A Morphology-Adaptive Random Feature Method for Inverse Source Problem of the Helmholtz Equation

Haijun Yu; Jingrun Chen; Xinwei Hu

REVIEW 5 major objections 5 minor 1 cited by

A Morphology-Adaptive Random Feature Method for Inverse Source Problem of the Helmholtz Equation

T0 review · 5 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A two-phase random feature method reconstructs sharp and disjoint Helmholtz sources at a fraction of traditional cost.

desk verdict Solid two-phase RFM paper with a real 'shape-prior-assisted' caveat and a repairable proof gap — worth a serious referee. read the letter →

arxiv 2510.09213 v2 pith:ON5XS4RC submitted 2025-10-10 math-ph math.MP

Xinwei Hu , Jingrun Chen , Haijun Yu This is my paper · ORCID

classification math-phmath.MP MSC 35R3065N35

keywords HelmholtzequationinversesourceproblemrandomfeaturemethodTikhonovregularizationadaptivequadraturemorphologybasismulti-frequencyintegral

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes the Morphology-Adaptive Random Feature Method (MA-RFM) for the inverse source problem of the Helmholtz equation, targeting sources with discontinuous or geometrically complex supports. The method first reformulates the problem as an integral equation using the fundamental solution, then applies an adaptive iteration that concentrates quadrature points where the source magnitude or gradient is large. In a second phase, it adds morphology-matched smooth basis functions, such as sigmoid or Gaussian bumps, to represent sharp interfaces. The authors report relative L2 errors around 10–16% even with 10% measurement noise, and computational speedups of two to three orders of magnitude over uniform-grid Fourier or fixed random-feature approaches. If correct, this offers a mesh-free, convex alternative to neural-network solvers for a class of ill-posed inverse problems.

What carries the argument

The central machinery is a two-phase basis- and mesh-adaptive random feature method built on the integral equation formulation of the Helmholtz equation. The fundamental solution (Hankel function in 2D, exponential over distance in 3D) serves as a kernel that makes the radiation condition a hard constraint and reduces the unknown to the source term alone. The first phase uses a dual-criterion indicator combining the absolute value and gradient of the current source reconstruction to iteratively refine a tensor-product Gauss quadrature mesh, concentrating samples where the source is large or varies rapidly. The second phase enriches the random-feature span with smooth approximations to level-

What would settle it

Run the method on a source whose support is a shape not in the paper's families, such as a thin spiral arm or a disconnected set of random polygons, with 10% noise; if the relative L2 error does not improve over the plain IA-RFM (or stays above roughly 20%), the adaptive-morphology claim loses its support.

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Extended reading notes

Core claim

The authors claim that the multi-frequency Helmholtz inverse source problem, even when the source has discontinuous or complicated geometry, can be solved accurately and cheaply by a two-stage random feature algorithm. In the first stage, IA-RFM iteratively refines a Gauss quadrature mesh using the absolute value and gradient of the current source approximation, concentrating integration points near the true support. In the second stage, based on the detected support and its boundary, additional basis functions are constructed from signed-distance level sets of simple shapes (circle, rectangle, kidney, torus) with sigmoid, tanh, Gaussian, or ReLU profiles. The resulting linear system is Tikh

Load-bearing premise

The second phase assumes that the user-chosen template family of shapes, together with the detection thresholds, identifies the true support well enough that the added basis functions can represent it; if the true shape is not in the family, the enrichment cannot improve the reconstruction.

Editorial extensions

If this is right

If correct, discontinuous and complex-shaped sources can be recovered from multi-frequency Cauchy data with only boundary measurements, without dense volumetric meshes.
The two to three orders of magnitude reduction in integration points would make 3D inverse source problems practical on ordinary hardware.
Because the discretized problem is strictly convex, the method avoids local minima that plague neural-network solvers and provides a unique solution for any regularization parameter.
The stability bound gives a principled way to choose the regularization parameter from the estimated noise level and model error, which could be extended to other linear inverse problems.
The morphology-adaptation idea suggests that a small dictionary of level-set-based basis functions can vastly improve random feature methods for PDEs with singular or piecewise-smooth coefficients.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The method's success depends on the user choosing a template family of shapes that matches the true source geometry; the paper tests only shapes from the chosen families, so a source outside these families might not benefit from the second phase at all.
The integral-equation reformulation is not limited to the Helmholtz equation: any linear PDE with a known fundamental solution could be treated similarly, though the adaptive morphology step would need a shape dictionary for that problem's singularities.
A natural testable extension is to let the second phase search over a broader dictionary of level-set functions (for example, arbitrary implicit curves) rather than pre-selected simple shapes, and to measure how the reconstruction improves when the dictionary contains the true shape versus when it does not.
The stability theorem suggests that the reconstruction error scales almost linearly with noise when the source-condition parameter is near one; this could guide practical error bars for reconstructions in medical or acoustic imaging.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a useful numerical paper that does what it claims on the tested benchmarks, but the 'morphology-adaptive' part is closer to 'shape-prior-assisted' — in every experiment the template family is chosen to match the true support, and there is no out-of-family test. The proof of Theorem 2.1 has a gap, though it is easily repairable.

