pith. sign in

arxiv: 2604.19843 · v1 · submitted 2026-04-21 · 🧮 math.NA · cs.NA

Mapping-based Hard-constrained Physics-Informed Neural Networks for unbounded wave problems

Tao Zhang , Hanshu Chen , Ilia Marchevsky , Zhuojia Fu This is my paper

Pith reviewed 2026-05-10 02:08 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords physics-informed neural networksunbounded domainswave propagationcoordinate mappinghard constraintsacoustic scatteringelastic wavesfar-field radiation
0
0 comments X

The pith

A coordinate mapping combined with hard physics constraints lets neural networks solve wave problems over infinite domains without boundary loss terms or artificial truncation.

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

The paper presents MH-PINN as a way to handle wave equations that extend to infinity, where standard neural networks struggle with sampling and convergence. A coordinate transformation shrinks the infinite physical space into a finite computational box, removing the need for artificial outer boundaries like perfectly matched layers. Inside this mapped domain the network is structured so that inner boundary conditions and far-field radiation conditions are satisfied exactly by construction, eliminating separate boundary loss terms. The resulting method shows fast convergence on high-frequency acoustic radiation, scattering, and elastic problems while remaining adaptable to different geometries through an inverse factor correction on boundary coefficients.

Core claim

The MH-PINN compactifies an unbounded physical domain into a finite computational domain via coordinate mapping and embeds the governing physics into a hard-constrained network architecture that automatically satisfies both the inner boundary conditions and the far-field radiation conditions, thereby removing all boundary loss terms and the associated truncation errors.

What carries the argument

Coordinate mapping that compactifies the infinite domain together with a physics-based hard-constrained network structure that enforces inner boundary and far-field radiation conditions by architecture.

If this is right

  • High-frequency wave problems converge faster because boundary loss terms are removed.
  • The method applies directly to acoustic radiation, scattering, and elastic wave problems without domain truncation.
  • Geometric adaptability is achieved through the inverse factor correction on boundary coefficients.
  • Artificial truncation errors from perfectly matched layers or other outer boundary treatments are avoided.
  • The approach yields both efficiency gains and exact satisfaction of radiation conditions at infinity.

Where Pith is reading between the lines

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

  • The same mapping-plus-hard-constraint idea could be tested on other unbounded PDEs such as electromagnetic or fluid problems.
  • Real-time engineering simulations over infinite domains become more feasible if the training cost stays low.
  • Accuracy on highly irregular or multiply-connected geometries would test the limits of the current mapping choice.
  • Coupling the mapped network with time-stepping schemes could extend the method to transient unbounded waves.

Load-bearing premise

The chosen coordinate mapping and inverse factor correction will correctly capture asymptotic factors and far-field behavior for arbitrary geometries without introducing significant mapping-induced errors.

What would settle it

Run MH-PINN on a canonical unbounded problem such as plane-wave scattering by a sphere or cylinder for which an exact series solution is known, then check whether the computed far-field pattern matches the analytic result to within a small tolerance at large distances when no boundary loss is used.

Figures

Figures reproduced from arXiv: 2604.19843 by Hanshu Chen, Ilia Marchevsky, Tao Zhang, Zhuojia Fu.

Figure 1
Figure 1. Figure 1: Visualization of spatial mapping transformation from computational domain [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Framework diagram of the proposed mapping-based hard-constrained PINN. [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Predicted real part of sound pressure by Physics-Informed Neural [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Predicted real part of sound pressure by mapping-based hard [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of computational performance between standard PINN and MH [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Predicted solution, exact solution, and error for the MH-PINN global computa [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of calculation results of MH-PINN and FDM in the test area [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of computational performance between standard PINN and MH [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the real sound pressure part of MH-PINN and the MFS reference [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the real sound pressure part of MH-PINN and the MFS reference [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Numerical results for the square boundary scatterer at different wavenumbers. [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Acoustic scattering by a sound-soft sphere ( [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Comparison results of the ellipsoidal scattering field on a three-dimensional [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of surface displacement amplitudes [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
read the original abstract

