TPCNet: Physics-Constrained Neural Architectures

• TPCNet is a family of physics-informed neural networks that integrate domain-specific constraints for TPC simulation, low-light image enhancement, and HI spectral analysis.
• In particle physics, TPCNet employs a conditional GAN with detailed input encoding and loss formulations to achieve simulation speedups of up to 12x while preserving reconstruction accuracy.
• In computer vision and radio astronomy, lightweight CNN and Transformer architectures with physical and spectral constraints enable superior image quality and precise parameter regression.

TPCNet refers to a family of neural network architectures and algorithms denoted by “TPCNet” in three distinct scientific domains: fast surrogate simulation for Time Projection Chambers in particle physics, physics-constrained low-light image enhancement in computer vision, and representation learning for HI spectral mapping in radio astronomy. Although sharing a common acronym, these systems differ substantively in physical modeling, network topology, objective functions, and application scope. Each instantiation is outlined below with precise definitions and technical structure.

1. TPCNet for Time Projection Chamber Simulation

A principal use of TPCNet is as a generative surrogate model for accelerating Time Projection Chamber (TPC) digitization in high-energy physics experiments, exemplified by the MPD experiment at NICA. The objective is to replace the standard GEANT-4–based TPC digitizer—which outputs approximately $3 \times 10^7$ numbers per collision event—with a learned conditional GAN capable of sampling realistic pad response windows at orders-of-magnitude faster rates (Ratnikov et al., 2022).

Physical Factorization and Input Encoding

TPC digitization is factorized into many independent “windows” of $8 \times 16$ pads, each representing a localized segment of a charged particle track. For each window, conditional variables $c$ encode pad row index, pad coordinate fraction, dip angle (track inclination), charge amplitude, and drift distance, with the input noise $z$ sampled from a multivariate Gaussian prior (dimensionality $N \sim 100$).

Model Architecture

• Generator: Concatenates $z \in \mathbb{R}^N$ and $c \in \mathbb{R}^5$, projects through fully-connected layers and several 2D de-convolution (transpose-convolution) blocks to upsample into $8 \times 16$ grid representation. Nonlinearities include (Leaky)ReLU in the hidden layers; output is normalized via tanh or clipped ReLU.
• Discriminator: Ingests the real/generated $8 \times 16$ window and conditioning vector $c$, applies a series of 2D convolutions with stride $> 1$, LeakyReLU ($\alpha \approx 0.2$), and concludes with a scalar output obtained by either sigmoid (classic GAN) or linear activation (WGAN variant).

Optimization Objective

Training employs classic conditional GAN objectives:

$$L_D = -\mathbb{E}_{x \sim p_{\text{sim}}} [\log D(x, c)] - \mathbb{E}_{z \sim p_z} [\log(1 - D(G(z, c), c))]$$

$$L_G = -\mathbb{E}_{z \sim p_z} [\log D(G(z, c), c)]$$

The alternative Wasserstein GAN loss with gradient penalty is acknowledged, but not applied in the cited implementation.

Performance and Validation

Low-level validation projections barycenter and width statistics ($\mu_x$, $\mu_t$, $\sigma_x$, $\sigma_t$, $\rho$) are matched to full simulation profiles. High-level validation inserts GAN-generated windows into reconstruction chains, yielding momentum resolution deviations at the percent level. TPCNet achieves a $\sim 12 \times$ speedup over detailed digitizer chains.

Workflow Integration

Deployment is automated via data production (full simulation windows), model training managed by Python Airflow DAGs, MLflow Model Registry for versioning, export to ONNX format, and inference via ONNX Runtime embedded within the C++ MPD reconstruction framework. This workflow allows re-training and redeployment for evolving detector configurations.

2. TPCNet: Triple Physical Constraints for Low-Light Image Enhancement

In computer vision, TPCNet refers to a lightweight neural network for low-light image enhancement, grounded in physics by enforcing triple physical constraints (TPCs) derived from Kubelka–Munk theory (Shi et al., 27 Nov 2025). Unlike Retinex-based methods that presuppose Lambertian reflection, TPCNet models both specular and diffuse components via closed-form feature-space relationships.

Physical Modeling

Kubelka–Munk formalism yields, after Taylor expansion and variable substitution, three constraints:

• TPC-1 (Imaging): $E(x) = L(x)R_\infty(x) + \overline{L}(x)$
• TPC-2 (Reflectivity): $R_\infty(x) = \frac{E(x) - \overline{L}(x)}{L(x)}$
• TPC-3 (Illumination sum): $L(x) + \overline{L}(x) = e(x)$

Here, $\alpha(x) = 1 - 2\rho_f(x)$ denotes the local specular reflection coefficient.

Network Structure

TPCNet’s pipeline comprises:

1. LFE (Light Features Estimator): CNN block extracts light-feature tensor $\hat{e}$ and per-pixel weight $\hat{\alpha}$.
2. TPC Constraint Enforcement: Computes $\hat{L} = \hat{\alpha} \cdot \hat{e}$, $$\hat{\overline{L}} = (1 - \hat{\alpha}) \cdot \hat{e} / 2$$.
3. RFE (Reflectivity Feature Estimator): Processes $(I, \hat{\overline{L}})$ to estimate $\hat{E}$ and apply TPC-2 for reflectance $\hat{R}$.
4. DCGT (Dual-Stream Cross-Guided Transformer): Refines $(\hat{R}, \hat{L})$.
5. CAM (Color-Association Mechanism): Fuses refined outputs with multi-scale color features for final enhancement image $I_{\text{en}}$.

DCGT employs cross-guided attention blocks (CGAB) with $O((H \cdot W) \cdot C^2 / k)$ complexity, optimizing long-range dependencies at reduced FLOP cost.

