⚛️ phenomenology

Reliable Hybrid Neural Surrogates for Multidimensional Grids: Application to Double Parton Distributions

This paper presents a hybrid neural-network method implemented in C++ that significantly reduces the storage, memory, and initialization costs of multidimensional double parton distribution grids by compressing most data into a neural model while storing only a sparse set of high-accuracy residuals, achieving a 10.6-fold reduction in disk footprint with minimal impact on evaluation speed.

Original authors: R. Kord Valeshabadi, S. Rezaie

Published 2026-09-15

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: R. Kord Valeshabadi, S. Rezaie

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

Inside the protons that race around the Large Hadron Collider, the universe is a chaotic soup of tiny particles called quarks and gluons. To understand what happens when these protons smash together, physicists must know exactly how these internal particles are arranged and how they move. They rely on mathematical maps known as parton distribution functions, which tell them the probability of finding a particle with a specific speed and energy. For decades, these maps have been stored as massive grids of numbers, much like a detailed spreadsheet where every cell holds a specific value. This method works well for simple collisions involving one particle from each proton. However, nature is often more complex. Sometimes, two distinct pairs of particles interact simultaneously within a single crash, a phenomenon known as double parton scattering. To predict these rare events, scientists need a much more complicated map that tracks two particles at once, along with two different energy scales for each. This creates a four-dimensional data structure that is exponentially larger and harder to manage than the standard maps, quickly overwhelming computer memory and storage systems.

A team of researchers at the Institute for Research in Fundamental Sciences in Tehran has developed a clever new way to shrink these massive data maps without losing the precision needed for high-energy physics. They created a hybrid system that combines a compact, intelligent computer model with a small, targeted list of exceptions. Instead of storing every single number in the massive four-dimensional grid, they trained a neural network—a type of artificial intelligence designed to recognize patterns—to learn the general shape and behavior of the data. The network acts as a highly efficient guesser, capable of reproducing the vast majority of the grid's values with high accuracy. However, because no model is perfect, the researchers identified the specific points where the network's guess was not quite good enough. For these difficult spots, they kept the original, exact numbers in a tiny, sparse list. When a physicist needs to look up a value, the system first checks if the point is one of the difficult exceptions; if not, it uses the fast, compact neural network to generate the answer.

The results of this approach are striking. By using this hybrid method, the researchers reduced the size of the data file needed to store the information by a factor of 10.6 compared to the standard compressed grid. More importantly, the amount of computer memory required to run the calculations dropped by a factor of 31.6, shrinking from over four gigabytes down to just 138 megabytes. This massive reduction in memory usage means that complex simulations can run on standard computers rather than requiring specialized, high-end hardware. The system also loads nearly 11 times faster, cutting the startup time from over two and a half seconds to less than a quarter of a second. While the speed of the actual calculations slowed slightly to about 85 percent of the original rate, the trade-off is overwhelmingly positive, offering a practical solution to a problem that had previously made these complex calculations prohibitively expensive and slow.

The key to this success lies in how the neural network was taught. Rather than trying to memorize every single number from scratch, the system was given a head start by using a simple, known approximation as a baseline. The network was then tasked only with learning the subtle differences and correlations that make the real data unique. This allowed the model to remain small and fast while still capturing the intricate details of the particle interactions. The researchers tested this system rigorously, scanning millions of data points to ensure that the neural network's predictions were accurate enough for scientific use. They found that only 0.694 percent of the data points needed to be stored in the exception list, confirming that the network could handle the bulk of the work on its own. The final software package is written in a standard programming language used by physicists, meaning it can be used immediately without needing complex external tools or programming environments. This work provides a new, efficient tool for exploring the most complex collisions in the universe, turning a data-heavy bottleneck into a streamlined process that opens the door to more detailed studies of how matter behaves at its most fundamental level.

Technical Summary: Reliable Hybrid Neural Surrogates for Multidimensional Grids

Problem Statement
The storage and evaluation of Double Parton Distributions (DPDs) present a significant computational challenge compared to ordinary Parton Distribution Functions (PDFs). While a collinear PDF depends on one longitudinal momentum fraction ( $x$ ) and one factorization scale ( $\mu^2$ ), an unequal-scale DPD depends on two momentum fractions ( $x_1, x_2$ ) and two independent factorization scales ( $\mu_1^2, \mu_2^2$ ) for every parton-flavor combination. Direct tabulation of this four-dimensional dependence results in massive data requirements. For the specific GS09-based reference grid studied, the ASCII representation occupies approximately 2.6 GB, and the compressed binary version still requires roughly 740 MB on disk. More critically, runtime memory usage for the dense grid is approximately 4.27 GiB, as the full nodal payload must be decompressed and held in memory for interpolation. This high memory footprint and slow initialization time (approx. 2.53 s) create a bottleneck for phenomenological applications, particularly in Monte Carlo event generators requiring repeated evaluations.