What is genuinely new: combining the integral-equation formulation of the Helmholtz inverse source problem with RFM and an adaptive quadrature (IA-RFM), then enriching the basis with morphology functions whose parameters are estimated from the rough first-stage solution. That two-phase synthesis is not in the cited RFM literature. The benchmark set is broad — disc, two circles, rectangle+circle, kidney, 3D cones, torus, limited aperture — and the error tables are honestly reported. The Tikhonov analysis in Theorem 3.1 is standard but clean, and the controlled verification in Example 4.1 is a nice addition.

The soft spots, in order of importance. First, the template family F in Algorithm 2 is user-chosen per test case and always matches the true shape. The first stage does detect location and scale — even though that detection is biased, e.g. r̂=0.219 vs 0.2 in Example 4.3 — but it does not choose the topology of the support. That makes 'morphology-adaptive' an overstatement; the real claim is 'adaptive within a shape family the user provides.' An out-of-family test (say an ellipse or star-shaped support) is needed before the general claim can be trusted. Second, no seeds or variance are reported for a method that is random by construction. Third, data are generated with the same integral operator used in the inversion (mild inverse crime); independent forward data would strengthen the numbers. Fourth, the proof of Theorem 2.1 as written is wrong: knowing the restriction of \hat{S} to one complex line vanishes at z=k_j does not imply \hat{S} vanishes as a function of several variables. The fix is to run the same argument along every direction, so the theorem is likely true, but the current text is not a proof. Fifth, the conclusion's 'two to three orders of magnitude' cost reduction is supported mainly by one comparison against the Fourier method, and 10–16% relative errors at 10% noise are decent, not 'high accuracy.'

Who should read it: anyone working on random feature methods or mesh-free inverse problems. It deserves a serious referee. I would send it to review with a request for multi-seed statistics, code/data release, an out-of-family morphology test, and a corrected proof.

Referee Report

5 major / 5 minor

Summary. The manuscript proposes a two-phase random feature method for the multi-frequency Helmholtz inverse source problem. The first phase (IA-RFM) uses an iterative adaptive integration mesh, driven by the current source estimate's magnitude and gradient, to concentrate quadrature points near the support. The second phase (MA-RFM) adds hand-designed 'morphology' basis functions -- circle sigmoids, Gaussian bumps, cone-ReLUs, torus SDFs, etc. -- chosen after inspecting the IA-RFM solution, and then retrains a convex Tikhonov-regularized least-squares problem. Theoretical sections prove conditional uniqueness for multi-frequency Cauchy data (Theorem 2.1) and an optimal convergence-rate bound for the Tikhonov solution (Theorem 3.1). Numerical experiments cover smooth, discontinuous, disjoint, and 3D sources, reporting relative L2 errors of roughly 1.35%--16% at 5--10% noise, and compare against IRFM, a Fourier method, and PINN.

Significance. If the numerical results are reproducible, the integral-equation formulation with a hard radiation condition and a convex random-feature solve is a useful and efficient alternative to PINN-type approaches for this inverse-source class. The paper's honest reporting of noise sensitivity, its inclusion of 3D benchmarks, and its theoretical stability bound are strengths. However, the central "morphology-adaptive" claim is not yet substantiated: every experiment supplies the template family that matches the true source, no out-of-family shape is tested, and the reported gains are not isolated from the adaptive-integration component. The uniqueness proof in Appendix A also has a serious gap. The manuscript is a promising contribution, but it needs substantial revision before the advertised claims can be accepted.

major comments (5)

