Physics-Informed Self-Supervised Learning for Joint Wire Calibration and Interaction Position Reconstruction in Multi-Wire Parallel Plate Avalanche Counters

• Conventional calibration can require dedicated measurements or labelled reference data, and may become invalid as detector characteristics evolve during extended campaigns.Source: §1 · p. 2
• The framework jointly estimates global wire calibration coefficients and event-wise interaction positions using losses derived from detector geometry and charge-energy consistency.Source: §1 · p. 3
• A detector-independent network learns sub-wire offsets from local charge distributions without assuming an analytical induction profile.Source: §1 · pp. 2–3
• The framework is presented as enabling continuous self-calibration during normal operation and adaptation to aging, electronic drift, or changing conditions.Source: §1 · p. 3

• Earlier VAMOS++ neural-network work addressed trajectory reconstruction and fractionally labelled ion identification, not the present joint wire-calibration problem.Source: §2 · p. 4
• Existing multi-wire position methods generally fit predefined analytical charge profiles, whose validity can depend on geometry, operating conditions, and electronic response.Source: §2 · p. 4
• The proposed approach learns the relation between neighboring-wire charges and sub-wire position from observations, constrained by detector geometry and charge-energy consistency.Source: §2 · p. 4
• The authors identify this as the first MWPPAC framework jointly optimizing wire calibration and position reconstruction using physical constraints without labelled positions or dedicated calibration data.Source: §2 · p. 4

3.1 Detector Geometry

• For particles originating near the target and traveling along straight trajectories between detectors, the interaction position on one MWPPAC predicts the corresponding position on the other.Source: §3.1 · p. 6

3.2 Latent Variables

• Wire calibration coefficients are global parameters shared across events, while interaction positions are event-specific latent variables.Source: §3.2 · p. 6
• The reconstructed coordinate combines the position of the maximum-charge wire with a sub-wire interaction offset, and the offset is inferred from a local neighborhood of wires.Source: §3.2 · p. 7

3.3 Physical Consistency Constraints

• Geometric consistency links the reconstructed positions across the two MWPPAC detectors, while charge-energy consistency relates calibrated plane charges to ionization-chamber energy.Source: §3.3 · p. 7
• The fixed charge-energy scaling coefficient sets wire-gain dynamic range, but its absolute value is not physically relevant because position depends on relative neighboring-wire calibration.Source: §3.3 · p. 7
• Geometry and charge-energy consistency provide supervisory information for jointly estimating calibration and positions without labelled experimental data.Source: §3.3 · p. 7

3.4 Joint Optimization Problem

• The objective minimizes physics-derived consistency losses over calibration vectors and the mapping from locally calibrated charges to sub-wire positions.Source: §3.4 · pp. 7–8

4.1 Calibration Networks

• Four independent calibration subnetworks estimate plane-specific wire gains, while one shared network reconstructs sub-wire positions from locally calibrated charges.Separate plane models reflect each plane's distinct response; Fig. 2 encodes each submodel by layer count, units per layer, and output units.Source: §4 · p. 8, §4.1 · p. 8, Figure 2 · p. 9

4.2 Universal Position Reconstruction Network

• A shared position network maps local calibrated charge vectors to sub-wire offsets without detector identity, wire number, or absolute coordinates, learning induction profiles from data and transferring across detector sizes.The input neighborhood is centered on the maximum-charge wire; the learned mapping requires no analytical induction-profile assumption.Source: §4.2 · pp. 10–11

4.3 Event Processing Pipeline

• For each event, plane-specific calibration is followed by local-neighborhood extraction, shared-network offset prediction, position reconstruction, and evaluation of the physical objective.The reconstructed positions and calibrated charge sums enter the consistency constraints.Source: §4.3 · p. 11

4.4 End-to-End Optimization

• Joint gradient-based optimization updates global calibration parameters and event-specific positions together, with no intermediate calibration stages or alternating procedure.The physics-informed objective sends gradients through both network types; calibration and position estimates mutually improve through coupled charge and geometry constraints.Source: §4 · p. 8, §4.4 · p. 11

