⚛️ phenomenology

Quantitative Understanding of PDF Fits and their Uncertainties

This paper establishes a theoretical framework based on the Neural Tangent Kernel to analytically describe the training dynamics and uncertainty propagation of neural network-based Parton Distribution Function fits, thereby providing a powerful diagnostic tool to assess the robustness of current methodologies and bridge particle physics with machine learning theory.

Original authors: Amedeo Chiefa, Luigi Del Debbio, Richard Kenway

Published 2026-07-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Amedeo Chiefa, Luigi Del Debbio, Richard Kenway

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

The Big Picture: Tuning a Radio to Find a Signal

Imagine you are trying to tune a radio to find a specific, clear song (the true structure of a proton) amidst a lot of static and noise (experimental data from particle colliders). The "song" is a complex curve called a Parton Distribution Function (PDF).

The problem is that the radio station (the data) only gives you a few scattered notes. You have to guess the rest of the melody. If you guess wrong, the song sounds terrible. If you guess right, you get a perfect hit.

The authors of this paper are studying how a very smart computer (a Neural Network) learns to guess this melody. They want to understand exactly how the computer changes its mind as it listens to the data, and how to measure how confident it should be in its final guess.

The Main Characters

The Neural Network (The Student): This is a computer program designed to be a "blank slate." It doesn't know the song yet. It starts with random guesses, like a student who has never studied music.
The Data (The Teacher): This is the experimental data from the Large Hadron Collider. It tells the student, "You were close here, but way off there."
The NTK (The Map): This is the paper's star invention. Think of the Neural Tangent Kernel (NTK) as a dynamic map that shows the student which directions are easy to learn and which are impossible. It tells the student, "You can easily move your guess in this direction, but don't bother trying to move in that direction; the data doesn't support it."

The Story of the Training Process

The paper breaks the learning process into two distinct phases, like a student taking a test.

Phase 1: The "Rich" Learning Phase (The Struggle)

At the very beginning, the student is confused. The "Map" (NTK) is changing rapidly.

What happens: The student tries out wild new ideas. The computer's internal structure is shifting dramatically. It discovers new "features" of the song it didn't know existed.
The Analogy: Imagine a sculptor chipping away at a massive block of stone. At first, the chisel moves everywhere, and the shape of the stone changes wildly. The student is learning what it is allowed to learn.
The Finding: The authors found that during this phase, the computer actually does change its internal "Map." It grows new capabilities to understand more complex parts of the data.

Phase 2: The "Lazy" Training Phase (The Smooth Ride)

After a while, the student settles down. The "Map" (NTK) stops changing and becomes fixed.

What happens: The computer is no longer reinventing the wheel. It just slides smoothly down a hill toward the best answer. The path is now predictable.
The Analogy: The sculptor has found the general shape of the statue. Now, they are just polishing the surface. The movements are small, smooth, and follow a straight line.
The Finding: This is the most important part of the paper. Once the computer enters this "Lazy" phase, the authors can write a mathematical formula that predicts exactly what the computer will output at any moment. They don't need to run the simulation; they can just do the math.

The "Lazy" Formula: A Recipe for the Future

Because the computer becomes "lazy" (predictable) in the second phase, the authors derived a simple equation (Equation 60 in the paper) that looks like this:

Final Answer = (What we started with) + (What the data taught us)

What we started with: This is the "prior." It's the random guess the computer made before seeing any data.
What the data taught us: This is the new information gained from the experiment.

This formula is powerful because it separates the two ingredients. It shows exactly how much the final result depends on the computer's initial random guess versus the actual experimental data.

Why This Matters (Without the Jargon)

Understanding Uncertainty: In science, knowing how sure you are is just as important as the answer itself. This paper shows how the "uncertainty" (the fuzziness of the guess) travels from the raw data into the final result. It's like tracking how a ripple in a pond spreads out.
Checking the Tools: The authors built a "diagnostic tool." Instead of just trusting the computer to do its job, they can now look at the "Map" (NTK) to see if the computer is actually learning the physics or just memorizing the noise.
It's Not Just for Protons: While they used this for particle physics, the math works for any problem where you try to guess a curve based on scattered points. It's a universal way to understand how machines learn.

The Bottom Line

The paper says: "We figured out that after a short period of chaotic learning, neural networks settle into a predictable, 'lazy' mode. In this mode, we can write down a perfect mathematical recipe that tells us exactly how the network combines its starting guess with the new data to produce the final answer. This helps us trust the results and understand exactly where the uncertainty comes from."

They didn't just say "the computer works"; they explained the engine under the hood and showed us the blueprint.

Technical Summary: Quantitative Understanding of PDF Fits and their Uncertainties

Problem Statement

Parton Distribution Functions (PDFs) are essential for describing experimental data at hadron colliders and understanding nucleon structure. As the Large Hadron Collider (LHC) enters an era of high-precision measurements, the demand for robust PDF determinations with reliable uncertainty quantification has become critical. The extraction of PDFs is an ill-posed inverse problem: inferring a continuous function from a finite set of data points. This process inherently lacks a unique solution, relying on assumptions and prior knowledge to regularize the problem.

While state-of-the-art determinations often use parametric approaches with specific functional forms, the NNPDF collaboration pioneered the use of Machine Learning (ML), specifically Neural Networks (NNs), to parametrize PDFs in a flexible, non-parametric manner. However, a rigorous theoretical understanding of the training dynamics of these networks, particularly regarding how uncertainties propagate from data to the fitted function and how the network architecture influences the learning process, remains elusive. Existing literature often relies on empirical validation (e.g., closure tests) rather than analytical frameworks to diagnose the robustness of these fitting methodologies.