[Appendix A, proof of Theorem 2.1] The final step of the proof is not valid as written. After showing that the Fourier transform \hat S vanishes on the spheres |\xi|=k_j, the proof restricts to one complex line f(z)=\hat S(z,0,\dots,0) and then concludes that '\hat S is zero on the whole complex domain C^d.' Vanishing of f on a set with an accumulation point only implies that the marginal \int S(x_1,x') dx' is zero, not that S is identically zero. To obtain S=0 one would need to repeat the argument in every direction (or use a genuinely multi-dimensional uniqueness theorem). Since Theorem 2.1 is stated as the theoretical basis for the multi-frequency setup, this gap must be repaired before the theorem can be used.
[§4, Tables 2 and 5] The accuracy improvement is not isolated to the morphology stage. MA-RFM differs from the IRFM baseline in two ways: adaptive integration (IA-RFM) and the added morphology basis functions. For discontinuous sources, Tables 2 and 5 compare IRFM directly with MA-RFM, but no IA-RFM-only column is reported. Thus the gains (e.g., 22.78% to 10.50% in Table 2) conflate the effect of adaptive quadrature with the effect of the morphology bases. An ablation with IA-RFM alone on the same examples is necessary to support the paper's claim that the second stage is responsible for the improved reconstruction of discontinuities.
[Algorithm 2, line 8; Examples 4.3–4.8] The 'morphology-adaptive' property is currently conditional on the user supplying a template family F that contains the true shape. In every benchmark, the basis family is chosen to match the unknown source: circle-sigmoid (Ex. 4.3), truncated Gaussian (Ex. 4.4), circle+rectangle (Ex. 4.5), kidney SDF (Ex. 4.6), cone-ReLU (Ex. 4.7), and torus SDF (Ex. 4.8). In Example 4.8 the text explicitly says the choice was prompted by the detected point cloud forming a 'toroidal donut-like structure.' No test uses a source whose shape is not in F, and Algorithm 2 line 8 does not describe a mechanism for selecting among candidate families. The claimed adaptivity therefore reduces to a user-supplied shape prior. Please provide either an automatic family-selection rule or an out-of-family benchmark that demonstrates robustness when the template is misspecified.
[§5 Conclusion] The claim that the method 'reduces the computational cost by two to three orders of magnitude compared to traditional methods' is not supported by the data. Table 1 reports no wall-clock time; the reported metric n_integral is ambiguous (entries like '800 2' suggest a formatting error for powers). The number of integration points alone does not determine complexity, since the linear solve cost depends on the number of basis functions and observation points as well. Please report actual timings or a clearly specified operation count for each method, and make the units/format of Table 1 unambiguous.
[§4 (all numerical experiments)] Random Feature Methods are stochastic, but every reported error is a single run with no seed specified and no variance estimate. The claimed accuracy differences between methods could be within run-to-run variability, especially at low noise levels. At minimum, provide mean and standard deviation over several random seeds (e.g., 5--10 seeds) for the main comparisons in Tables 2, 5, and 6. Without this, the results are not statistically reproducible.

minor comments (5)

[Throughout] The radiation condition is repeatedly spelled 'Sommerfield'; it should be 'Sommerfeld'.
[Example 4.8] The domain definitions are 2D: V0 = [-0.5,0.5]×[-0.5,0.5] and Ω = [-0.75,0.75]×[-0.75,0.75], but the example is 3D. These should be Cartesian products of three intervals.
[Proof of Theorem 3.1] In the displayed calculation for the second term, the source condition is written as s* = Ψ_M^T Ψ_M w (without the exponent ν). Also 'singular value deposition' should be 'singular value decomposition'.
[Example 4.9, Table 7] The relative error at θ_max=π/2 is 22.79%, which is large. The phrase 'still roughly determine the locations' is fair, but the abstract's claim of superior performance under limited aperture should be tempered.
[Algorithm 1, line 1] The notation 'n_d Gauss points in the reference cell' is later used as n^d; please clarify in the text or notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: MA-RFM's adaptive enrichment is data-driven basis selection, not a reduction of the result to its inputs.

full rationale

The paper's derivation chain is: (1) reformulate the Helmholtz inverse source problem as a linear integral equation using the fundamental solution (Eqs. (2)-(3), (11)-(13)); (2) discretize with RFM and adaptive quadrature (IA-RFM, Algorithm 1, Eqs. (26)-(29)); (3) enrich the basis using morphology detected from the preliminary solution (Eqs. (31)-(36), Algorithm 2); and (4) prove uniqueness and Tikhonov stability (Theorems 2.1 and 3.1). None of these steps defines the output in terms of the target quantity or fits a parameter and then reports it as an independent prediction. The morphology basis family F is a user-supplied input: Algorithm 2 says 'Select the basis type T_k in F based on the characteristics of Q', and Example 4.8 states that the detected structure 'prompts our choice of the similarly shaped level set function'. Thus the method is adaptive only within the supplied family, and every benchmark chooses a family containing the true morphology. That is an external-validity/generalization limitation, not circularity: the final coefficients are still obtained by fitting the boundary data in the enriched space, and no equation in the paper reduces the reconstruction or the error bound to the detected parameters alone. Example 4.1 is a synthetic self-consistency check constructed to satisfy the assumptions of Theorem 3.1, not an independent prediction. Theorem 2.1 relies on external results (Holmgren's theorem, Paley-Wiener), and Theorem 3.1 is a standard source-condition regularization bound. The self-citations to the RFM framework are to a method that is reproduced in the paper and used as a baseline; they are not load-bearing circularity. The honest non-finding is therefore appropriate.