5 Physics-Informed Objective Function

• Training requires no labelled positions, calibration coefficients, or reference trajectories; supervisory information comes from physical relationships in detector measurements.The objective combines geometric consistency and charge-energy consistency, with targets generated implicitly by the measurement process.Source: §5 · p. 12, §5.4 · p. 14

5.1 Geometric Consistency Loss

• The geometric loss encourages reconstructed positions to satisfy detector-layout relationships and the assumption that particle trajectories originate from the target region.The two MWPPAC detectors measure the same particle trajectory, linking their reconstructed positions through known geometry.Source: §5.1 · p. 12

5.2 Charge-Energy Consistency Loss

• Position reconstruction uses only relative neighboring-wire calibration, so geometric consistency alone cannot uniquely determine the calibration's overall normalization.A charge-energy constraint supplements geometry by requiring calibrated charge sums to be proportional to ionization-chamber energy.Source: §5.2 · pp. 12–13

5.3 Joint Self-Supervised Objective

• The joint objective weights geometry-based position constraints and charge-energy constraints on calibration coefficients to seek a solution satisfying both physical relationships.The two loss weights control the relative importance of the constraints; neither requires labelled training data.Source: §5.3 · p. 13

5.4 Why the Problem is Self-Supervised ?

• No measured positions or reference calibration coefficients are supplied during training; geometry and ionization-chamber measurements provide the physical constraints instead.The optimization targets are generated implicitly from the measurement process, placing the method in self-supervised scientific machine learning.Source: §5.4 · pp. 13–14

6.1 VAMOS++ MWPPAC Detector

• The evaluation uses the VAMOS++ dual position-sensitive MWPC setup, with two detectors between the target and spectrometer entrance for two-dimensional position readout.The front and back active areas are 40×61 mm2 and 65×93 mm2, respectively.Source: §6.1 · p. 14
• The detector uses orthogonal X and Y anode planes, with 39, 64, 60, and 92 wires in X1, X2, Y1, and Y2, respectively.Anode wires have a 1.0 mm pitch; cathode wires provide timing information.Source: §6.1 · p. 14

6.2 Experimental Data

• The experimental evaluation uses VAMOS++ data from the E850 GANIL experiment, which tracked reaction products from a 238U beam on a 12C target.The fission fragments were detected and isotopically identified in VAMOS++; light nuclei were detected in the PISTA array.Source: §6.2 · p. 14

6.3 Implementation Details

• Training uses 4 × 10^6 events, split 80%/20% into training and validation sets, and monitors residual RMSD and mean calibration coefficients for stabilization.An epoch takes approximately 6 seconds, and convergence is typically reached in approximately 0.5 minute; Figures 3 and 4 show residual and coefficient behavior.Source: §6.3 · p. 15, Figure 3 · p. 15, Figure 4 · p. 15

7.1 Optimization Behavior

• With σXbeam = 0.40 mm, improving detector resolution by approximately 30% decreases residual width by several percent and reconstructed target-beam width by approximately 9%.For detector resolutions σXdet = 0, 0.06, and 0.09 mm, the reconstructed target widths are σXtarget = 0.40, 0.44, and 0.48 mm, respectively.Source: §7.1 · pp. 16–17

7.2 Optimization Convergence

• The proposed optimization converges within a few epochs for N = 3 and N = 5, with validation curves tracking training curves and no overfitting.N = 5 yields no significant improvement over N = 3; tested N = 7, 9, and 11 models and greater model complexity likewise produce no substantial improvement.Source: §7.2 · p. 17

7.3 Spatial distribution of Residuals

• δσ = 0.017 mm mean improvement in residual widths accompanies removal of the conventional method’s local discontinuities and wire-adjacent clustering.The paper attributes the artifacts to mismatch between the assumed analytical induction profile and the true detector response; the presented framework learns the profile implicitly.Source: §7.3 · pp. 17–18, Table 1 · p. 18

