Learning Stable Latent Dynamics for Plasma Simulation

Nuclear Fusion: The Elusive Dream of Clean Energy

The Sun produces enormous amounts of energy through nuclear fusion: light atomic nuclei collide and fuse into heavier ones, releasing energy in the process. If we could reproduce fusion efficiently and controllably on Earth, it could provide a powerful source of energy. The hard part is getting atomic nuclei close enough together to fuse. Because positively charged nuclei repel one another, the fuel must be heated to extraordinary temperatures, creating plasma, a state of matter in which electrons are stripped away from atoms.

One way to create these extreme conditions is inertial confinement fusion (ICF). Instead of trying to hold hot plasma for a long time, ICF starts with a tiny capsule of fusion fuel and compresses it extremely rapidly. Powerful lasers deposit energy around the capsule, driving its outer layers outward and the fuel itself inward. For a fraction of a second, the center can become hot and dense enough for fusion reactions to occur.

Understanding this process requires simulations that track many interacting pieces of physics at once: fluid motion, radiation, heat transport, atomic processes, and more. These are known as radiation–hydrodynamics simulations. And buried inside these already enormous calculations is another expensive problem: figuring out exactly what the atoms and ions in the plasma are doing.

When Atoms Are Far From Equilibrium

In an ordinary system at equilibrium, temperature largely determines how particles are distributed among their possible states. But laser-produced plasmas evolve so quickly that atoms often do not have time to settle into equilibrium. This regime is called non-local thermodynamic equilibrium, or NLTE.

Under NLTE conditions, we need to keep track of how populations move among potentially thousands or even millions of atomic states as electrons collide with atoms, electrons are stripped away, ions recombine, and radiation is absorbed or emitted. These atomic populations determine quantities such as opacity—how easily radiation travels through the plasma—and emissivity—how strongly the plasma emits radiation. Those quantities then feed back into the larger radiation–hydrodynamics simulation.

The problem is computational cost. An NLTE calculation may need to be performed at every time step of a plasma simulation, and solving this becomes one of the dominant computational bottlenecks.

Why a Black-Box Neural Network Is Not Enough

Machine-learning surrogate models have already been used to accelerate NLTE calculations. A common approach is to treat the problem as a direct mapping: give a neural network the plasma conditions, such as temperature and density, and ask it to predict the desired atomic or radiative quantities.

But atomic populations evolve through time. A model might reproduce individual training points very accurately while still learning dynamics that eventually become unstable, diverge, or settle into a physically incorrect state. This is especially problematic when we want to run the model longer than the trajectories it saw during training.

Instead of simply asking a neural network to predict the answer, we wanted to learn a small dynamical system that still behaves like physics.

Compress the Physics, Then Learn How It Evolves

We propose a framework called physics-informed Latent Space Dynamics Identification (pLaSDI). The basic idea is simple. The full atomic state can be enormous. In our test problem, each moment in time is represented by 1,583 atomic populations. We first use an autoencoder to compress those 1,583 numbers into just three latent variables.

But instead of allowing those three numbers to evolve through an arbitrary neural network, we learn an explicit differential equation describing how they change as the plasma temperature and density evolve. This matters because an explicit equation gives us something a black box does not: we can mathematically inspect its behavior.

In particular, we build physical constraints directly into training:

1. Conservation and atomic consistency: reconstructed populations should preserve physically meaningful quantities such as total population and charge-state distributions.
2. Stability: the learned dynamics should not suddenly explode when integrated forward in time.
3. Correct steady state: if temperature and density stop changing, the atomic populations should eventually settle into the physically correct equilibrium rather than merely some stable state.

Why Physics Constraints Matter

One of the most revealing experiments was simply removing the stability constraint.

The resulting model could fit local changes in the training data reasonably well. But when we actually integrated its learned equations forward in time, small errors accumulated and the predicted atomic populations diverged.

Adding the mathematical stability constraint changed the behavior completely: trajectories remained bounded and continued toward a stable state even beyond the time window used for training. There was another subtlety. Stability by itself only guarantees that the system eventually goes somewhere. It does not guarantee that it goes to the correct physical equilibrium. For that, we added a separate steady-state constraint. Together, these constraints let the model reproduce not only short-time trajectories but also physically meaningful long-time behavior.

This points to a broader lesson for scientific machine learning:

It is not enough for a model to fit the data. For dynamical systems, we also need to constrain how the learned system behaves.

Up to 100,000× Faster

The resulting model compresses the original 1,583-dimensional atomic state into only three dimensions, a reduction of more than 500×. Despite that compression, it reproduces the mean charge state with about 2% relative error and closely tracks the evolving charge-state distribution.

The computational difference is much larger. A conventional time-dependent SCFLY NLTE calculation can take tens of minutes to hours. Once trained, our surrogate evaluates an entire trajectory in less than approximately 0.038 seconds, corresponding to a speedup of roughly 50,000–100,000×. That kind of speedup matters because plasma simulations are rarely run just once. Researchers may need thousands of simulations for design optimization, uncertainty quantification, parameter searches, or eventually autonomous experimental workflows.

What's Next?

This work is still a proof of concept. The experiments use an optically thin, laser-heated tin plasma and a reduced atomic representation, which lets us isolate the atomic-kinetics problem without including the full complexity of radiation transport.

The next steps are to incorporate radiation transport, move toward more detailed atomic models, improve reconstruction of individual atomic populations, and ultimately integrate the surrogate directly into radiation–hydrodynamics codes. The paper also leaves explicit spectral and opacity calculations for future work.

For fusion and other extreme-plasma applications, that is ultimately the goal: not simply to build an ML model that approximates an expensive calculation, but to build a fast reduced model whose dynamics remain constrained by the physics we already know.