Assumptions & free parameters 8 free parameters · 8 assumptions · 1 invented entities

The central claim rests on standard uniqueness machinery (Rellich, Holmgren, Paley–Wiener), the problem's modeling assumptions (compactly supported L² source separated from the boundary), standard Tikhonov theory with a source condition, and three unproven heuristics: the adaptive cell-indicator/error correlation, the morphology template bank matching the unknown source, and the invalid analytic-continuation step in Appendix A. Free parameters are numerous (λ_reg, thresholds, tolerances, basis counts, K ranges, FWHM decay rates) and are tuned per example; measured performance is therefore partly a function of hand-tuning.

free parameters (8)

Tikhonov parameter λ_reg = 1e-34 to 1e-1, chosen per example via L-curve (Tables 2–7)
Selected using the measured data itself; controls the balance between data fit and solution norm throughout the experiments.
Mesh refinement threshold δ (and constant c) = δ_k = c · mean indicator, c not reported
Algorithm 1, Step 3; determines which cells are refined in IA-RFM; hand-set.
Detection thresholds t_abs, t_grad = e.g., t_grad=1/2, 1/2.8, 1/3; t_abs=1/2.2
Eq. (33) and Algorithm 2; decide which points seed the morphology basis. Changed per example.
Shape perturbation tolerances ε_c, ε_r, ε_width, ε_height = 3%–20% depending on example
Control the random spread of added basis centers, radii, widths/heights around detected features (§3.2).
Basis hardness K = K ~ U(1000, 20000–50000)
Sigmoid/tanh sharpness of the jump templates; sampled randomly and fixed; affects conditioning and accuracy.
Basis counts M_0, M_1, M_2 = 800–6400 per example
Hand-set network widths; accuracy saturates with M for IRFM (Table 2), so M is a free capacity parameter.
Refinement weights γ_abs, γ_grad = not numerically reported
Algorithm 1, Step 2: relative weighting of amplitude vs gradient indicators in the refinement criterion.
Gaussian decay rates v_k^(j) = v ~ U(0.5 v_min, 2 v_max), v from FWHM of first-stage solution
Example 4.7: decay rates of added Gaussian/conical bases are derived from the IA-RFM solution, i.e., fitted to the data.

assumptions (8)

standard math Exterior uniqueness theorem for radiating solutions (Rellich-type, McLean Thm 9.10)
Used in Appendix A: radiating solution with zero Dirichlet data on Γ vanishes outside Ω.
standard math Holmgren's uniqueness theorem
Used in Appendix A case (d) to extend vanishing from an open subset Γ0 to the exterior.
standard math Paley–Wiener theorem for compactly supported L² sources
Appendix A: \hat S extends to an entire function of exponential type.
ad hoc to paper Vanishing of \hat S on one complex line implies vanishing on all of C^d
Appendix A, final step: 'From unique continuation for analytic functions, \hat S(ξ) is zero on the whole complex domain C^d.' This inference is false as written (an entire function of several variables can vanish on a line); the theorem is repairable by slicing direction-by-direction, but the stated step is an unproven (invalid) assumption.
domain assumption Source is compactly supported, separated from Γ, and contained in the quadrature box V_0; S ∈ L²
Used throughout Eqs. (1)–(2) and in the numerical integration over V_0 (§3.1).
domain assumption Tikhonov source condition s* = (Ψ^T Ψ)^ν w, 0 < ν ≤ 1
Theorem 3.1(b): the convergence rate O((δ+η)^{2ν/(2ν+1)}) holds only under this standard regularization-theory condition, which the paper assumes without verifying for the actual problems.
ad hoc to paper Cells with large |S| or |∇S| dominate quadrature error
Algorithm 1, Steps 2–3: the adaptive integration strategy rests on an unproven heuristic correlation between solution magnitude/gradient and quadrature error.
ad hoc to paper The morphology template bank F contains the true source's shape
§3.2 and Algorithm 2, line 8: second-stage gains assume the user selects the matching family (circle-sigmoid, rectangle-sigmoid, truncated Gaussian, cone-ReLU, torus SDF). No test covers an unknown morphology.

invented entities (1)

Morphology template basis functions (Φ_jump, Φ_Gauss, Φ_noise, cone, torus SDF)
purpose: Posterior morphology basis enriching the RFM approximation space V into V' (Eq. 36) to represent sharp source boundaries.
These are algorithmic basis constructs, not new physical entities. Their numerical support in Section 4 is real, but their selection is user-driven and their success depends on the template family matching the unknown source; hence independent_evidence is false.