7.4 Reconstructed Beam Profile on the Target

• σXtarget = 0.468(2) mm versus 0.433(2) mm and σYtarget = 0.592(2) mm versus 0.572(2) mm for conventional and presented methods, respectively.The presented framework removes a systematic reconstruction bias in the beam profile.Source: Figure 6 · p. 18, §7.4 · p. 19

7.5 Position Resolution

• 33% and 44%: position resolution improves for the front and rear reference wires, respectively, in an independent reference-wire validation.The front-wire values are σW1 = 0.09(2) mm for the conventional method and σW1 = 0.06(1) mm for the presented method; rear-wire values are σW2 = 0.09(2) mm and σW2 = 0.05(1) mm, respectively.Source: §7.5 · pp. 19–20

7.6 Implicit Learning of the Induction Profile

• The learned representation captures the induction profile without a prescribed analytical form, reducing modeling bias and improving position uniformity across the detector surface.The paper cites the absence of local discontinuities in Fig. 5 as evidence for this representation.Source: §7.6 · p. 20

7.7 Transferability and Generalization

• The reconstruction network uses local calibrated charge distributions and is independent of detector geometry, absolute wire numbering, and active area.The same reconstruction model can be applied across detector dimensions without changing its architecture; only detector-specific calibration subnetworks require re-optimization.Source: §7.7 · p. 20

8 Discussion

• Improved wire calibration sharpens local charge distributions, while better positions strengthen geometric consistency between the two MWPPACs, reinforcing both optimizations.The coupled process continues through training until a physically consistent solution is reached.Source: §8 · p. 21
• Preliminary investigations indicate accurate position reconstruction may remain possible after removing saturated maximum-charge input, by exploiting neighboring-wire signal redundancy.A detailed analysis of this capability is beyond the work’s scope.Source: §8 · p. 21
• The detector-independent network uses detector-specific physical constraints: geometric consistency for entrance MWPPACs, ionic charge-state and mass consistency for the ionization chamber, and atomic-mass dependence for focal-plane MWPPACs.The framework is modular, combining differentiable detector models with the appropriate physical consistency constraints.Source: §8 · pp. 21–22
• Within VAMOS++, only timing calibration currently requires a conventional procedure; focal-plane MWPPAC calibration is expected to use atomic-mass constraints.Trajectory reconstruction is already absolute-coordinate, and the ionization chamber and entrance MWPPACs can be calibrated by self-supervised optimization.Source: §8 · p. 22
• Continuous adaptation during operation could improve measurement reliability, reduce downtime, and simplify operation as detector conditions change.The discussion identifies changing operating conditions, electronic drift, and aging as potential drivers for adaptation.Source: §8 · p. 22
• Success depends on informative physical constraints; inadequate geometric redundancy or an incomplete model may require added supervision, while nonlinear responses need richer parameterizations and potentially stronger regularization.The demonstrated work uses linear gain calibration.Source: §8 · p. 22

9 Conclusions

• The method jointly estimates global wire gains and event-wise positions without dedicated calibration measurements or labelled interaction positions, using geometry and charge-energy consistency.Source: §9 · pp. 22–23
• Experiments on VAMOS++ entrance MWPPACs demonstrated stable optimization, improved spatial homogeneity, and enhanced position reconstruction performance during continuous normal-operation calibration.Source: §9 · p. 23
• A detector-independent network learns sub-wire positions directly from local charge distributions, avoiding assumed analytical induction profiles.Source: §9 · p. 23
• The modular framework applies where redundant measurements or established physical relationships provide sufficient supervisory information, with detector-specific knowledge entering through differentiable consistency constraints.Source: §9 · p. 23
• The remaining focal-plane MWPPACs could use atomic-mass reconstruction consistency instead of geometric consistency as their supervisory constraint.Source: §9 · p. 23
• The paper presents continuous adaptation to detector aging, electronic drift, and changing conditions without interrupting operation as a step toward fully self-supervised instrumentation.Source: §9 · p. 23
• Future work will examine signal saturation, nonlinear calibration models, and additional detector subsystems.Source: §9 · p. 23