The aim of this paper is to introduce a Mapping-based Hard-constrained Physics-Informed Neural Network (MH-PINN) for efficiently and accurately solving unbounded wave problems. First, we propose a coordinate mapping technique that compactifies the infinite physical domain into a finite computational space. This effectively resolves the sampling difficulties inherent to standard PINNs in unbounded regions. Additionally, it avoids the artificial truncation errors introduced by traditional methods such as perfectly matched layers. Second, we design a physics-based hard-constrained network structure that automatically satisfies both the inner boundary conditions and the far-field radiation conditions. This structure eliminates boundary loss terms, yielding high computational efficiency and fast convergence, which effectively addresses the challenges of high-frequency problems. Third, we introduce an inverse factor correction for boundary coefficients to address the influence of asymptotic factors,which makes the method highly geometrically adaptable. Finally, we present numerical examples covering various acoustic radiation and scattering scenarios as well as elastic dynamics scenarios to demonstrate the efficiency and accuracy of our algorithm.It highlights its potential for broader applications in the field of computational wave dynamics.

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 / 2 minor

Summary. The manuscript introduces the Mapping-based Hard-constrained Physics-Informed Neural Network (MH-PINN) for unbounded wave problems. It proposes a coordinate mapping to compactify the infinite physical domain into a finite computational domain, a physics-based hard-constrained network architecture that exactly satisfies inner boundary conditions and far-field radiation conditions (eliminating boundary loss terms), and an inverse factor correction for boundary coefficients to handle asymptotic factors and enable geometric adaptability. These are demonstrated through numerical examples on acoustic radiation/scattering and elastic dynamics problems, with claims of improved efficiency and accuracy over standard approaches.

Significance. If the central claims are verified with quantitative evidence, the method could provide a notable advance for PINN-based solvers of unbounded wave problems by removing artificial truncation and boundary-loss penalties while enforcing radiation conditions exactly, potentially improving convergence for high-frequency cases and extending applicability to complex geometries without per-problem tuning.

major comments (3)
  1. [Abstract and Method] Abstract and Method section: the claim that the hard-constrained architecture plus inverse factor correction exactly enforces the Sommerfeld (or equivalent) radiation condition after compactification is load-bearing for the elimination of boundary loss terms, yet no derivation or explicit verification is provided that the correction preserves exact far-field behavior in mapped coordinates for general (non-spherical) geometries or commutes with the coordinate transformation without introducing phase/amplitude errors.
  2. [Numerical Examples] Numerical Examples section: the abstract asserts accuracy and efficiency via numerical examples, but the description provides no quantitative error metrics (e.g., L2 or relative errors against analytic solutions), baseline comparisons (e.g., to standard PINNs, FEM with PML, or other mapping methods), or convergence studies with respect to frequency or network size, leaving the central performance claims without substantiation.
  3. [Method] Method section on coordinate mapping: the construction of the mapping for arbitrary scatterer geometries and its interaction with the hard constraints and inverse correction are not shown to be free of mapping-induced errors in the far field; this is required to support the asserted geometric adaptability and exact satisfaction of unbounded conditions.
minor comments (2)
  1. [Abstract] The abstract would benefit from explicitly naming the governing equations (e.g., Helmholtz or time-harmonic elastic wave equation) and the precise form of the far-field condition being enforced.
  2. [Method] Notation for the inverse factor correction and the mapped coordinates could be clarified with an explicit equation or diagram to aid reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and will revise the manuscript accordingly to strengthen the derivations, quantitative validations, and methodological details.

read point-by-point responses

  1. Referee: [Abstract and Method] Abstract and Method section: the claim that the hard-constrained architecture plus inverse factor correction exactly enforces the Sommerfeld (or equivalent) radiation condition after compactification is load-bearing for the elimination of boundary loss terms, yet no derivation or explicit verification is provided that the correction preserves exact far-field behavior in mapped coordinates for general (non-spherical) geometries or commutes with the coordinate transformation without introducing phase/amplitude errors.