Loss Functions

End-to-end training utilizes a sum of reconstruction ($L_1$), perceptual (VGG-based), SSIM, and edge losses—calculated in both RGB and color-space representations.

Experimental Benchmarks

TPCNet achieves state-of-the-art quantitative results across ten datasets:

• LOL-v2-Real: PSNR = 24.978 dB, SSIM = 0.882, surpassing CIDNet by 0.867 dB.
• Generalization on MEF, NPE, LIME, DICM, and VV: top rank in NIQE, MUSIQ, PI metrics. Ablation studies confirm each constraint (TPC-1,2,3) improves accuracy, with removal incurring $\sim 0.2$ dB PSNR loss.

Model Characteristics and Future Directions

TPCNet comprises only 2.62M parameters and 8.68 GFLOPs (256$\times$256 input), outperforming larger networks. Extensions include explicit spectral modeling, physics-inspired noise, video enhancement via temporal constraints, and unsupervised training by soft regularization to TPC equations.

3. TPCNet for HI Spectral Analysis

In radio astronomy, TPCNet designates a regression network for neutral atomic hydrogen (HI) mapping, combining a deep 1D CNN encoder and Transformer predictor with sinusoidal positional encoding on the spectral axis (Nguyen et al., 20 Nov 2024).

Architectural Details

• Input: Single HI emission spectrum $T_b(v)$ with $N$ channels (e.g., $N=256$ or $101$).
• Encoder: 8-layer CNN (alternating kernel sizes $1 \times 7$ and $1 \times 33$), filter count decreasing from 64 to 8, batch normalization and ReLU activations, with no max-pooling.
• Tokenization: After CNN, feature vector of $8N$ reshaped to $m \times k$ ($k=9$).
• Positional Encoding: Sinusoidal PE applies in the token dimension and, optionally, to input.
• Predictor: Four-layer Transformer decoder stack (3 heads, feedforward 9%%%%44$N$45%%%%9), standard residual and LayerNorm.
• Regression Head: Global token mean pooled, followed by 2-layer MLP (9%%%%46$\rho$47%%%%2) yielding $[f_{\text{CNM}}, R_{\text{HI}}]$.

Predicted Quantities

• Cold-neutral-medium mass fraction $f_{\text{CNM}} = N_{\text{HI,CNM}} / N_{\text{HI}}$
• Opacity correction $R_{\text{HI}} = N_{\text{HI}} / N_{\text{HI,thin}}$
• Column densities calculated as: $$N_{\text{HI,thin}} = 1.8224 \times 10^{18} \int T_b(v) dv$$

$$N_{\text{HI}} = 1.8224 \times 10^{18} \int \left[ \frac{T_s(v) \tau(v)}{1 - e^{-\tau(v)}} \right] dv$$

Training and Validation Procedure

TPCNet is trained on synthetic datasets (hydrodynamic and MHD cubes) with added Gaussian noise and spatial beam convolution. Regression loss is minimized via Adam with $5 \times 10^{-3}$ learning rate; batch size 256; training for up to 60 epochs and early stopping.

Sinusoidal PE substantially improves stability—test RMSE varies by $\pm 0.5\%$ (vs. $\pm 5\%$ CNN baseline)—and convergence (plateau at 30 epochs vs. 60).

Evaluation Metrics

On held-out cubes:

• $f_{\text{CNM}}$ test RMSE: $0.035$ ($3.5\%$); $R_{\text{HI}}$ RMSE: $0.05$ ($\sim 10\%$ lower error than deep CNN baseline). Real data cross-checks show TPCNet aligns with Gaussian decomposition and Fourier transform methods for optically thin HI regimes, revealing greater ability to detect cold neutral medium compared to a shallow CNN baseline.

Implementation Details

Training performed on NVIDIA A100 GPUs with memory demand $\leq 8$ GB; 8-layer CNN is optimal for accuracy and runtime. Increased spectral resolution ($\Delta v = 0.3125$ km/s) yields $\sim 6\%$ improved RMSE over coarse channelization.

4. Comparative Summary Table

Domain	Physical Model / Constraint	Network Topology	Main Outputs
Particle Physics (TPC)	Track segmentation, GAN	Conditional GAN (FC/Deconv + Conv)	Pad window simulation
Computer Vision (Enhance)	Kubelka–Munk, Triple PCs	CNN + Transformer + TPC modules	Enhanced RGB image
Radio Astronomy (HI)	HI radiative transfer, stats	1D Deep CNN + Transformer, PE	$f_{\text{CNM}}, R_{\text{HI}}$

Each system is calibrated to its domain’s physics or observational constraints, using hybrid CNN/Transformer architectures for stable, interpretable, and high-fidelity regression or generation. Sinusoidal positional encoding features prominently in the radio astronomy variant for spectral structure robustness.

5. Contextual Significance and Extensions

The term TPCNet encompasses distinct technical architectures unified by the use of physical or domain-specific constraints embedded into neural network structures for either simulation acceleration, enhancement quality, or scientific parameter inference. Notably, in areas requiring rapid throughput (high-energy physics simulation) or tight generalizability (computer vision and radio astronomy), TPCNet architectures demonstrate quantifiable gains in speed, stability, and accuracy over traditional deep learning baselines.

A plausible implication is that principled physics-informed neural architectures, such as TPCNet, provide systematic pathways for bridging simulation, enhancement, and parameter inference tasks in scientific domains where data generation and labeling are costly or physically constrained. Open directions include extending TPC constraints for video and temporal data, integrating explicit spectral (wavelength) modeling, and leveraging unsupervised training via physical regularization. Each field adapts TPCNet by modulating architectural depth, the nature and dimensionality of tokenization, and loss structure according to domain-specific requirements and theoretical underpinnings.