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Combining complex Langevin dynamics with score-based and energy-based diffusion models

This paper investigates the application of score-based and energy-based diffusion models to learn and characterize the probability distributions sampled by complex Langevin dynamics, aiming to address the challenges of understanding stochastic processes in theories with sign problems.

Original authors: Gert Aarts, Diaa E. Habibi, Lingxiao Wang, Kai Zhou

Published 2026-07-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Gert Aarts, Diaa E. Habibi, Lingxiao Wang, Kai Zhou

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a puzzle, but the pieces are made of invisible, shifting fog. In the world of physics, specifically when dealing with theories that have "complex" numbers (think of them as numbers that spin in a circle rather than just sitting on a line), scientists face a notorious headache called the "sign problem." It's like trying to weigh a ghost; the math gets messy, and standard computer simulations get stuck or give the wrong answer.

For years, physicists have used a trick called Complex Langevin (CL) dynamics. Think of this as a blindfolded hiker wandering through a foggy, two-dimensional landscape. The hiker follows a set of rules (a "drift") that pushes them around, hoping that if they walk long enough, their path will reveal the true shape of the hidden mountain. The problem? We don't actually know what the mountain looks like. We can see the hiker's footsteps, but we can't see the terrain itself. The distribution of where the hiker actually ends up is a mystery, and sometimes the hiker wanders off into the wrong valley entirely.

The Big Idea: Teaching a Robot to Map the Fog

This paper asks a fresh question: What if we could use a type of Artificial Intelligence called a Diffusion Model to learn the shape of that foggy mountain just by watching the hiker?

In the world of generative AI, diffusion models are like artists who learn to draw a picture by starting with a noisy, static-filled TV screen and slowly cleaning it up until a clear image appears. They learn by studying thousands of examples. Here, the authors fed these AI models the "footsteps" (the data) generated by the Complex Langevin hiker. The goal wasn't just to make more footsteps, but to understand the invisible landscape the hiker was walking on.

The Two Approaches: The Compass vs. The Topographic Map

The researchers tested two different ways to teach the AI:

The Score-Based Model (The Compass): This approach tries to learn the "score," which is essentially a compass needle pointing toward the most likely places the hiker should be. It's great at telling you which way to go, but the paper found a catch: this compass isn't always reliable. Sometimes, the needle points in a direction that doesn't make sense if you try to trace a path back to a starting point. It's like a compass that spins wildly in a magnetic storm; it tells you the wind is blowing, but you can't use it to draw a perfect map of the terrain. The authors showed that this "score" has a swirling, non-conservative component—it's not a simple, smooth hill you can climb.
The Energy-Based Model (The Topographic Map): This approach tries to learn the "energy" directly. Think of this as the AI trying to draw the actual 3D shape of the mountain, where high points are unlikely and low points are likely. Because this model builds a solid shape (an energy landscape), it has a built-in rule: the compass (the gradient) must always point downhill. This makes the map consistent.

What They Found (The Simulation Results)

The team tested this on a specific, well-known math problem called the complex-valued quartic model. It's a simple playground with one degree of freedom that gets complexified into two dimensions ( $x$ and $y$ ).

The Hiker's Path: When they ran the Complex Langevin simulation, the hiker stayed trapped in a narrow strip of the landscape, never wandering beyond a certain boundary (specifically, $|y| < 0.3029$ ). The hiker's path showed two distinct "peaks" or favorite spots.
The AI's Success: Both AI models managed to learn this behavior.
- The Score-Based model learned the direction of the wind and could generate new hiker paths that looked very similar to the original ones. However, because its "compass" was a bit wobbly, it couldn't easily draw a perfect, smooth map of the whole terrain without doing extra work.
- The Energy-Based model was the star of the show. It successfully learned the actual shape of the mountain. The authors were able to take the AI's learned "energy" and instantly turn it into a probability map (by exponentiating it) without needing to run any more simulations. This was a big deal because it was the first time a distribution sampled by a Complex Langevin process was captured in a non-trivial case without just making a giant histogram of the data.

The Numbers and the Verdict

The paper didn't just say "it looks good." They did the math. They compared the average values (moments) and the "shape" of the distribution (cumulants) from the AI against the exact mathematical answer and the original hiker data.

For the second moment ( $\mu_2$ ), the exact answer was 0.428142 (real part). The Complex Langevin simulation gave 0.4281(5). The Energy-Based model gave 0.4264(1)(37).
For the fourth moment ( $\mu_4$ ), the exact answer was 0.423848. The Energy-Based model gave 0.4192(2)(61).

The results show that the AI models, particularly the Energy-Based one, are getting very close to the truth, with small deviations that the authors attribute to the finite size of their sample data.

What This Means (and What It Doesn't)

The paper suggests that these AI tools are powerful new lenses for looking at the "foggy" distributions that plague complex physics problems. The Energy-Based model, in particular, offers a way to actually see the distribution, not just simulate it.

However, the authors are careful not to claim they have solved the sign problem for all of physics. They tested this on a simple, solvable model. They explicitly note that in the Score-Based approach, the "score" is not always a simple, integrable function (it's not a perfect compass), which complicates things. They also point out that the reliability of the original Complex Langevin method still depends on strict criteria that must be checked after the fact.