    Authors: We agree that an explicit derivation is necessary to rigorously support the exact enforcement claim. In the revised manuscript, we will add a dedicated derivation subsection in the Method section. This will mathematically show that the inverse factor correction preserves the Sommerfeld condition in mapped coordinates, commutes with the transformation without phase/amplitude errors, and holds for general (non-spherical) geometries, including supporting analysis and verification steps. revision: yes

  2. Referee: [Numerical Examples] Numerical Examples section: the abstract asserts accuracy and efficiency via numerical examples, but the description provides no quantitative error metrics (e.g., L2 or relative errors against analytic solutions), baseline comparisons (e.g., to standard PINNs, FEM with PML, or other mapping methods), or convergence studies with respect to frequency or network size, leaving the central performance claims without substantiation.

    Authors: We acknowledge that the current numerical examples lack sufficient quantitative substantiation. We will revise this section to include L2 and relative error metrics against analytic solutions, direct baseline comparisons to standard PINNs and FEM with PML (and other mapping approaches where relevant), and convergence studies with respect to frequency and network size. These additions will directly support the efficiency and accuracy claims. revision: yes

  3. Referee: [Method] Method section on coordinate mapping: the construction of the mapping for arbitrary scatterer geometries and its interaction with the hard constraints and inverse correction are not shown to be free of mapping-induced errors in the far field; this is required to support the asserted geometric adaptability and exact satisfaction of unbounded conditions.

    Authors: We will expand the Method section with a detailed exposition of the coordinate mapping construction for arbitrary scatterer geometries. We will explicitly analyze and demonstrate its interaction with the hard constraints and inverse correction, including proofs and tests confirming the absence of mapping-induced far-field errors. This will bolster the claims of geometric adaptability and exact unbounded condition satisfaction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained and independent of inputs by construction.

full rationale

The paper's core chain—coordinate mapping to compactify the unbounded domain, followed by a physics-based hard-constrained network architecture that enforces inner boundary and far-field radiation conditions by design, plus an introduced inverse factor correction for asymptotic coefficients—does not reduce any prediction or result to a fitted parameter or self-referential definition. No equations or steps are shown to be equivalent to their inputs by construction, and no load-bearing claims rely on self-citations, uniqueness theorems from the same authors, or smuggled ansatzes. Numerical examples provide external validation rather than tautological confirmation. This is the expected non-finding for a methods paper extending standard PINN and mapping techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit details on free parameters, axioms, or invented entities; assessment is limited to high-level description.

pith-pipeline@v0.9.0 · 5489 in / 971 out tokens · 34673 ms · 2026-05-10T02:08:22.263500+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

35 extracted references · 35 canonical work pages

  1. [1]

    P. Li, J. Shen, Analysis of the scattering by an unbounded rough surface, Mathematical Methods in the Applied Sciences 35 (18) (2012) 2166– 2184

    work page 2012

  2. [2]

    Chandler-Wilde, I

    S. Chandler-Wilde, I. Graham, S. Langdon, E. Spence, Numerical- asymptotic boundary integral methods in high-frequency acoustic scat- tering, Acta Numerica 21 (2012) 89–305

    work page 2012

  3. [3]

    Alves, P

    C. Alves, P. Antunes, Wave scattering problems in exterior domains with the method of fundamental solutions, Numerische Mathematik 156 (2024) 375–394

    work page 2024

  4. [4]

    Zienkiewicz, The Finite Element Method, McGraw-Hill Inc, New York, NY, 1977

    O. Zienkiewicz, The Finite Element Method, McGraw-Hill Inc, New York, NY, 1977

    work page 1977

  5. [5]

    Jiang, P

    X. Jiang, P. Li, W. Zheng, Numerical solution of acoustic scattering by an adaptive dtn finite element method, Communications in Computa- tional Physics 13 (5) (2013) 1277–1244

    work page 2013

  6. [6]

