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arxiv: 2604.15615 · v1 · submitted 2026-04-17 · 📡 eess.SP

Discovery of unobservable parameters via physical embedding

Le Cheng (1) , Xiaoran Liu (1) , Lingjin Kong (1) , Haitao Zhao (1) , Jun Xiong (1) , Fanglin Gu (1) , Xiaoying Zhang (1) , Baoquan Ren (2)
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Jibo Wei (1) Hao Yin (1 2) ((1) College of Electronic Science Technology National University of Defense Technology Changsha China (2) Academy of Military Science Beijing China)
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Pith reviewed 2026-05-10 08:56 UTC · model grok-4.3

classification 📡 eess.SP
keywords Physics-Embedded Inverse Learningunobservable parametersinverse problemssignal reconstructionwireless communicationsparallel MRIcoil sensitivity mapsnon-identifiability
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The pith

A neural estimator paired with a fixed physics inverse operator learns unobservable parameters by optimizing only the final signal reconstruction.

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

The paper introduces Physics-Embedded Inverse Learning, in which a learned model predicts latent parameters such as channel states or coil sensitivities, then passes them to an unchanging physics-based operator that produces the source signal. Supervision comes solely from how well the reconstructed signal matches the available data, so the learner never sees ground-truth values for the hidden parameters. When many different parameter sets can produce identical reconstructions, the method selects those that offset inaccuracies inside the physics operator. Experiments show this yields better wireless performance than methods given true parameters, plus strong generalization with far less data, and it recovers physically plausible coil maps in MRI without any calibration step.

Core claim

In PEIL a learned estimator outputs unobservable parameters that a fixed physics-based inverse operator then uses to recover the source signal; the training objective is defined only on reconstruction quality using the source signal as the sole form of supervision. Because multiple parameter combinations can yield the same reconstruction, the estimator exploits this non-identifiability to choose values that compensate for residual modeling errors rather than match the true parameters. In high-mobility wireless communications the resulting configurations outperform baselines that have access to ground-truth parameters, enable zero-shot generalization, and require more than twenty times less训练

What carries the argument

Physics-Embedded Inverse Learning (PEIL) framework, which couples a trainable parameter estimator to a non-trainable physics-based inverse operator so that the loss is computed on the reconstructed signal.

If this is right

  • In high-mobility wireless settings, PEIL configurations outperform even methods supplied with ground-truth parameters.
  • Training data requirements drop by more than a factor of twenty while zero-shot generalization across scenarios is retained.
  • In parallel MRI the method produces coil-sensitivity maps that are physically interpretable and usable without separate calibration scans.
  • Reconstructions remain grounded entirely in the acquired measurements rather than in external parameter labels.
  • Non-identifiability is converted from a modeling obstacle into a mechanism for compensating modeling inaccuracies.

Where Pith is reading between the lines

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

  • The same embedding strategy could be tested on other inverse problems such as radar imaging or seismic inversion where a known forward model exists but calibration data are expensive.
  • If the physics operator is updated periodically to reduce its residual error, the learned estimator might converge to parameters closer to the true values while preserving the original reconstruction advantage.
  • The approach may break down when the mapping from parameters to measurements becomes highly nonlinear or when the non-identifiable set is too small to allow useful compensation.
  • Extending PEIL to time-varying physics operators could reveal whether the compensation mechanism remains stable under changing conditions.

Load-bearing premise

The fixed physics-based inverse operator is accurate enough that any remaining modeling errors can be absorbed into the learned parameters without lowering reconstruction quality.

What would settle it

Applying PEIL in a new domain where the physics operator is known to contain large systematic errors and observing that reconstruction quality falls below that of supervised parameter-estimation baselines would falsify the central claim.

read the original abstract

Recovering a source signal from indirect measurements often requires estimating latent parameters, such as wireless channel states or MRI coil sensitivities, that cannot be directly observed. Here, we introduce Physics-Embedded Inverse Learning (PEIL), in which a learned estimator predicts these parameters and a fixed, physics-based inverse operator uses them to reconstruct the signal, so that training requires only the source signal as supervision. In systems where multiple parameter combinations can reconstruct the signal equally well, the estimator exploits this freedom to coordinate parameters that compensate for residual modelling errors rather than match ground-truth parameters. In high-mobility wireless communications, PEIL discovers task-optimal configurations that outperform baselines given access to ground-truth parameters, enabling zero-shot generalisation and over 20-fold reduction in training data relative to supervised baselines. To test whether these properties extend across physical domains, we apply PEIL to parallel MRI, where it discovers physically interpretable coil sensitivity maps without calibration scans, yielding reconstructions grounded purely in acquired measurements. These results demonstrate that non-identifiability, conventionally a liability, becomes a resource when the learning objective targets reconstruction quality rather than parameter accuracy.