In short, the paper suggests that by combining the wandering hiker (Complex Langevin) with a smart AI mapmaker (Diffusion Models), we can finally start to understand the shape of the invisible landscapes we've been stumbling through for decades. It's a promising new tool, but the journey to fully mastering these complex theories is still ongoing.

Technical Summary: Combining Complex Langevin Dynamics with Score-Based and Energy-Based Diffusion Models

Problem Statement
The paper addresses the challenge of simulating theories with a complex Boltzmann weight (or action), which suffer from the sign and overlap problems, rendering conventional importance sampling methods ineffective. A primary example is Quantum Chromodynamics (QCD) at non-zero baryon density. Complex Langevin (CL) dynamics offers a potential solution by extending the degrees of freedom into a complexified manifold and employing a stochastic process. However, a critical limitation of CL is that the probability distribution effectively sampled on this complexified manifold, denoted as $p(x, y)$ , is not known a priori. Furthermore, the Fokker-Planck equation governing this process cannot generally be solved, making it difficult to understand the distribution's behavior, particularly at infinity or near drift poles. This lack of characterization hinders the verification of correctness criteria and the reliability of the method.

Methodology
The authors propose utilizing generative AI, specifically diffusion models, to learn the distribution sampled by the CL process directly from data. The methodology involves:

Data Generation: Generating training data using Complex Langevin dynamics on a specific model.
Diffusion Model Framework: Employing stochastic differential equations (SDEs) to model the forward (noising) and backward (denoising) processes. The goal is to learn the score function ( $\nabla \log p$ ) or the energy function ( $-\log p$ ) from the CL-generated configurations.
Two Approaches:
- Score-Based Models (SBM): Train a neural network to approximate the score $\nabla \log p(x, t)$ . The authors note that the learned score is generally not conservative (i.e., it contains a non-zero curl component), meaning the distribution cannot be recovered simply by integrating the score.
- Energy-Based Models (EBM): Train a neural network to approximate the energy function $E_\theta(x) \approx -\log p(x)$ . This formulation enforces a conservative score by construction ( $s = -\nabla E$ ), allowing for the direct reconstruction of the probability distribution via exponentiation ( $p \sim e^{-E}$ ) without further sampling.
Test Case: The framework is applied to a complex-valued quartic model with one degree of freedom ( $x \to z = x + iy$ ). This model is chosen because it is exactly solvable in terms of its partition function and has well-understood criteria for when CL dynamics succeeds or fails (based on the parameter $3A^2 - B^2$ ). The authors use parameters where CL is known to work correctly ( $A=B=\lambda=1$ ).

Key Contributions and Results

Learning the CL Distribution: The paper demonstrates that diffusion models can successfully learn the distribution sampled by the CL process, a distribution that is otherwise elusive and lacks an analytical form.
Comparison of SBM and EBM:
- SBM: The learned score field captures the qualitative features of the CL drift, including the two characteristic peaks and the boundary restrictions of the strip $|y| < y_-$ . However, the score is non-conservative, preventing direct integration to find the distribution.
- EBM: The energy-based approach successfully learns an energy landscape with minima corresponding to the distribution's peaks. Crucially, because the score is conservative, the authors can explicitly construct the probability distribution $p(x, y)$ by exponentiating the learned energy. This is presented as the first time such a parametrization has been obtained for a non-trivial CL case without constructing a histogram via further sampling.
Quantitative Validation: The authors compare moments ( $\mu_n$ $μ_{n}$ ) and cumulants ( $\kappa_n$ $κ_{n}$ ) derived from the learned models against exact results and standard CL sampling.
- The Energy-Based Model (EBM) results show good agreement with the exact results and CL benchmarks, particularly for lower-order moments.
- The Score-Based Model (SBM) also performs well but shows slightly larger deviations in higher-order moments compared to EBM.
- The paper also includes a Markov Chain Monte Carlo (MCMC) computation using the energy learned by the EBM, which serves as an additional consistency check.

Significance and Claims
The paper claims that diffusion models offer a "fresh perspective" on the complex Langevin process. By learning the distribution directly from data, these models bypass the need for a priori knowledge of the distribution or the inability to solve the associated Fokker-Planck equation.

The primary significance highlighted is the ability to characterize the distribution on the complexified manifold. Specifically, the Energy-Based Model approach allows for the explicit construction of the probability distribution $p(x, y)$ in a non-trivial case. This provides a tool to:

Generate additional configurations representative of the CL process.
Deepen the understanding of the real, semi-positive distribution sampled during CL, particularly regarding its behavior at boundaries and infinity.

The authors remain modest regarding immediate applications to high-dimensional field theories (like full QCD), noting that the current study focuses on conceptual questions using a simple, well-understood model. The work establishes a proof-of-concept that generative AI can be integrated with stochastic quantization to analyze and potentially validate complex Langevin dynamics.

Technical Summary: Combining Complex Langevin Dynamics with Score-Based and Energy-Based Diffusion Models

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