    E. Li, Z. He, X. Xu, et al., Hybrid smoothed finite element method for acoustic problems, Computer Methods in Applied Mechanics and Engineering 283 (2015) 664–688

    work page 2015

  7. [7]

    van der Eerden, Finite element method for acoustic radiation and scattering in an unbounded domain, no

    F. van der Eerden, Finite element method for acoustic radiation and scattering in an unbounded domain, no. NLR-TR 96603 L in Aerospace Engineering Reports, National Aerospace Laboratory, NLR, 1996

    work page 1996

  8. [8]

    Zhang, Z

    Y. Zhang, Z. Ling, H. Du, et al., Finite-difference frequency-domain scheme for sound scattering by a vortex with perfectly matched layers, Mathematics 11 (18) (2023) 3959. 31

    work page 2023

  9. [9]

    Thomas, Numerical partial differential equations: finite difference methods, Springer Science & Business Media, 2013

    J. Thomas, Numerical partial differential equations: finite difference methods, Springer Science & Business Media, 2013

    work page 2013

  10. [10]

    Rabczuk, J

    T. Rabczuk, J. Song, X. Zhuang, et al., Extended finite element and meshfree methods, Academic Press, 2019

    work page 2019

  11. [11]

    Engquist, B

    A. Engquist, B. Majda, Absorbing boundary conditions for the numer- ical simulation of waves, Mathematics of Computation 31 (139) (1977) 629–651

    work page 1977

  12. [12]

    Bérenger, A perfectly matched layer for the absorption of elec- tromagnetic waves, Journal of Computational Physics 114 (2) (1994) 185–200

    J.-P. Bérenger, A perfectly matched layer for the absorption of elec- tromagnetic waves, Journal of Computational Physics 114 (2) (1994) 185–200

    work page 1994

  13. [13]

    Z. Hao, S. Liu, Y. Zhang, C. Ying, Y. Feng, H. Su, J. Zhu, Physics- informed machine learning: A survey on problems, methods and appli- cations, arXiv preprint arXiv:2211.08064 (2022)

    work page arXiv 2022

  14. [14]

    Wu, A direct boundary element method for acoustic radiation and scatteringfrommixedregularandthinbodies, TheJournaloftheAcous- tical Society of America 97 (1) (1995) 84–91

    T. Wu, A direct boundary element method for acoustic radiation and scatteringfrommixedregularandthinbodies, TheJournaloftheAcous- tical Society of America 97 (1) (1995) 84–91

    work page 1995

  15. [15]

    Chandler-Wilde, S

    S. Chandler-Wilde, S. Langdon, L. Ritter, A high-wavenumber boundary-element method for an acoustic scattering problem, Philo- sophical Transactions of the Royal Society of London. Series A: Mathe- matical, Physical and Engineering Sciences 362 (1816) (2004) 647–671

    work page 2004

  16. [16]

    D. Xiu, J. Shen, An efficient spectral method for acoustic scattering from rough surfaces, Communications in Computational Physics 2 (1) (2007) 54–72

    work page 2007

  17. [17]

    Cho, Spectrally-accurate numerical method for acoustic scattering from doubly-periodic 3d multilayered media, Journal of Computational Physics 393 (2019) 46–58

    M. Cho, Spectrally-accurate numerical method for acoustic scattering from doubly-periodic 3d multilayered media, Journal of Computational Physics 393 (2019) 46–58

    work page 2019

  18. [18]

    Shen, L.-L

    J. Shen, L.-L. Wang, Some recent advances on spectral methods for unbounded domains, Communications in Computational Physics 5 (2–

    work page

  19. [19]

    Y. Liu, S. Mukherjee, N. Nishimura, M. Schanz, W. Ye, A. Sutradhar, E. Pan, N. Dumont, A. Frangi, A. Saez, Recent advances and emerg- ing applications of the boundary element method, Applied Mechanics Reviews 64 (3) (2011) 030802

    work page 2011

  20. [20]