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

0 major / 3 minor

Summary. The paper introduces Physics-Embedded Inverse Learning (PEIL), in which a learned estimator predicts latent unobservable parameters (such as wireless channel states or MRI coil sensitivities) while a fixed physics-based inverse operator reconstructs the source signal; training uses only the source signal as supervision. The method exploits non-identifiability to select parameter configurations that compensate for residual modeling errors rather than matching ground truth, yielding task-optimal parameters. Empirical results are reported in high-mobility wireless communications (outperforming ground-truth baselines, zero-shot generalization, >20-fold training-data reduction) and parallel MRI (interpretable coil maps without calibration scans).

Significance. If the central claims hold, the work reframes non-identifiability as a constructive resource for inverse problems, enabling superior reconstruction quality, data efficiency, and cross-domain generalization while grounding learning in a fixed physics operator. This could influence parameter estimation in communications, imaging, and other signal-processing domains where direct supervision on latent variables is unavailable.

minor comments (3)
  1. Abstract: the quantitative claims (outperformance relative to ground-truth baselines, 20-fold data reduction, zero-shot generalization) would benefit from explicit numerical values or metrics in the abstract itself rather than qualitative description.
  2. The manuscript should clarify in the methods section how the fixed physics operator is constructed and verified to remain unchanged during training, to directly address the assumption that non-identifiability compensates for modeling residuals without introducing implicit fitting.
  3. Figure captions and experimental sections: error bars, number of trials, and statistical tests for the reported performance gains are needed to support the empirical superiority claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work on Physics-Embedded Inverse Learning (PEIL) and for recommending minor revision. The referee's description accurately reflects the core idea of exploiting non-identifiability in a physics-embedded reconstruction loop to learn unobservable parameters from source-signal supervision alone. No specific major comments were provided in the report, so we have no individual points to rebut or revise at this stage. We remain available to address any minor issues or clarifications the editor or referee may identify.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The PEIL construction trains a parameter estimator using only source-signal supervision through a fixed, externally specified physics-based inverse operator and reconstruction loss. This supplies independent grounding via the physics model and loss, without any step reducing the learned parameters or predictions to the inputs by construction. The deliberate use of non-identifiability to compensate for modeling residuals is an explicit design objective rather than an implicit fit or self-definition. No self-citation load-bearing steps, uniqueness theorems imported from the authors, ansatzes smuggled via citation, or renaming of known results appear in the abstract or described method. The derivation chain remains self-contained against the stated physics operator and loss.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a reliable fixed physics-based inverse operator and the utility of non-identifiability for error compensation; no explicit free parameters or invented physical entities are named in the abstract.

axioms (2)
  • domain assumption A fixed, accurate physics-based inverse operator exists that can reconstruct the signal from predicted parameters.
    Stated in the description of PEIL where the learned estimator outputs parameters for use by this operator.
  • domain assumption Multiple parameter combinations can produce equivalent reconstructions, allowing coordination to compensate for modeling errors.
    Invoked when explaining how the estimator exploits non-identifiability.

pith-pipeline@v0.9.0 · 5570 in / 1519 out tokens · 55907 ms · 2026-05-10T08:56:34.369438+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

45 extracted references · 5 canonical work pages · 2 internal anchors

  1. [1]

    You, X.et al.Towards 6g wireless communication networks: Vision, enabling technologies, and new paradigm shifts.Science China information sciences64, 110301 (2021)

    2021

  2. [2]

    Z., Cauley, S

    Zhu, B., Liu, J. Z., Cauley, S. F., Rosen, B. R. & Rosen, M. S. Image reconstruction by domain-transform manifold learning.Nature555, 487–492 (2018)

    2018

  3. [3]

    Magnetic resonance in medicine79, 3055–3071 (2018)

    Hammernik, K.et al.Learning a variational network for reconstruction of accelerated mri data. Magnetic resonance in medicine79, 3055–3071 (2018)

    2018

  4. [4]

    & Sch¨ onlieb, C.-B

    Arridge, S., Maass, P., ¨Oktem, O. & Sch¨ onlieb, C.-B. Solving inverse problems using data-driven models.Acta Numerica28, 1–174 (2019)

    2019

  5. [5]

    T., Jin, K

    McCann, M. T., Jin, K. H. & Unser, M. Convolutional neural networks for inverse problems in imaging: A review.IEEE Signal Processing Magazine34, 85–95 (2017)

    2017

  6. [6]

    & Katsaggelos, A

    Lucas, A., Iliadis, M., Molina, R. & Katsaggelos, A. K. Using deep neural networks for inverse problems in imaging: beyond analytical methods.IEEE Signal Processing Magazine35, 20–36 (2018)

