Physics-informed continuous normalizing flows to learn the electric field within a time-projection chamber

TL;DR

We address the challenge of reconstructing interaction positions in noble-element TPCs where surface charging distorts the electric field, limiting vertex accuracy. We propose a physics-informed continuous normalizing flow that learns a curl-free, conservative electric-field mapping by representing the drift with a scalar potential via $\boldsymbol{f}'_{\boldsymbol{\phi}}(\boldsymbol{s},t)=-\nabla_{\boldsymbol{s}} g_{\boldsymbol{\phi}}(\boldsymbol{s},t)$ and training with a simulation-based negative log-likelihood that incorporates the electron survival probability $p_{surv}$ and observed hit patterns. The method achieves superior position reconstruction with only $6\times 10^5$ calibration events, about an order of magnitude data reduction relative to histogram-based FDC maps, and yields a differentiable scalar-potential map and learned electric-field lines. This enables monthly field monitoring, probabilistic position reconstruction with uncertainty propagation, and improved background discrimination for next-generation rare-event searches.

Abstract

Accurate position reconstruction in noble-element time-projection chambers (TPCs) is critical for rare-event searches in astroparticle physics, yet is systematically limited by electric field distortions arising from charge accumulation on detector surfaces. Conventional data-driven field corrections suffer from three fundamental limitations: discretization artifacts that break smoothness and differentiability, lack of guaranteed consistency with Maxwell's equations, and statistical requirements of $\mathcal{O}(10^7)$ calibration events. We introduce a physics-informed continuous normalizing flow architecture that learns the electric field transformation directly from calibration data while enforcing the constraint of field conservativity through the model structure itself. Applied to simulated $^{83\mathrm{m}}$Kr calibration data in an XLZD-like dual-phase xenon TPC, our method achieves superior reconstruction accuracy compared to histogram-based corrections when trained on identical datasets, demonstrating viable performance with only $6\times10^5$ events$\unicode{x2013}$an order of magnitude reduction in calibration requirements. This approach enables practical monthly field monitoring campaigns, propagation of position uncertainties through differentiable transformations, and enhanced background discrimination in next-generation rare-event searches.

Figures (9)

• Figure 1: Previous state-of-the-art: here we show the previous method for field distortion corrections. Our new method removes these unphysical artifacts in the depicted slices of the field distortion correction (FDC) map when correcting for field distortions. Slices in $z=-130$ cm and $\phi=0 ^{\circ}$ of the interpolated FDC map are shown following XENONnT's binning scheme of 95 $z$ bins, 100 $r^2$ bins, and 180 $\phi$ bins.
• Figure 2: Diagram of continuous normalizing flow model. We represent a point within the box (TPC) with its transverse position $\boldsymbol{s}$ and its time $t$ which corresponds to depth $z(t)=v_{d} t$ such that $v_d$ is the field drift velocity. The neural network takes point as its input and returns how that position transforms over time, as governed by the neural network which represents the negative transverse gradient of a function $g_{\phi}$ that corresponds to an approximation of a scaled scalar potential.
• Figure 3: Final radial position at the liquid-gas interface $z = 0$ cm as a function of initial radius and depth, showing the distortion caused by nonuniform electric fields.
• Figure 4: Electron survival probability map, accounting for radial diffusion and loss to the detector walls. Zero-probability regions define the charge-insensitive volume.
• Figure 5: The continuous normalizing flow achieves a comparable reconstruction error with approximately an order of magnitude less calibration data than a traditional field distortion correction (FDC) map. The flow shown in this figure was trained on $6 \times 10^{5}$ events and it performs comparably with a field distortion map that was fit on $5 \times 10^6$ events. The reconstruction error displayed in this figure is the mean squared error on an independent test dataset of $5 \times 10^{5}$ events, comparing corrected positions with their corresponding ground truth interaction vertices.