PMT Waveform Simulation and Reconstruction with Conditional Diffusion Network

TL;DR

This work tackles PMT waveform reconstruction under multi-photon, high-overlap conditions by introducing a bidirectional conditional diffusion framework that jointly simulates waveforms from PE sequences and reconstructs PE sequences from waveforms in a weakly supervised setting. The approach comprises two diffusion modules, DFA and DFB, operating in a loop that iteratively refines waveform realism and PE inference, with ResNet50 provided as a performance benchmark. Supervised results establish strong baselines for both waveform simulation and nPE/timing reconstruction, while the weakly supervised regime demonstrates practical viability when ground-truth labels are unavailable, achieving about 99% of the supervised nPE resolution and around 0.5 ns timing precision under favorable data distributions. The method offers a data-driven alternative for detector-level waveform modeling, reducing reliance on exact PE labels and enabling broader applicability to vertex and energy reconstruction in neutrino experiments.

Abstract

Photomultiplier tubes (PMTs) are widely employed in particle and nuclear physics experiments. The accuracy of PMT waveform reconstruction directly impacts the detector's spatial and energy resolution. A key challenge arises when multiple photons arrive within a few nanoseconds, making it difficult to resolve individual photoelectrons (PEs). Although supervised deep learning methods have surpassed traditional methods in performance, their practical applicability is limited by the lack of ground-truth PE labels in real data. To address this issue, we propose an innovative weakly supervised waveform simulation and reconstruction approach based on a bidirectional conditional diffusion network framework. The method is fully data-driven and requires only raw waveforms and coarse estimates of PE information as input. It first employs a PE-conditioned diffusion model to simulate realistic waveforms from PE sequences, thereby learning the features of overlapping waveforms. Subsequently, these simulated waveforms are used to train a waveform-conditioned diffusion model to reconstruct the PE sequences from waveforms, reinforcing the learning of features of overlapping waveforms. Through iterative refinement between the two conditional diffusion processes, the model progressively improves reconstruction accuracy. Experimental results demonstrate that the proposed method achieves 99% of the normalized PE-number resolution averaged over 1-5 p.e. and 80% of the timing resolution attained by fully supervised learning.

Figures (14)

• Figure 1: Left: The averaged sPE waveform comprises three components: the main peak, overshoot, and reflection peaks. Right: Probability density function (PDF) of the LS fluorescence time, obtained by convolving a four-exponential scintillation decay model with the instrument response function.
• Figure 2: Schematic diagram of the proposed bidirectional conditional diffusion network framework. The framework network consists of two conditional DDPMs: Diffusion-A (left) performs waveform simulation by conditioning the latent vector $\mathbf{z}$ on the PE sequence $\mathbf{y}$, while Diffusion-B (right) reconstructs the PE sequence by conditioning $\mathbf{z}$ on the observed waveform $\mathbf{x}$. Synthetic waveforms $\mathbf{x}'$ generated by DFA are employed to train DFB, and DFB's output $\mathbf{y}'$ is subsequently fed back to retrain DFA, progressively enhancing waveform simulation quality and reconstruction accuracy. The top arrow indicates the iterative training loop between the two diffusion models, enabling self-optimization and joint convergence.
• Figure 3: Comparison of training and validation loss curves for different models: DFA(top), DFB(middle), and ResNet50(bottom). All curves highlight key performance points marked by asterisks.
• Figure 4: Comparison of averaged waveforms and charge spectra between EMC and DFA models. The relative deviation deviations are defined relative to EMC. (a) sPE samples: All four models exhibit waveform deviations below 2% relative to EMC, with both the mean and resolution deviations of the charge spectra under 2%, demonstrating accurate reproduction of characteristics of sPE charge. (b) UT-UPE samples: Synthetic waveforms show uniform temporal distribution across all models, with overall waveform shapes consistent with EMC, indicating successful learning of the strong correlation between PE times and peak positions. Comparisons of charge spectra reveals that LT-0.1PE-DFA's waveform deviation primarily stems from systematically lower charges in the high nPE region. (c) LT-UPE samples: Compared to EMC, UT-UPE-DFA, LT-UPE-DFA, and LT-1PE-DFA show waveform deviations below 6%, while LT-0.1PE-DFA exhibits a maximum deviation of approximately 10%. Relative to the UT-UPE case in (b), all models demonstrate reduced deviation in both waveform and charge spectrum for LT-UPE samples.
• Figure 5: The upper panel shows the charge linearity comparison between EMC and DFA models at different $\Delta T$: $\bar{Q}_n/\bar{Q}_1$ as a function of nPE $n$. Dashed lines show linear fits. The lower panel shows the charge resolution comparison between EMC and DFA models at different $\Delta T$: $\sigma_{Q_n}/\sigma_{Q_1}$ as a function of $n$. Dashed lines indicate power-law fits.