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Pith. "Pith review of A Morphology-Adaptive Random Feature Method for Inverse Source Problem of the Helmholtz Equation." pith.science (2026). https://pith.science/paper/ON5XS4RC

@misc{pith2026251009213,
  author       = {Pith},
  title        = {Pith review of: A Morphology-Adaptive Random Feature Method for Inverse Source Problem of the Helmholtz Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ON5XS4RC}},
  note         = {Machine review of arXiv:2510.09213}
}

read the original abstract

The inverse source problem for the Helmholtz equation poses significant challenges, particularly when sources exhibit complex or discontinuous geometries. Traditional numerical methods suffer from prohibitive computational costs, while machine learning-based approaches such as Physics-Informed Neural Networks (PINNs) and the Random Feature Method (RFM) -- though computationally efficient for inverse problems -- lack the intrinsic machinery to handle the sharp morphological features in such singular problems, leading to inaccurate solutions. To address this issue, we propose the Morphology-Adaptive Random Feature Method (MA-RFM), a novel two-phase framework that adaptively locates critical regions and adds morphology activation functions for tackling the multi-frequency inverse source problem with complex geometry. Our framework recasts the ill-posed inverse problem into a well-posed, strictly convex optimization problem by reformulating the governing Helmholtz equation as a Tikhonov-regularized integral equation via its fundamental solution. In the first stage, the Integral Adaptive RFM (IA-RFM), employs an adaptive algorithm to rapidly localize the source support, thereby reducing computational overhead and accelerating convergence. In the second stage, posterior geometric information is progressively integrated into the solver via hybrid basis functions, enabling a precise reconstruction of complex morphologies. The MA-RFM extends the capabilities of RFM to handle PDEs with singular solutions while preserving its mesh-free efficiency. We demonstrate the superior performance of our approach through ample challenging 2D and 3D benchmark problems, even under limited and noisy measurement conditions, highlighting its robustness and accuracy in reconstructing complex and disjoint sources.

Figures

Figures reproduced from arXiv: 2510.09213 by the authors.

**Figure 2.** Example 4.1: Left) Prior parameters: ηM versus δall; Middle) λ 2 reg, ∥s δ − s ∗∥2 and the theoretical error bounds; Right) ∥S δ − S ∗ M∥2 for the training set versus the test set. Example 4.2. Mountain shape source function In the example, we aim to reconstruct a mountain-shaped source function S(x1, x2) = 1.1e −200((x1−0.01)2 )+(x2−0.12)2 − 100(x 2 2 − x 2 1 )e −90(x 2 1+x 2 2 ) . We adopt the data generation proc… view at source ↗

**Figure 3.** Example 4.1: Prior parameters: IRFM results with [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

Figures from the paper (13 more)

**Figure 4.** Figure 4: Example 4.2: Mountain shape source function: (a). PINN solution with [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

**Figure 5.** Figure 5: Example 4.2: Mountain shape source function: IA-RFM results with [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: Example 4.3: Discontinuous source: MA-RFM results with [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 7.** Figure 7: Example 4.4: Two circle sources: grid division diagram for IA-RFM with [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]

**Figure 8.** Figure 8: Example 4.4: Two circle sources: MA-RFM results with [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]

**Figure 9.** Figure 9: Example 4.5: One rectangle and one circle: grid division diagram for IA-RFM with [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]

**Figure 10.** Figure 10: Example 4.5: One rectangle and one circle: (a), (b) show the gradient of points from [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]

**Figure 11.** Figure 11: Example 4.6: Complex geometry: Adaptive meshing Cell [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]

**Figure 12.** Figure 12: Example 4.6: Complex geometry: MA-RFM results with [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]

**Figure 13.** Figure 13: M0 = 3600, Mk,ReLU = Mk,exp = 900, v (j) k ∼ U(5.775, 25.2). r (j) k ∼ U(0.18, 0.26), cˆ (j) k ∼ U(ˆck − 0.02, cˆk + 0.02), k = 1, 2. The initial mesh Nx1 = Nx2 = 4 with nx1 = nx2 = 3, max iter=10. See [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]