Methodology
The authors propose a hybrid neural-network surrogate to compress the DPD grid while maintaining numerical accuracy and fast evaluation. The approach does not replace the underlying reference calculation or the interpolation algorithm but rather compresses the nodal data stored in the grid.

Hybrid Architecture: The method employs a feed-forward neural network to approximate the bulk of the DPD grid. For grid nodes where the neural prediction fails to meet a prescribed accuracy tolerance, the original reference values are stored in a sparse correction table. During evaluation, the network reconstructs the nodal values, and sparse entries replace the neural predictions where necessary before standard interpolation is applied.
Input Representation: The network accepts 12 kinematic features derived from the four variables ( $x_1, x_2, \mu_1^2, \mu_2^2$ ). These include logarithmic mappings of the variables, features describing the longitudinal support boundary ( $x_1+x_2 \le 1$ ), asymmetry between parton legs, and heavy-flavor thresholds (charm and bottom).
Target Representation: Instead of predicting the absolute DPD values, the network learns the relative correction to a physically motivated baseline. The baseline is constructed from the product of single PDFs ( $f_{a1}(x_1, \mu_1^2)f_{a2}(x_2, \mu_2^2)$ ) multiplied by a phase-space suppression factor $\rho_0(x_1, x_2)$ . The target is the relative deviation, transformed via an inverse hyperbolic sine ( $\text{asinh}$ ) function to handle the wide dynamic range of corrections.
Network Architecture: A compact multilayer perceptron (MLP) with three hidden layers of width 256 (12 $\to$ 256 $\to$ 256 $\to$ 256 $\to$ 121) is used. The output layer produces 121 values corresponding to all ordered parton-flavor pairs. The compact size is chosen to ensure fast inference, as a single off-grid evaluation may require up to 16 neighboring node predictions.
Training Strategy: The network is trained on the reference grid nodes using PyTorch. The loss function is a weighted mean-squared error, with higher weights assigned to difficult regions (near the longitudinal endpoint, heavy-flavor thresholds) and specific flavor channels (e.g., gluon-gluon). Data augmentation via exchange symmetry is applied.
Implementation: The trained model is exported to a native C++ implementation within the PDFxTMDLib framework. Runtime evaluation requires neither Python nor PyTorch, utilizing oneDNN for efficient inference.

Key Results
Using the GS09-based dense grid ( $N_x=56$ , $N_\mu=41$ , 121 flavor channels) as a benchmark, the hybrid method achieves the following:

Compression Efficiency: Only 0.694% of the active flavor values (3,845,458 out of 554,383,007) require storage in the sparse correction table.
Storage Reduction: The hybrid file size is 66.65 MiB, representing a reduction by a factor of 10.6 compared to the 707.12 MiB compressed dense grid.
Memory Reduction: The median peak resident memory drops from 4.27 GiB (dense) to 138.20 MiB (hybrid), a reduction factor of 31.6.
Initialization Speed: Initialization time decreases from approximately 2.53 s to 0.238 s, a factor of 10.6 improvement, primarily due to avoiding the decompression of the full dense payload.
Evaluation Throughput: The hybrid evaluator achieves a throughput of approximately $2.6 \times 10^4$ kinematic points per second, which is about 85% of the dense-grid rate ( $3.1 \times 10^4$ points/s).
Accuracy: The method satisfies a mixed absolute-relative tolerance criterion. The sparse table ensures that nodes with large deviations are replaced by exact reference values, while the network handles the remaining bulk with high fidelity.

Significance and Claims
The paper claims that this hybrid scheme offers a practical solution to the storage and memory bottlenecks associated with multidimensional DPD grids. By encoding the scale dependence directly into a compact surrogate, the method eliminates the need for on-the-fly double-DGLAP evolution during inference, which is computationally expensive. The authors emphasize that the method maintains an evaluation speed comparable to dense-grid interpolation while drastically reducing disk footprint, memory usage, and startup costs.

The work demonstrates that a neural network can effectively capture the non-factorized correlation structures of DPDs when guided by a physics-informed baseline. The implementation is fully integrated into the PDFxTMDLib C++ library, making it immediately usable for phenomenological studies without runtime dependencies on machine learning frameworks. While the current study is restricted to $y$ -independent DPDs, the authors note that the strategy is theoretically extensible to distributions with explicit transverse-separation dependence.

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