pith. sign in

arxiv: 2605.25640 · v1 · pith:IQ3JKUH4new · submitted 2026-05-25 · ⚛️ physics.ins-det · cs.LG· hep-ex· nucl-ex

3D Magnetic Field Reconstruction and Mapping with Physics-Informed Neural Networks

Haohan Yu , Zhanxu Hao , Bingzhi Li , Zejia Lu , Xiang Chen , Liang Li This is my paper

Pith reviewed 2026-06-29 19:37 UTC · model grok-4.3

classification ⚛️ physics.ins-det cs.LGhep-exnucl-ex
keywords magnetic field reconstructionphysics-informed neural networksMaxwell equations3D field mappingdivergence-freecurl-freePINN
0
0 comments X

The pith

A physics-informed neural network reconstructs 3D magnetic fields to 10^{-4} accuracy by embedding Maxwell's equations into the loss function.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to demonstrate that a neural network can recover three-dimensional magnetic fields in regions without direct sensor access by training on both sparse measurements and the physical constraints from Maxwell's equations. A sympathetic reader would care because many precision physics experiments need accurate field maps where placing sensors is impractical, and older methods like spherical harmonic expansions introduce truncation errors that limit precision. The key step is adding explicit physics-residual terms at the actual measurement locations rather than relying only on random collocation points. Simulated tests reach 10^{-4} accuracy, an order of magnitude better than prior PINN results, while a custom coil experiment under ambient conditions reaches 10^{-3} relative accuracy.

Core claim

The central claim is that a PINN framework which directly incorporates the divergence-free and curl-free conditions from Maxwell's equations into the loss function, together with explicit physics-residual losses evaluated at the measurement sites, produces high-precision 3D magnetic field reconstructions that outperform existing PINN benchmarks by a factor of ten in simulation and achieve sub-percent accuracy in experiment.

What carries the argument

The PINN loss that combines data-matching terms with explicit enforcement of zero magnetic divergence and zero magnetic curl throughout the domain, plus additional residual penalties computed exactly at the sensor locations.

If this is right

  • The approach supplies a practical method for field monitoring in complex setups where sensor placement is restricted.
  • Reconstruction accuracy reaches 10^{-4} on simulated data, ten times better than previous PINN results.
  • Experimental tests under ambient conditions achieve 10^{-3} relative accuracy.
  • Explicit residuals at measurement locations enforce physical consistency beyond standard collocation sampling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Fewer physical sensors may suffice for a given target accuracy if the physics constraints are strong enough.
  • The same loss-construction pattern could be tried on other vector fields that obey similar differential constraints.
  • Adding time dependence would require extending the loss with the remaining Maxwell equations to handle dynamic fields.

Load-bearing premise

The true magnetic field is exactly divergence-free and curl-free everywhere, so that sparse measurements plus the physics loss terms are enough for the network to recover the field without being misled by noise or model mismatch.

What would settle it

Collect independent field measurements at additional test points withheld from training; if the network predictions deviate from these measurements by more than the claimed error levels, the reconstruction method does not recover the true field.

Figures

Figures reproduced from arXiv: 2605.25640 by Bingzhi Li, Haohan Yu, Liang Li, Xiang Chen, Zejia Lu, Zhanxu Hao.

Figure 2
Figure 2. Figure 2: Spatial sampling strategies for experimental validation. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Training convergence history for the ideal synthetic data, il [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Spatial reconstruction quality at the y = 0 section for sim￾ulated data with sparse boundary sampling (Nsample = 3 per face, noise-free). The panels show (left) PINN prediction, (middle) Biot￾Savart ground truth, and (right) relative error distribution, demon￾strating ∼ 10−4 relative error despite severely limited training data [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Field prediction error vs. sampling density under Gaussian [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of reconstructed magnetic-field magnitudes [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
read the original abstract

Accurate reconstruction of magnetic fields in inaccessible regions is vital for many high-precision experiments in physics. Traditional methods, such as spherical harmonic expansion, often suffer from truncation errors that limit their precision. This study proposes an advanced Physics-Informed Neural Network (PINN) framework for high-precision 3D magnetic field mapping. Unlike conventional data-driven models, the proposed PINN integrates Maxwell's equations directly into the loss function, enforcing divergence-free and curl-free conditions across the entire domain. A key innovation is the inclusion of explicit physics-residual losses at measurement locations, ensuring rigorous physical consistency beyond random collocation sampling. Validation using simulated data achieves a reconstruction accuracy of $10^{-4}$, a tenfold improvement over existing PINN benchmarks. Furthermore, experimental validation using a custom coil assembly demonstrates robust reconstruction with sub-percent relative accuracy, reaching the $10^{-3}$ level under ambient conditions. This AI-driven methodology provides a robust, high-precision solution for field monitoring and measurement in complex experimental environments where direct sensor placement is restricted.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 0 minor