**Figure 14.** Figure 14: Example 4.7: Three-dimensional C 0 source: MA-RFM results with δ = 5%, M0 = 3600, Mk,ReLU = Mk,exp = 900, λ 2 reg = 1e-4 yield 1.35% El 2 (S). Example 4.8. Three-dimensional donut Next, consider a 3-D “donut” segment source, which can be parameterized as follows:  …

**Figure 15.** Figure 15: Example 4.8: Three-dimensional donut: (a),(b),(c) show the gradient of points from [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]

**Figure 16.** Figure 16: Example 4.9 : Limited aperture: effect of the measurement aperture [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]

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Reference graph

Works this paper leans on

41 extracted references · 2 linked inside Pith · cited by 1 Pith paper

[1]

The inverse source problem for Maxwell’s equations

Richard Albanese and Peter B Monk. The inverse source problem for Maxwell’s equations. Inverse Problems, 22(3):1023, 2006

2006
[2]

Optical tomography in medical imaging.Inverse Problems, 15(2):R41, 1999

Simon R Arridge. Optical tomography in medical imaging.Inverse Problems, 15(2):R41, 1999

1999
[3]

The unique determination of neuronal currents in the brain via magnetoencephalography.Inverse Problems, 20(4):1067, 2004

AS Fokas, Y Kurylev, and V Marinakis. The unique determination of neuronal currents in the brain via magnetoencephalography.Inverse Problems, 20(4):1067, 2004

2004
[4]

Application of inverse source concepts to photoacoustic tomography.Inverse Problems, 23(6):S21, 2007

Mark A Anastasio, Jin Zhang, Dimple Modgil, and Patrick J La Rivi` ere. Application of inverse source concepts to photoacoustic tomography.Inverse Problems, 23(6):S21, 2007

2007
[5]

Inverse source problem in nonhomoge- neous background media.SIAM Journal on Applied Mathematics, 67(5):1353–1378, 2007

Anthony J Devaney, Edwin A Marengo, and Mei Li. Inverse source problem in nonhomoge- neous background media.SIAM Journal on Applied Mathematics, 67(5):1353–1378, 2007

2007
[6]

Multidimensional inverse scattering problems

Alexander G Ramm. Multidimensional inverse scattering problems. InDIPED-99. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory. Proceedings of 4th Interna- tional Seminar/Workshop (IEEE Cat. No. 99TH8402), page 22. IEEE, 1999

1999
[7]

Thermoacoustic tomography with variable sound speed.Inverse Problems, 25(7):075011, 2009

Plamen Stefanov and Gunther Uhlmann. Thermoacoustic tomography with variable sound speed.Inverse Problems, 25(7):075011, 2009

2009
[8]

Inverse scattering.Sect

NN Bojarski. Inverse scattering.Sect. II, Third Quarterly Company Report, Naval Air Systems Command Contract N00019–73-C-0312, pages 19–73, 1973

1973

Show all 41 references

[9]

Helmholtz surface wave tomography for isotropic and azimuthally anisotropic structure.Geophysical Journal International, 186(3):1104–1120, 2011

Fan-Chi Lin and Michael H Ritzwoller. Helmholtz surface wave tomography for isotropic and azimuthally anisotropic structure.Geophysical Journal International, 186(3):1104–1120, 2011

2011
[10]

Image formation with arbitrary holographic type surfaces.Physics Letters A, 29(4):193–194, 1969

RP Porter. Image formation with arbitrary holographic type surfaces.Physics Letters A, 29(4):193–194, 1969

1969
[11]

Diffraction-limited, scalar image formation with holograms of arbitrary shape.Journal of the Optical Society of America, 60(8):1051–1059, 1970

Robert P Porter. Diffraction-limited, scalar image formation with holograms of arbitrary shape.Journal of the Optical Society of America, 60(8):1051–1059, 1970

1970
[12]

Nonuniqueness in the inverse source problem in acoustics and electromagnetics.Journal of Mathematical Physics, 18(2):194–201, 1977

Norman Bleistein and Jack K Cohen. Nonuniqueness in the inverse source problem in acoustics and electromagnetics.Journal of Mathematical Physics, 18(2):194–201, 1977

1977
[13]

Number 34 in CBMS-NSF Regional Conference Series in Applied Mathematics

Victor Isakov.Inverse Source Problems. Number 34 in CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1990

1990
[14]

A multi-frequency inverse source problem.Journal of Differential Equations, 249(12):3443–3465, 2010

Gang Bao, Junshan Lin, and Faouzi Triki. A multi-frequency inverse source problem.Journal of Differential Equations, 249(12):3443–3465, 2010

2010
[15]

Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data.Contemp

Gang Bao, Junshan Lin, Faouzi Triki, et al. Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data.Contemp. Math, 548:45–60, 2011. 29