Methodology

This work develops a theoretical framework based on the Neural Tangent Kernel (NTK) to analyze the training dynamics of neural networks used in PDF fits. The authors adopt a simplified setting to ensure analytical tractability:

Data: A single PDF flavor combination ( $T_3$ ) using Deep Inelastic Scattering (DIS) data, where theoretical predictions depend linearly on the PDFs.
Optimization: Vanilla Gradient Descent (GD) with a continuous flow approximation.
Architecture: The standard NNPDF architecture (2-25-20-8 layers with tanh activation) initialized via Glorot normal distribution.
Approach: The training process is reformulated in functional space rather than parameter space. The evolution of the network output is governed by the NTK, which encodes the dependence on the network architecture and parameters.

The study utilizes "closure tests" with synthetic data (Level-0, Level-1, and Level-2) to track the evolution of the NTK spectrum and eigenvectors during training. The analysis distinguishes between two regimes: an initial "rich" training phase where the NTK evolves significantly, and a subsequent "lazy" training regime where the NTK stabilizes.

Key Contributions

1. Analytical Description of Training Dynamics

The paper derives an analytical solution for the network output $f_t$ during the lazy training regime (where the NTK $\Theta_t$ becomes approximately constant). The solution is expressed as:
$f_t = U(t)f_0 + V(t)Y$
where $f_0$ is the initial network output, $Y$ represents the training data, and $U(t)$ and $V(t)$ are time-dependent matrices derived explicitly from the NTK and the data covariance. This expression decomposes the final result into a contribution from the initial condition (prior information) and a contribution from the data (experimental measurements), making the role of priors and data explicit.

2. Spectral Analysis of the NTK

The authors provide a detailed spectral analysis of the NTK:

Initialization: The NTK spectrum is highly hierarchical, with only a few large eigenvalues significantly different from zero. This implies that initially, the network explores a small subspace of the vast functional space.
Training Evolution: During the "rich" phase, subleading eigenvalues grow by one or two orders of magnitude. This indicates that the network dynamically discovers new features and expands the subspace of learnable functions.
Alignment: The study demonstrates an alignment between the eigenvectors of the NTK (model-dependent) and the eigenvectors of the matrix $M$ (data-dependent). As training progresses, the learnable directions of the network align with the directions that contribute most to the observables (the loss function).

3. Quantitative Uncertainty Propagation

By deriving the analytical evolution of the network output, the authors provide a quantitative description of how uncertainties propagate from the experimental data to the fitted PDFs. The framework allows for the computation of the covariance of the NN output during training, bridging the gap between parametric regression (NNPDF) and other methods for solving inverse problems.

4. Regularization Mechanism

The paper elucidates how the parametrization (the neural network) regularizes the ill-posed inverse problem. In the functional space, the data matrix $M$ is singular, with a kernel containing directions to which data are insensitive. The neural network parametrization restricts the solution to the image of the NTK. In the lazy regime, the component of the solution in the kernel of the NTK is "frozen" at its initial value, meaning the uncertainty in these unconstrained directions is dictated entirely by the prior distribution (initialization) rather than the data.

Results

Deviation from Standard Lazy Training: Contrary to the "lazy training" picture often discussed in ML literature (where the NTK is constant from the start), the NNPDF training exhibits a distinct "rich" phase. During this initial phase (approx. first 20,000 epochs), the NTK undergoes significant changes, and subdominant eigenvalues grow, allowing the network to learn new features.
Stability of the Lazy Regime: After the initial transient, the NTK stabilizes, and the training enters the lazy regime. In this regime, the analytical solution holds, and the loss function decreases smoothly.
Reconstruction Capability: The study shows that the subset of NTK eigenvectors orthogonal to the kernel can effectively reconstruct the input function, provided the modes discovered during the rich phase are included.
Architecture Independence: The hierarchical structure of the NTK spectrum and the qualitative behavior of the training dynamics are largely independent of the network width, although the variance of the NTK decreases with larger network sizes.

Significance and Claims

The authors position this work not as a replacement for standard PDF fitting, but as a powerful diagnostic tool to assess the robustness of current fitting methodologies.

Unified Framework: The methodology provides a unified framework to compare different PDF determinations (e.g., those based on specific functional forms vs. NNs) by analyzing their respective NTKs and training dynamics.
Transparency: The analytical approach clarifies the role of the NN architecture and the impact of experimental data in a transparent way, demystifying the "black box" nature of the training process.
Testbed for ML Theory: The paper claims that the analysis of PDF determinations serves as a testbed to apply theoretical ideas about the learning process developed in the ML community to high-energy physics.
Modest Scope: The authors explicitly state that their results are derived in a simplified setting (single flavor, linear dependence, vanilla GD) and are intended as an exploration of foundational theoretical issues. Extensions to the full complexity of modern global PDF fits (multiple flavors, non-linear dependencies, advanced optimizers) are left for future work.

In summary, the paper establishes that the training dynamics of NNPDF can be understood analytically through the lens of the NTK, revealing a dynamic interplay between the network's ability to learn new features and the data's ability to constrain the solution, ultimately providing a rigorous basis for understanding uncertainty quantification in ML-driven PDF fits.