    2018

  7. [7]

    & Hinton, G

    LeCun, Y., Bengio, Y. & Hinton, G. Deep learning.Nature521, 436–444 (2015)

    2015

  8. [8]

    & Hoydis, J

    O’shea, T. & Hoydis, J. An introduction to deep learning for the physical layer.IEEE Transactions on Cognitive Communications and Networking3, 563–575 (2017). 12

    2017

  9. [9]

    Tobin, J.et al.Domain randomization for transferring deep neural networks from simulation to the real world.2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)23–30 (2017)

    2017

  10. [10]

    & Viswanath, P.Fundamentals of wireless communication(Cambridge university press, 2005)

    Tse, D. & Viswanath, P.Fundamentals of wireless communication(Cambridge university press, 2005)

    2005

  11. [11]

    P., Weiger, M., Scheidegger, M

    Pruessmann, K. P., Weiger, M., Scheidegger, M. B. & Boesiger, P. Sense: sensitivity encoding for fast mri.Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine42, 952–962 (1999)

    1999

  12. [12]

    & Mitra, U

    Choudhary, S. & Mitra, U. Identifiability scaling laws in bilinear inverse problems.arXiv preprint arXiv:1402.2637(2014)

    work page arXiv 2014

  13. [13]

    & Bresler, Y

    Li, Y., Lee, K. & Bresler, Y. Identifiability in blind deconvolution with subspace or sparsity constraints.IEEE Transactions on information Theory62, 4266–4275 (2016)

    2016

  14. [14]

    & Kolter, J

    de Avila Belbute-Peres, F., Smith, K., Allen, K., Tenenbaum, J. & Kolter, J. Z. End-to- end differentiable physics for learning and control.Advances in neural information processing systems31(2018)

    2018

  15. [15]

    & Wyffels, F

    Degrave, J., Hermans, M., Dambre, J. & Wyffels, F. A differentiable physics engine for deep learning in robotics.Frontiers in neurorobotics13, 6 (2019)

    2019

  16. [16]

    & Thuerey, N

    Holl, P., Koltun, V. & Thuerey, N. Learning to control pdes with differentiable physics.arXiv preprint arXiv:2001.07457(2020)

    work page arXiv 2001

  17. [17]

    & Willett, R

    Gilton, D., Ongie, G. & Willett, R. Deep equilibrium architectures for inverse problems in imaging.IEEE Transactions on Computational Imaging7, 1123–1133 (2021)

    2021

  18. [18]

    & Salzo, S

    Grazzi, R., Franceschi, L., Pontil, M. & Salzo, S. On the iteration complexity of hypergradient computation.Proceedings of the 37th International Conference on Machine Learning119, 3748–3758 (2020)

    2020

  19. [19]

    Yu, H., Fessler, J. A. & Jiang, Y. Bilevel optimized implicit neural representation for scan- specific accelerated mri reconstruction.arXiv preprint arXiv:2502.21292(2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  20. [20]

    & Karniadakis, G

    Raissi, M., Perdikaris, P. & Karniadakis, G. E. Physics-informed neural networks: A deep learn- ing framework for solving forward and inverse problems involving nonlinear partial differential equations.Journal of Computational physics378, 686–707 (2019)

    2019

  21. [21]

    E.et al.Physics-informed machine learning.Nature Reviews Physics3, 422–440 (2021)

    Karniadakis, G. E.et al.Physics-informed machine learning.Nature Reviews Physics3, 422–440 (2021)

    2021

  22. [22]

    Hemachandra, A., Lau, G. K. R., Ng, S.-K. & Low, B. K. H. Pied: Physics-informed experimen- tal design for inverse problems.International Conference on Representation Learning2025, 99807–99837 (2025)

    2025

  23. [23]

    & Eldar, Y

    Monga, V., Li, Y. & Eldar, Y. C. Algorithm unrolling: Interpretable, efficient deep learning for signal and image processing.IEEE Signal Processing Magazine38, 18–44 (2021)

    2021

  24. [24]

    & LeCun, Y

    Gregor, K. & LeCun, Y. Learning fast approximations of sparse coding.Proceedings of the 27th international conference on international conference on machine learning399–406 (2010)

    2010

  25. [25]

    & Ghanem, B

    Zhang, J. & Ghanem, B. Ista-net: Interpretable optimization-inspired deep network for image compressive sensing.Proceedings of the IEEE conference on computer vision and pattern recognition1828–1837 (2018)

    2018

  26. [26]

