PMT Waveform Simulation and Reconstruction with Conditional Diffusion Network

This paper proposes a fully data-driven, weakly supervised bidirectional conditional diffusion network that iteratively simulates and reconstructs photomultiplier tube waveforms to accurately resolve overlapping photoelectrons without requiring ground-truth labels.

The Gist

Imagine you are trying to listen to a crowded party where everyone is shouting at once. Your goal is to figure out exactly how many people are speaking and when each person started talking. This is essentially the challenge faced by scientists studying subatomic particles, specifically using devices called Photomultiplier Tubes (PMTs).

These tubes detect tiny flashes of light (photons) created by particles. When a particle hits the detector, it might create a single flash, or it might create a rapid-fire burst of many flashes arriving within a few billionths of a second. The detector records this as a "waveform"—a squiggly line on a graph.

The problem? When the flashes happen too close together, their waves overlap and mash into a single, messy blob. It's like trying to count individual raindrops hitting a tin roof during a heavy downpour; you just hear one continuous roar.

The Old Way vs. The New Way

The Traditional Approach:
Scientists used to try to "untangle" these messy waves using math formulas (fitting and deconvolution). It's like trying to un-mix a smoothie back into strawberries and bananas. It works okay if the ingredients are separate, but if they are blended perfectly, the math gets confused and fails.

The "Supervised" AI Approach:
Recently, scientists tried teaching computers to do this by showing them millions of examples where they already knew the answer (e.g., "This messy wave came from exactly 3 flashes"). This worked great, but there's a catch: in real life, we never actually know the exact answer. We can't see the individual flashes to count them. So, we can't teach the computer with "real" data, only with fake data from simulations.

The New Solution: The "Two-Way Mirror" (Bidirectional Diffusion Network)
This paper introduces a clever new method called a Bidirectional Conditional Diffusion Network. Think of it as a two-way learning loop between two AI "artists":

1. Artist A (The Simulator): This AI is given a list of numbers (e.g., "3 flashes at these times") and asked to draw a waveform. It learns to create realistic-looking messy waves from clean instructions.
2. Artist B (The Detective): This AI is given a messy waveform and asked to guess the list of numbers (how many flashes and when).

The Magic Loop:
Here is the genius part. Usually, Artist B needs perfect "answer keys" to learn. But in the real world, we don't have them. So, the scientists created a weakly supervised loop:

• Artist A draws a wave based on a rough guess of the flashes.
• Artist B looks at that drawing and tries to guess the flash count back.
• If Artist B's guess is better than the original rough guess, that new, better guess is fed back to Artist A.
• Artist A then learns from this improved guess to draw even better waves.

They keep passing the baton back and forth, refining each other's skills until they both get incredibly good at the job, all without needing a human to tell them the "true" answer for every single wave.

The Analogy: The "Blind Painter and the Sculptor"

Imagine a Blind Painter (Artist A) who can only paint if you tell them, "Paint 3 dots here."
Imagine a Sculptor (Artist B) who can only carve a statue if you hand them a painting and say, "Tell me how many dots were in this."

• The Problem: The Sculptor needs to know the truth to learn, but no one knows the truth for real statues.
• The Solution: The Sculptor starts with a bad guess. They look at the painting, guess "Maybe 3 dots," and tell the Painter. The Painter paints a new picture based on "3 dots." The Sculptor looks at the new picture, realizes, "Ah, that looks like it should have been 3.5 dots," and updates their guess.
• The Result: They repeat this cycle. The Painter gets better at capturing the feel of overlapping dots, and the Sculptor gets better at counting them. Eventually, the Sculptor can look at a real, messy painting and count the dots with near-perfect accuracy, even though they never saw the "correct" answer key.

What Did They Find?

The researchers tested this system with different types of "messy" data:

1. The "Sparse" Crowd: When the flashes are far apart (like people talking one by one), the system works almost perfectly.
2. The "Dense" Crowd: When the flashes are bunched up tight (like a shouting crowd), it gets harder.
   • They found that if they trained the system on data where the flashes were moderately overlapping (not too sparse, not too chaotic), the system learned the best.
   • If they trained it on data that was too chaotic, the system got confused because the initial guesses were too wrong.

The Final Score:

• Counting Accuracy: The new method achieved 99% of the accuracy of the "perfect" supervised method (the one that had all the answer keys).
• Timing Accuracy: It achieved 80% of the timing accuracy of the perfect method.

Why This Matters

This is a breakthrough because it allows scientists to analyze real-world particle data with high precision without needing to know the "true" answer beforehand. It's like teaching a student to solve a complex puzzle by having them practice on puzzles they can solve, then gradually moving to harder ones, rather than forcing them to solve a puzzle they can't see the solution to.

In short, they built a self-improving AI loop that can untangle the "noise" of particle physics experiments, helping us understand the universe better, all while working with the messy, incomplete data we actually have.