Summary. The manuscript proposes a Physics-Informed Neural Network (PINN) framework for 3D magnetic field reconstruction that incorporates Maxwell's equations (div B = 0 and curl B = 0) directly into the loss function, with an explicit innovation of physics-residual losses evaluated at the measurement locations rather than only at random collocation points. It reports a reconstruction accuracy of 10^{-4} on simulated data, claimed to represent a tenfold improvement over existing PINN benchmarks, and sub-percent relative accuracy reaching the 10^{-3} level on experimental data from a custom coil assembly under ambient conditions.

Significance. If the accuracy claims can be substantiated with complete details on baselines, metrics, and robustness, the method could offer a practical advance over traditional approaches such as spherical harmonic expansions for mapping fields in regions inaccessible to dense sensor placement, which is relevant for high-precision physics experiments. The explicit physics residuals at data points represent a standard but potentially useful refinement to PINN loss formulations.

major comments (3)
  1. [Abstract] Abstract: The claim of 10^{-4} simulated accuracy and a 'tenfold improvement over existing PINN benchmarks' is load-bearing for the central contribution but provides no information on the baseline methods, the precise error metric (e.g., relative L2 or pointwise), data splits, training/validation protocols, or whether hyperparameter choices were post-hoc; without these the improvement cannot be evaluated.
  2. [Abstract] Abstract / Validation: The reported accuracies presuppose that the true field satisfies div B = 0 and curl B = 0 exactly over the entire domain and that sparse measurements plus the physics loss suffice to recover the field without significant degradation from noise or local model mismatch (e.g., near coils); the manuscript must include quantitative tests of sensitivity to violations of these assumptions or to realistic noise levels.
  3. [Abstract] Abstract: The experimental validation claims 'sub-percent relative accuracy reaching the 10^{-3} level' but supplies no details on sensor count and placement, noise characteristics of the measurements, how ground truth was obtained for comparison, or the domain size relative to the coil assembly; these omissions prevent assessment of whether the physics constraints alone compensate for sparsity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments on the abstract and validation sections. We address each point below and have revised the manuscript to incorporate additional details and quantitative tests where the comments identify gaps.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim of 10^{-4} simulated accuracy and a 'tenfold improvement over existing PINN benchmarks' is load-bearing for the central contribution but provides no information on the baseline methods, the precise error metric (e.g., relative L2 or pointwise), data splits, training/validation protocols, or whether hyperparameter choices were post-hoc; without these the improvement cannot be evaluated.

    Authors: We agree the abstract is insufficiently specific. The full manuscript details the baseline (standard PINN without measurement-point residuals) in Section 3.2, uses domain-averaged relative L2 error, employs an 80/20 simulated data split with validation-based early stopping, and reports hyperparameter selection via grid search. The tenfold claim is relative to accuracies in the cited PINN literature for comparable 3D field tasks. We have revised the abstract to state 'relative L2 error of 10^{-4}, a tenfold improvement over standard PINN benchmarks (Section 4)' to make the claim traceable. revision: yes

  2. Referee: [Abstract] Abstract / Validation: The reported accuracies presuppose that the true field satisfies div B = 0 and curl B = 0 exactly over the entire domain and that sparse measurements plus the physics loss suffice to recover the field without significant degradation from noise or local model mismatch (e.g., near coils); the manuscript must include quantitative tests of sensitivity to violations of these assumptions or to realistic noise levels.

    Authors: The simulated data is generated to satisfy Maxwell's equations exactly and the experimental coil data satisfies them to high approximation away from the conductors. We acknowledge that explicit sensitivity quantification would strengthen the claims. We have added Section 4.4 containing quantitative tests: Gaussian noise injection (0.1–5% levels) on measurements and controlled small curl violations (amplitude 10^{-5}) in simulations. Reconstruction error remains below 4 imes 10^{-3} for noise up to 1%, supporting robustness. These results are now summarized in the abstract. revision: yes

  3. Referee: [Abstract] Abstract: The experimental validation claims 'sub-percent relative accuracy reaching the 10^{-3} level' but supplies no details on sensor count and placement, noise characteristics of the measurements, how ground truth was obtained for comparison, or the domain size relative to the coil assembly; these omissions prevent assessment of whether the physics constraints alone compensate for sparsity.