    UTOPY: Unrolling Algorithm Learning via Fidelity Homotopy for Inverse Problems

    Jacome, R., Gualdr´ on-Hurtado, R., Suarez-Rodriguez, L. & Arguello, H. Utopy: Unrolling algorithm learning via fidelity homotopy for inverse problems.arXiv preprint arXiv:2509.14394 (2025). 13

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  27. [27]

    Shlezinger, N., Whang, J., Eldar, Y. C. & Dimakis, A. G. Model-based deep learning. Proceedings of the IEEE111, 465–499 (2023)

    2023

  28. [28]

    M.et al.gradsim: Differentiable simulation for system identification and visuomotor control.International Conference on Learning Representations (ICLR)(2021)

    Jatavallabhula, K. M.et al.gradsim: Differentiable simulation for system identification and visuomotor control.International Conference on Learning Representations (ICLR)(2021)

    2021

  29. [29]

    & Zhang, S

    Yang, Y., Gao, F., Ma, X. & Zhang, S. Deep learning-based channel estimation for doubly selective fading channels.IEEE Access7, 36579–36589 (2019)

    2019

  30. [30]

    Evolved universal terrestrial radio access (e-utra); user equipment (ue) radio transmission and reception

    3GPP. Evolved universal terrestrial radio access (e-utra); user equipment (ue) radio transmission and reception. Tech. Rep. TS 36.101, 3rd Generation Partnership Project (3GPP) (2024)

    2024

  31. [31]

    M.Fundamentals of statistical signal processing: estimation theory(Prentice-Hall, Inc., 1993)

    Kay, S. M.Fundamentals of statistical signal processing: estimation theory(Prentice-Hall, Inc., 1993)

    1993

  32. [32]

    & Bahai, A

    Coleri, S., Ergen, M., Puri, A. & Bahai, A. Channel estimation techniques based on pilot arrangement in ofdm systems.IEEE Transactions on Broadcasting48, 223–229 (2002)

    2002

  33. [33]

    Chen, Z., Liu, Z., Geng, X., Zhao, Y. & Wu, H. Attention guided multi-task network for joint cfo and channel estimation in ofdm systems.IEEE Transactions on Wireless Communications 23, 321–333 (2024)

    2024

  34. [34]

    & Gal, Y

    Kendall, A. & Gal, Y. What uncertainties do we need in bayesian deep learning for computer vision?Advances in neural information processing systems30(2017)

    2017

  35. [35]

    Liu, J.et al.Towards out-of-distribution generalization: A survey.arXiv preprint arXiv:2108.13624(2021)

    work page arXiv 2021

  36. [36]

    Knoll, F.et al.fastmri: A publicly available raw k-space and dicom dataset of knee images for accelerated mr image reconstruction using machine learning.Radiology: Artificial Intelligence 2, e190007 (2020)

    2020

  37. [37]

    Uecker, M.et al.Espirit—an eigenvalue approach to autocalibrating parallel mri: where sense meets grappa.Magnetic resonance in medicine71, 990–1001 (2014)

    2014

  38. [38]

    Yaman, B.et al.Self-supervised learning of physics-guided reconstruction neural networks without fully sampled reference data.Magnetic resonance in medicine84, 3172–3191 (2020)

    2020

  39. [39]

    Sriram, A.et al.End-to-end variational networks for accelerated mri reconstruction.Medical Image Computing and Computer Assisted Intervention (MICCAI)12261, 64–73 (2020)

    2020

  40. [40]

    Compressed sensing.IEEE Transactions on Information Theory52, 1289–1306 (2006)

    Donoho, D. Compressed sensing.IEEE Transactions on Information Theory52, 1289–1306 (2006)

    2006

  41. [41]

    Gull, S. F. & Daniell, G. J. Image reconstruction from incomplete and noisy data.Nature272, 686–690 (1978)

    1978

  42. [42]

    & Brox, T

    Ronneberger, O., Fischer, P. & Brox, T. U-net: Convolutional networks for biomedical image segmentation.International Conference on Medical Image Computing and Computer-Assisted Intervention234–241 (2015)

    2015

  43. [43]

    R., Stiefel, E.et al.Methods of conjugate gradients for solving linear systems

    Hestenes, M. R., Stiefel, E.et al.Methods of conjugate gradients for solving linear systems. Journal of research of the National Bureau of Standards49, 409–436 (1952)

    1952

  44. [44]

    Arvinte, M., Vishwanath, S., Tewfik, A. H. & Tamir, J. I. Deep j-sense: Accelerated mri reconstruction via unrolled alternating optimization.Medical Image Computing and Computer Assisted Intervention – MICCAI 2021350–360 (2021)

    2021

  45. [45]

    Kingma, D. P. & Ba, J. Adam: A method for stochastic optimization.International Conference on Learning Representations (ICLR)(2015). 14

    2015