⚛️ phenomenology

Know What You Don't Flow

This paper demonstrates how heteroscedastic and Bayesian normalizing flows can learn and propagate calibrated systematic and statistical uncertainties for generative neural networks in LHC physics, using both toy models with explicit likelihoods and classifier-reweighted approaches for top pair events.

Original authors: Anja Butter, Sascha Diefenbacher, Tilman Plehn, Lorenz Vogel

Published 2026-08-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Anja Butter, Sascha Diefenbacher, Tilman Plehn, Lorenz Vogel

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

In the heart of Europe, the Large Hadron Collider smashes protons together at speeds close to light, creating a shower of new particles that fly out in every direction. Physicists have spent decades mapping these collisions, turning the chaotic spray of debris into precise measurements of how the universe works. To make sense of this data, they rely on two things: the actual measurements from the detectors and the theoretical predictions from complex computer simulations. For years, these two have matched up beautifully, but as the collider becomes more powerful and collects more data, the margin for error shrinks. The measurements are becoming so precise that the theoretical predictions must be equally sharp. If the computer models are slightly off, or if the scientists cannot tell exactly how much they might be off, the entire search for new physics could be misled. The challenge is no longer just about getting the right answer, but about knowing exactly how confident we can be in that answer.

This is where a new approach from a team at the University of Heidelberg comes in. They are working on a way to teach artificial intelligence to not only predict the outcome of these particle collisions but also to admit when it is unsure. In the world of machine learning, a common tool called a generative network acts like a master forger. It learns the patterns of real particle collisions and then creates new, fake events that look indistinguishable from the real thing. This is incredibly useful because simulating real collisions is slow and expensive, while the AI can generate millions of fake events in seconds. However, until now, these forgers have been silent about their own mistakes. They would produce a result without telling the physicist whether the answer was rock-solid or a wild guess. The Heidelberg team has developed a method to fix this, giving these networks a voice to express their uncertainty.

The researchers focused on two distinct types of uncertainty that can plague these simulations. The first is statistical uncertainty, which happens simply because the training data is limited. If a network has only seen a few examples of a rare event, it should be less confident when predicting that event again. The second type is systematic uncertainty, which is more subtle. This arises from flaws in the network itself, the way it was trained, or the noise in the data it learned from. Even with infinite data, a network might still have a built-in bias or a blind spot. The team wanted to know if they could teach a single AI to track both of these sources of error simultaneously and report them accurately.

To test their idea, they started with a simple, made-up model where they knew the exact truth. They injected artificial noise into the data to simulate real-world imperfections and then watched how the AI reacted. They found that they could train the network to learn the size of the noise it was seeing. When they added more noise, the network's reported uncertainty grew larger, matching the reality of the situation. They also tested how the network handled the speed at which it learned. If the learning process was too fast and chaotic, the network developed a specific kind of error, and the team's method successfully identified this as a systematic flaw. Crucially, they showed that the network could distinguish between errors caused by a lack of data and errors caused by the training process itself, assigning the correct type of uncertainty to each.

The real test came when they applied this method to a complex, realistic scenario: the production of top quark pairs, one of the heaviest and most studied particles in the collider. In this case, they did not know the exact mathematical truth beforehand, which is the usual situation for physicists. To solve this, they used a clever trick involving a second AI, a classifier, to act as a guide. This classifier compared the rough, initial guesses of the generative network against the real data and helped correct the network's path. By using this corrected path as a target, the generative network could learn to produce accurate events and, at the same time, learn to quantify its own systematic errors. The results were striking. The network successfully mapped out where it was confident and where it was not, propagating these uncertainties through every direction of the particle collision.

The team also explored how these networks handle "nuisance parameters," which are variables in the simulation that are not perfectly known, such as the exact mass of the top quark. They trained the network to vary this mass and observed how the uncertainty in the final results changed. They discovered a clear, predictable relationship: as the uncertainty in the input mass grew, the uncertainty in the output events grew in a specific, measurable way. This means that in the future, physicists could use these networks to instantly see how a small change in a fundamental constant would ripple through their entire analysis.

The work demonstrates that it is possible to build generative networks that are not just fast forgers, but honest partners in scientific discovery. By teaching these systems to calibrate their own confidence, the researchers have provided a tool that can handle the extreme precision required by the next generation of particle physics experiments. The networks now do more than just generate data; they tell the scientists exactly how much they can trust that data, turning a black box of artificial intelligence into a transparent instrument for exploring the fundamental laws of nature.