    Authors: Section 2.3 of the manuscript describes the setup: an approximately 0.5 m cubic domain containing a custom coil assembly, instrumented with 12 three-axis Hall sensors at fixed locations, with measurement noise characterized at ~0.5% via repeated calibrations, and ground truth obtained from a separate dense robotic-arm scan. We have revised the abstract to include the clause 'using 12 sensors on a 0.5 m domain with 0.5% measurement noise, compared against dense reference mapping' to supply the missing context. revision: yes

Circularity Check

0 steps flagged

No significant circularity; Maxwell constraints external and validation independent

full rationale

The paper incorporates Maxwell's equations (divergence-free and curl-free conditions) directly into the PINN loss function as physics residuals at measurement points and collocation points. These are independent physical laws, not derived from or defined in terms of the network outputs or data. Reported accuracies (10^{-4} simulated, 10^{-3} experimental) are obtained from separate validation sets and experiments, not by construction from the training inputs. No self-citations, ansatzes smuggled via prior work, or renamings of known results appear in the abstract or described framework. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on the domain assumption that the field obeys Maxwell's equations everywhere and that the neural network can represent the solution; no new entities are introduced and free parameters are the usual neural network hyperparameters.

free parameters (1)
  • Neural network architecture and training hyperparameters
    Number of layers, neurons, learning rate, and weighting of loss terms are chosen but not quantified in the abstract.
axioms (1)
  • domain assumption Magnetic field satisfies div B = 0 and curl B = 0 in the source-free region
    Directly encoded as physics residual losses in the training objective.

pith-pipeline@v0.9.1-grok · 5724 in / 1171 out tokens · 34110 ms · 2026-06-29T19:37:22.999778+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

20 extracted references · 16 canonical work pages · 1 internal anchor

  1. [1]

    and curl-free (∇ ×B= 0) conditions throughout the source-free region: Lphys = 1 N X j∈{rf ,rd} ∥∇ ·B pred(rj)∥2 + 1 N X j∈{rf ,rd} ∥∇ ×B pred(rj)∥2. (5) Crucially, these laws are evaluated at both the internal collo- cation pointsr f and the boundary detector positionsr d, en- suring that physical laws are most rigorously enforced where measurement data i...

  2. [2]

    Venanzoni (for the J-PARC Muon g-2/EDM Collaboration), Status of the Muong−2/EDM Experiment at J-PARC.arXiv preprintarXiv:2512.20335 (2025)

    G. Venanzoni (for the J-PARC Muon g-2/EDM Collaboration), Status of the Muong−2/EDM Experiment at J-PARC.arXiv preprintarXiv:2512.20335 (2025). arXiv:2512.20335

    work page arXiv 2025

  3. [3]

    Zaid (for the Muon g-2 Collaboration), Run 2/3 measure- ment of the muon anomalous magnetic moment by the Muon g-2 experiment at Fermilab.arXiv preprintarXiv:2506.21219 (2025)

    E. Zaid (for the Muon g-2 Collaboration), Run 2/3 measure- ment of the muon anomalous magnetic moment by the Muon g-2 experiment at Fermilab.arXiv preprintarXiv:2506.21219 (2025). arXiv:2506.21219

    work page arXiv 2025

  4. [4]

    Muon (g-2) Technical Design Report

    J. Grange et al. (Muon g-2 Collaboration), Muong−2Tech- nical Design Report.arXiv preprintarXiv:1501.06858 (2015). arXiv:1501.06858

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  5. [5]

    Albahri et al

    T. Albahri et al. (Muon g-2 Collaboration), Magnetic-field measurement and analysis for the Muong−2Experiment at Fermilab.Physical Review A103(4), 042208 (2021). doi: 10.1103/PhysRevA.103.042208

    work page doi:10.1103/physreva.103.042208 2021

  6. [6]

    Isakov, ALICE ITS2: overview and performance, Journal of Instrumentation 20 (07) (2025) C07026

    N. Nouri et al., A prototype vector magnetic field monitoring system for a neutron electric dipole moment experiment.Jour- nal of Instrumentation10, P12003 (2015). doi: 10.1088/1748- 0221/10/12/p12003

    work page doi:10.1088/1748- 2015

  7. [7]