    Raissi, P

    M. Raissi, P. Perdikaris, G. Karniadakis, Physics-informed neural net- works: A deep learning framework for solving forward and inverse prob- lems involving nonlinear partial differential equations, Journal of Com- putational Physics 378 (2019) 686–707

    work page 2019

  21. [21]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, L. Yang, Physics-informed machine learning, Nature Reviews Physics 3 (6) (2021) 422–440

    work page 2021

  22. [22]

    Jeong, C

    H. Jeong, C. Batuwatta-Gamage, J. Bai, C. Rathnayaka, Y. Zhou, Y. Gu, An advanced physics-informed neural network-based framework for nonlinear and complex topology optimization, Engineering Struc- tures 322 (2025) 119194

    work page 2025

  23. [23]

    Zhongkai, J

    H. Zhongkai, J. Yao, C. Su, et al., Pinnacle: A comprehensive bench- mark of physics-informed neural networks for solving pdes, Advances in Neural Information Processing Systems 37 (2024) 76721–76774

    work page 2024

  24. [24]

    K. Luo, J. Zhao, Y. Wang, et al., Physics-informed neural networks for pde problems: a comprehensive review, Artificial Intelligence Review 58 (10) (2025) 323

    work page 2025

  25. [25]

    Z. Tang, Z. Fu, S. Reutskiy, An extrinsic approach based on physics- informed neural networks for pdes on surfaces, Mathematics 10 (16) (2022) 2861

    work page 2022

  26. [26]

    Z. Fu, W. Xu, S. Liu, Physics-informed kernel function neural networks for solving partial differential equations, Neural Networks 172 (2024) 106098

    work page 2024

  27. [27]

    M. Xia, L. Böttcher, T. Chou, Spectrally adapted physics-informed neu- ralnetworksforsolvingunboundeddomainproblems, MachineLearning: Science and Technology 4 (2) (2023) 025024. 33

    work page 2023

  28. [28]

    P. Ren, C. Rao, S. Chen, et al., Seismicnet: Physics-informed neural networks for seismic wave modeling in semi-infinite domain, Computer Physics Communications 295 (2024) 109010

    work page 2024

  29. [29]

    Krishnapriyan, A

    A. Krishnapriyan, A. Gholami, S. Zhe, R. Kirby, M. W. Mahoney, Char- acterizing possible failure modes in physics-informed neural networks, Advances in neural information processing systems 34 (2021) 26548– 26560

    work page 2021

  30. [30]

    Rahaman, A

    N. Rahaman, A. Baratin, D. Arpit, F. Draxler, M. Lin, F. Hamprecht, Y. Bengio, A. Courville, On the spectral bias of neural networks, in: Proceedings of the 36th International Conference on Machine Learning (ICML), 2019, pp. 5301–5310

    work page 2019

  31. [31]

    S. Wang, H. Wang, P. Perdikaris, On the eigenvector bias of fourier fea- ture networks: From regression to solving multi-scale pdes with physics- informedneuralnetworks, ComputerMethodsinAppliedMechanicsand Engineering 384 (2021) 113938

    work page 2021

  32. [32]

    S. Wang, Y. Teng, P. Perdikaris, Understanding and mitigating gradient flow pathologies in physics-informed neural networks, SIAM Journal on Scientific Computing 43 (5) (2021) A3055–A3081

    work page 2021

  33. [33]

    S. Nair, T. F. Walsh, G. Pickrell, F. Semperlotti, Acoustic scattering simulations via physics-informed neural network, in: Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems 2024, Vol. 12949, SPIE, 2024, pp. 138–145

    work page 2024

  34. [34]

    S. Wang, X. Yu, P. Perdikaris, When and why pinns fail to train: A neural tangent kernel perspective, Journal of Computational Physics 449 (2022) 110768

    work page 2022

  35. [35]

    P. M. Morse, K. U. Ingard, Theoretical acoustics, Princeton university press, 1986. 34

    work page 1986