Technical Summary: "Know What You Don't Flow"

Problem Statement
Generative machine learning models, such as normalizing flows (NFs), have become transformative tools in LHC physics for extracting phase-space densities and generating events faster than classical simulation chains. However, a critical weakness remains: the lack of robust, calibrated uncertainty quantification for these implicitly learned densities. While methods exist to learn statistical uncertainties (arising from limited training data) or systematic uncertainties (arising from model expressivity or noisy data) in supervised regression, these have not been comprehensively combined or calibrated for generative networks. Specifically, there is a gap in understanding how generative networks learn systematic uncertainties from training noise or optimization artifacts, and how to calibrate these uncertainties when the true underlying density is unknown.

Methodology
The authors propose a framework to distinguish and calibrate two distinct sources of uncertainty in generative networks:

Systematic Uncertainties ( $\sigma_{\text{syst}}$ ): Defined as uncertainties that do not vanish with infinite training data of the same kind (e.g., noise in the target density, network expressivity, or learning rate fluctuations).
Statistical Uncertainties ( $\sigma_{\text{stat}}$ ): Defined as uncertainties that vanish with more training data (e.g., lack of statistics in a specific phase-space region).

To address these, the paper introduces and compares two architectures:

Heteroscedastic Normalizing Flow (HNF): A flow network trained to predict the log-density $\log p_\theta(x)$ alongside an MLP that predicts a local uncertainty $\sigma_{\text{syst}}(x)$ . This is trained using a heteroscedastic Gaussian loss function. Crucially, for cases where the true density $p_{\text{truth}}(x)$ is unknown, the authors employ a classifier-reweighted target. An approximate density $p_{\text{expl}}(x)$ is reweighted by a classifier $C(x)$ distinguishing it from the true data, creating an explicit target $p_{\text{expl}}(x) \times \frac{C(x)}{1-C(x)}$ for the HNF.
Bayesian Normalizing Flow (BNF): A flow where network parameters are treated as probability distributions (posteriors) rather than fixed values. By sampling from the posterior, the ensemble of generated densities yields a statistical uncertainty $\sigma_{\text{stat}}(x)$ .

The methodology is validated first on a 2D toy model with an explicit known density, and subsequently on a realistic 16-dimensional top pair ( $t\bar{t}$ ) production process with fully leptonic decays.

Key Contributions and Results

Calibration of Systematic Uncertainties: Using the toy model, the authors demonstrate that the HNF correctly learns injected Gaussian noise as a systematic uncertainty. The learned uncertainty scales linearly with the injected noise level, and the resulting "pulls" (the difference between truth and prediction normalized by the learned uncertainty) follow a standard Gaussian distribution, indicating perfect calibration.
Learning Rate as a Systematic Source: The study reveals that the learning rate $\eta$ in stochastic optimization induces a systematic uncertainty. The minimum learned uncertainty scales as $\min \sigma_{\text{syst}} \propto \sqrt{\eta}$ , confirming that optimization noise is captured as a systematic effect by the HNF.
Distinction of Uncertainty Types: The paper clearly distinguishes the roles of HNF and BNF. The HNF learns stochastic or deterministic systematics from the data, network, or training, which remain constant regardless of training dataset size ( $N_{\text{train}}$ ). Conversely, the BNF learns the statistical limitations arising from the size of the training dataset, scaling as $1/\sqrt{N_{\text{train}}}$ .
Classifier-Reweighted Training: For the realistic $t\bar{t}$ application where the explicit likelihood is not available for direct training, the classifier-reweighting strategy successfully reconstructs the target density. The HNF trained on this reweighted target learns local systematic uncertainties that are well-calibrated, particularly when using a "statistics-motivated" approximate density (trained on a subset of data) rather than a "theory-motivated" factorized approximation.
Nuisance Parameter Propagation: The authors demonstrate that a conditional HNF can learn systematic uncertainties encoded in nuisance parameters (specifically the top quark mass $m_t$ ). By training the network conditional on $m_t$ , the model can propagate the prior uncertainty of the top mass to all phase-space observables, establishing a scaling law between the prior width and the learned systematic uncertainty.

Significance and Claims
The paper claims to provide the first comprehensive approach to learning and calibrating both statistical and systematic uncertainties for generative networks in particle physics. The authors emphasize that while previous studies have shown generative networks can learn statistical uncertainties, no prior particle physics study has demonstrated the correct calibration of learned statistical uncertainties or detailed the handling of systematic uncertainties on the learned density.

The significance lies in the ability to move from purely generative models to uncertainty-aware generative models. By combining HNF (for systematics) and BNF (for statistics), the framework offers a complete uncertainty budget. The authors note that this is particularly vital for LHC applications such as parameter inference, where calibrated uncertainties are a prerequisite. The work establishes that generative networks can be trained to "know what they don't flow," providing reliable error bars on the phase-space density even in the absence of an explicit analytical likelihood.

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