Technical Summary

Technical Summary: PMT Waveform Simulation and Reconstruction with Conditional Diffusion Network

Problem Statement
In particle and nuclear physics experiments, such as the Jiangmen Underground Neutrino Observatory (JUNO), Photomultiplier Tubes (PMTs) are critical for detecting faint Cherenkov or scintillation light. The accuracy of reconstructing PMT waveforms directly dictates the detector's spatial and energy resolution. A primary challenge arises when multiple photons arrive within a few nanoseconds, causing photoelectrons (PEs) to overlap in the waveform. While traditional methods (waveform fitting and deconvolution) and supervised deep learning approaches have improved performance, they face significant limitations. Traditional methods rely heavily on accurate prior knowledge of detector response and degrade with severe overlap. Supervised deep learning, though powerful, requires ground-truth PE labels which are generally inaccessible in real experimental data, limiting its practical applicability.

Methodology
The authors propose a Bidirectional Conditional Diffusion Network (BCDDPM) framework designed for synergistic waveform simulation and reconstruction under a weakly supervised learning paradigm. This approach is fully data-driven, requiring only raw waveforms and coarse initial estimates of PE information, rather than precise ground-truth labels.

The framework consists of two structurally identical conditional Denoising Diffusion Probabilistic Models (DDPMs) based on a modified 1D U-Net architecture:

1. Diffusion-A (DFA): A PE-conditioned model that simulates realistic waveforms ($x$) given a PE sequence ($y$). It learns the features of overlapping waveforms by mapping PE sequences to voltage waveforms.
2. Diffusion-B (DFB): A waveform-conditioned model that reconstructs PE sequences ($y$) from observed or simulated waveforms ($x$).

Key Contributions

• Bidirectional Conditional Framework: The paper introduces a novel architecture where the two diffusion models interact iteratively. In the weakly supervised setting, DFB reconstructs a refined PE sequence ($y'$) from raw waveforms. This refined sequence is then used to retrain DFA, which in turn generates higher-quality synthetic waveforms to train DFB. This iterative refinement loop allows the system to progressively improve both simulation fidelity and reconstruction accuracy without ground-truth labels.
• Weakly Supervised Learning Strategy: The method addresses the lack of ground-truth data by utilizing an iterative training process. It initializes with coarse PE estimates derived from peak-finding algorithms on filtered waveforms and refines these estimates through the bidirectional interaction of the diffusion models.
• Network Architecture Optimization: The authors adapt the standard U-Net for 1D waveform data, incorporating multi-source conditioning (noise level, time step, and physical conditions like PE sequences). They replace 2D convolutions with 1D, utilize Group Normalization for stability, and employ Swish activation functions.
• Comprehensive Benchmarking: The study evaluates the models against fully supervised learning benchmarks (using Monte Carlo truth) and traditional charge-based estimation across various PE multiplicity and time distribution scenarios (UT-UPE, LT-xPE, LT-UPE).

Results
Experimental results were evaluated using Electronics Monte Carlo (EMC) datasets simulating JUNO-like conditions:

• Waveform Simulation: The DFA models successfully learned the statistical properties of single-PE (sPE) and overlapping waveforms. Models trained on datasets with specific PE distributions (e.g., LT-UPE) demonstrated the ability to reproduce charge linearity and resolution characteristics close to the ideal EMC truth, particularly for sparse to moderately overlapping waveforms.
• Waveform Reconstruction:
  • Under supervised learning, the diffusion models achieved high accuracy, with nPE reconstruction resolution reaching approximately 99% of the ideal performance for 1–5 p.e. events and timing resolution within 80% of the supervised baseline.
  • Under weakly supervised learning, the iterative refinement proved effective. The LT-0.1PE-DFA-DFB model (trained on sparse PE data) achieved an average normalized nPE resolution of 0.18 p.e. (99% of the supervised value) for 1–5 p.e. and a timing resolution of 0.5 ns (80% of the supervised value).
  • The study found that the accuracy of the initial PE sequence labels is critical. Training on data with severe waveform overlap (e.g., high mean nPE) introduced biases in the initial labels, leading to degraded reconstruction performance in the weakly supervised regime. Conversely, training on data with mild overlap (e.g., ~0.1 p.e. mean) yielded optimal results by balancing the need for sPE characterization and overlap features without introducing large initial errors.

Significance and Claims
The paper claims that the proposed BCDDPM framework provides an effective and practical approach for waveform simulation and reconstruction in particle physics experiments where ground-truth labels are unavailable. By leveraging a bidirectional conditional diffusion network, the method significantly reduces dependence on precise labels while maintaining reconstruction accuracy comparable to fully supervised methods.

The authors emphasize that the success of this weakly supervised approach is contingent upon the selection of training data; specifically, using waveforms with an average intensity of ~0.1 p.e. allows the model to capture realistic overlap features without the severe errors associated with highly overlapping initial estimates. This work offers a pathway to enhance detector energy and vertex resolution in future neutrino experiments without the prohibitive cost of obtaining ground-truth PE labels for real data.