the motion (a "spectral gap"). When these conditions hold, the neural network can successfully approximate the leading Koopman eigenfunctions and modes, providing a profound theoretical justification for the practical power of these AI models. It is not a black box after all; it is an operator discovery machine.

Finally, Koopman theory gives us a universal language to describe motion itself. Just as a prism resolves white light into a spectrum of fundamental colors, the Koopman operator decomposes a complex, chaotic motion into a spectrum of fundamental frequencies and decay rates.

Consider any observable property of a system. We can measure its time-autocorrelation function, $C(t)$, which tells us how quickly the system "forgets" its current state. A rapidly decaying $C(t)$ signifies chaotic motion, while an oscillating one suggests periodic behavior. The spectral theorem, the foundation stone of Koopman theory, tells us that this correlation function is the Fourier transform of a spectral measure, $\mu(\lambda)$. This measure is the "power spectrum" of the dynamics. It tells us precisely which frequencies $\lambda$ are present in the motion and with what intensity.

If, for a given system, we observe a correlation function of a certain form, say $C(t) = (1+\gamma|t|)e^{-\gamma|t|}$, we can perform a Fourier transform to find the exact shape of its spectral density. This connects a macroscopic measurement (the rate of forgetting) to the fundamental frequencies of the system's modes, providing a complete spectral fingerprint of the dynamics.

This perspective is so powerful that it can even be applied to the seemingly intractable world of fluid mechanics, governed by partial differential equations. The inviscid Burgers' equation, for example, is a classic model for how shock waves form in a fluid—a highly nonlinear event. Yet, even for this system, we can define a Koopman operator and find eigenfunctions that evolve in a perfectly linear, predictable way, sailing smoothly through the formation of the shock.

From ensuring the stability of a drone, to discovering the forces that hold molecules together, to justifying the architecture of next-generation AI, to deciphering the spectrum of chaos—the applications of Koopman theory are as diverse as science itself. What unites them is a single, profound idea: complexity is often a matter of perspective. By searching for the right observables, the right set of "coordinates," the tangled web of nonlinear dynamics can often be unraveled into a simple collection of parallel, linear threads. This is more than just a useful trick; it reveals a hidden unity in the way the universe behaves and in the way we have learned to describe it.