    Nouri, B

    N. Nouri, B. Plaster, Systematic optimization of exterior mea- surement locations for the determination of interior magnetic field vector components in inaccessible regions.Nuclear In- struments and Methods in Physics Research A767, 92–98 (2014). doi: 10.1016/j.nima.2014.08.026

    work page doi:10.1016/j.nima.2014.08.026 2014

  8. [8]

    J. D. Jackson,Classical Electrodynamics, 3rd edn. (Wiley, New York, 1999), Chap. 4–5. doi: 10.1002/9783527618158

    work page doi:10.1002/9783527618158 1999

  9. [9]

    Carleo, I

    G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. V ogt-Maranto, L. Zdeborová, Machine learning and the physical sciences.Reviews of Modern Physics91(4), 045002 (2019). doi: 10.1103/RevModPhys.91.045002

    work page doi:10.1103/revmodphys.91.045002 2019

  10. [10]

    Boehnlein et al., Colloquium: Machine learning in nuclear physics.Reviews of Modern Physics94(3), 031003 (2022)

    A. Boehnlein et al., Colloquium: Machine learning in nuclear physics.Reviews of Modern Physics94(3), 031003 (2022). doi: 10.1103/RevModPhys.94.031003

    work page doi:10.1103/revmodphys.94.031003 2022

  11. [11]

    Radovic, M

    A. Radovic, M. Williams, D. Rousseau, M. Kagan, D. Bonacorsi, A. Himmel, A. Aurisano, K. Terao, T. Wongji- rad, Machine learning at the energy and intensity frontiers of particle physics.Nature560(7716), 41–48 (2018). doi: 10.1038/s41586-018-0361-2

    work page doi:10.1038/s41586-018-0361-2 2018

  12. [12]

    Raissi, P

    M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving for- ward and inverse problems involving nonlinear partial differen- tial equations.Journal of Computational Physics378, 686–707 (2019). doi: 10.1016/j.jcp.2018.10.045

    work page doi:10.1016/j.jcp.2018.10.045 2019

  13. [13]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, L. Yang, Physics-informed machine learning.Nature Reviews Physics3(6), 422–440 (2021). doi: 10.1038/s42254- 021-00314-5

    work page doi:10.1038/s42254- 2021

  14. [14]

    U. H. Coskun, B. Sel, B. Plaster, Magnetic field mapping of inaccessible regions using physics-informed neural networks. Scientific Reports12, 12858 (2022). doi: 10.1038/s41598-022- 15777-4 -9 Haohan Yu◦ et al

    work page doi:10.1038/s41598-022- 2022

  15. [15]

    Stoer, R

    J. Stoer, R. Bulirsch,Introduction to Numerical Analysis, 2nd edn., Texts in Applied Mathematics V ol. 12 (Springer, New York, 1993), pp. 559–620. doi: 10.1007/978-0-387-21738-3

    work page doi:10.1007/978-0-387-21738-3 1993

  16. [16]

    Sitzmann, J

    V . Sitzmann, J. Martel, A. Bergman, D. Lindell, G. Wet- zstein, Implicit neural representations with periodic activation functions.Advances in Neural Information Processing Systems (NeurIPS)33, 7462–7473 (2020). URL: neurips.cc/paper/2020

    2020

  17. [17]

    Tancik, P

    M. Tancik, P. Srinivasan, B. Mildenhall, S. Fridovich-Keil, N. Raghavan, U. Singhal, R. Ramamoorthi, J. Barron, R. Ng, Fourier features let networks learn high frequency functions in low dimensional domains.Advances in Neural Information Processing Systems (NeurIPS)33, 7537–7547 (2020). URL: neurips.cc/paper/2020

    2020

  18. [18]

    Loshchilov, F

    I. Loshchilov, F. Hutter, Decoupled weight decay regulariza- tion. InInternational Conference on Learning Representations (ICLR)(2019). OpenReview

    2019

  19. [19]

    1999Combustion

    J. Nocedal, S. J. Wright,Numerical Optimization, 2nd edn. (Springer, New York, 2006), pp. 177–180. doi: 10.1007/978- 0-387-40065-5

    work page doi:10.1007/978- 2006

  20. [20]

    C. Wu, M. Zhu, Q. Tan, Y . Kartha, L. Lu, A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks.Computer Methods in Applied Mechanics and Engineering403, 115671 (2023). doi: 10.1016/j.cma.2022.115671 -10

    work page doi:10.1016/j.cma.2022.115671 2023