2011
[16]

Acoustic source identification using multiple frequency information.Inverse Problems, 25(11):115005, 2009

Matthias Eller and Nicolas P Valdivia. Acoustic source identification using multiple frequency information.Inverse Problems, 25(11):115005, 2009

2009
[17]

Fourier method for solving the multi-frequency inverse source problem for the Helmholtz equation.Inverse Problems, 31(3):035007, 2015

Deyue Zhang and Yukun Guo. Fourier method for solving the multi-frequency inverse source problem for the Helmholtz equation.Inverse Problems, 31(3):035007, 2015

2015
[18]

Maziar Raissi, Paris Perdikaris, and George E Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.Journal of Computational Physics, 378:686–707, 2019

2019
[19]

DGM: A deep learning algorithm for solving partial differential equations.Journal of Computational Physics, 375:1339–1364, 2018

Justin Sirignano and Konstantinos Spiliopoulos. DGM: A deep learning algorithm for solving partial differential equations.Journal of Computational Physics, 375:1339–1364, 2018

2018
[20]

The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems.Communications in Mathematics and Statistics, 6(1):1–12, 2018

Bing Yu Wweinan E. The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems.Communications in Mathematics and Statistics, 6(1):1–12, 2018

2018
[21]

Ameya D Jagtap and George Em Karniadakis. Extended physics-informed neural networks (XPINNs): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations.Communications in Computational Physics, 28(5), 2020

2020
[22]

hp-VPINNs: Variational physics-informed neural networks with domain decomposition.Computer Methods in Applied Mechanics and Engineering, 374:113547, 2021

Ehsan Kharazmi, Zhongqiang Zhang, and George Em Karniadakis. hp-VPINNs: Variational physics-informed neural networks with domain decomposition.Computer Methods in Applied Mechanics and Engineering, 374:113547, 2021

2021
[23]

Solving the inverse source problem of the fractional Poisson equation by MC-fPINNs.arXiv preprint arXiv:2407.03801, 2024

Rui Sheng, Peiying Wu, Jerry Zhijian Yang, and Cheng Yuan. Solving the inverse source problem of the fractional Poisson equation by MC-fPINNs.arXiv preprint arXiv:2407.03801, 2024

2024 arXiv
[24]

Ham- precht, Yoshua Bengio, and Aaron Courville

Nasim Rahaman, Aristide Baratin, Devansh Arpit, Felix Draxler, Min Lin, Fred A. Ham- precht, Yoshua Bengio, and Aaron Courville. On the spectral bias of neural networks. In Proceedings of the 36th International Conference on Machine Learning, volume 97 ofProceed- ings of Machi...

2019
[25]

Frequency principle: Fourier analysis sheds light on deep neural networks.Commun

Zhi-Qin John Xu, Yaoyu Zhang, Tao Luo, Yanyang Xiao, and Zheng Ma. Frequency principle: Fourier analysis sheds light on deep neural networks.Commun. Comput. Phys., 28(5):1746– 1767, 2020

2020
[26]

Bi-GreenNet: learning Green’s functions by boundary integral network.Communications in Mathematics and Statistics, 11(1):103–129, 2023

Guochang Lin, Fukai Chen, Pipi Hu, Xiang Chen, Junqing Chen, Jun Wang, and Zuoqiang Shi. Bi-GreenNet: learning Green’s functions by boundary integral network.Communications in Mathematics and Statistics, 11(1):103–129, 2023

2023
[27]

Suchuan Dong and Zongwei Li. Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations.Computer Methods in Applied Mechanics and Engineering, 387:114129, 2021

2021
[28]

An extreme learning machine-based method for computa- tional PDEs in higher dimensions.Computer Methods in Applied Mechanics and Engineering, 418:116578, 2024

Yiran Wang and Suchuan Dong. An extreme learning machine-based method for computa- tional PDEs in higher dimensions.Computer Methods in Applied Mechanics and Engineering, 418:116578, 2024

2024
[29]

Bridging traditional and machine learning- based algorithms for solving PDEs: the random feature method.J Mach Learn, 1(3):268–298, 2022

Jingrun Chen, Xurong Chi, Zhouwang Yang, et al. Bridging traditional and machine learning- based algorithms for solving PDEs: the random feature method.J Mach Learn, 1(3):268–298, 2022

2022
[30]

Physics-informed extreme learning machine (PIELM)–a rapid method for the numerical solution of partial differential equations.Neurocomputing, 391:96–118, 2020

Vikas Dwivedi and Balaji Srinivasan. Physics-informed extreme learning machine (PIELM)–a rapid method for the numerical solution of partial differential equations.Neurocomputing, 391:96–118, 2020. 30

2020
[31]

The random feature method for solving interface problems.Computer Methods in Applied Mechanics and Engineering, 420:116719, 2024

Xurong Chi, Jingrun Chen, and Zhouwang Yang. The random feature method for solving interface problems.Computer Methods in Applied Mechanics and Engineering, 420:116719, 2024

2024
[32]

Adaptive feature capture method for solving partial differential equations with low regularity solutions.arXiv preprint arXiv:2507.12941, 2025

Yangtao Deng, Qiaolin He, and Xiaoping Wang. Adaptive feature capture method for solving partial differential equations with low regularity solutions.arXiv preprint arXiv:2507.12941, 2025

2025 arXiv
[33]

Theh,pandhpversions of the finite element method in 1 dimension: Part I

Wen-zhuang Gui and Ivo Babuˇ ska. Theh,pandhpversions of the finite element method in 1 dimension: Part I. the error analysis of thep-version.Numerische Mathematik, 49(6):577–612, 1986

1986
[34]

Theh,pandhpversions of the finite element method in 1 dimension: Part II

Wen-zhuang Gui and Ivo Babuˇ ska. Theh,pandhpversions of the finite element method in 1 dimension: Part II. the error analysis of theh-andhpversions.Numerische Mathematik, 49(6):613–657, 1986

1986
[35]

The p-version of the finite element method

Ivo Babuska, Barna A Szabo, and I Norman Katz. The p-version of the finite element method. SIAM Journal on Numerical Analysis, 18(3):515–545, 1981

1981
[36]

Thep-andhpversions of the finite element method, an overview

Ivo Babuˇ ska and Manil Suri. Thep-andhpversions of the finite element method, an overview. Computer Methods in Applied Mechanics and Engineering, 80(1-3):5–26, 1990

1990
[37]

Cambridge University Press, 2000

William Charles Hector McLean.Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, 2000

2000
[38]

Applied Numerical Mathematics, 26(1-2):97–104, 1998

Paul A Zegeling.r-refinement for evolutionary PDEs with finite elements or finite differences. Applied Numerical Mathematics, 26(1-2):97–104, 1998

1998
[39]

Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains.SIAM J

Jie Shen and Li-Lian Wang. Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains.SIAM J. Numer. Anal., 45(5):1954–1978, 2007

1954
[40]

Analysis of discrete ill-posed problems by means of the L-curve.SIAM Review, 34(4):561–580, 1992

Per Christian Hansen. Analysis of discrete ill-posed problems by means of the L-curve.SIAM Review, 34(4):561–580, 1992

1992
[41]

On the uniqueness theorem of Holmgren.Mathematische Zeitschrift, 281(1):357–378, 2015

Haakan Hedenmalm. On the uniqueness theorem of Holmgren.Mathematische Zeitschrift, 281(1):357–378, 2015. 31

2015

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

The inverse source problem for Maxwell’s equations

Richard Albanese and Peter B Monk. The inverse source problem for Maxwell’s equations. Inverse Problems, 22(3):1023, 2006

2006

[2] [2]

Optical tomography in medical imaging.Inverse Problems, 15(2):R41, 1999

Simon R Arridge. Optical tomography in medical imaging.Inverse Problems, 15(2):R41, 1999

1999

[3] [3]

The unique determination of neuronal currents in the brain via magnetoencephalography.Inverse Problems, 20(4):1067, 2004

AS Fokas, Y Kurylev, and V Marinakis. The unique determination of neuronal currents in the brain via magnetoencephalography.Inverse Problems, 20(4):1067, 2004

2004

[4] [4]

Application of inverse source concepts to photoacoustic tomography.Inverse Problems, 23(6):S21, 2007

Mark A Anastasio, Jin Zhang, Dimple Modgil, and Patrick J La Rivi` ere. Application of inverse source concepts to photoacoustic tomography.Inverse Problems, 23(6):S21, 2007

2007

[5] [5]

Inverse source problem in nonhomoge- neous background media.SIAM Journal on Applied Mathematics, 67(5):1353–1378, 2007

Anthony J Devaney, Edwin A Marengo, and Mei Li. Inverse source problem in nonhomoge- neous background media.SIAM Journal on Applied Mathematics, 67(5):1353–1378, 2007

2007

[6] [6]

Multidimensional inverse scattering problems

Alexander G Ramm. Multidimensional inverse scattering problems. InDIPED-99. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory. Proceedings of 4th Interna- tional Seminar/Workshop (IEEE Cat. No. 99TH8402), page 22. IEEE, 1999

1999

[7] [7]

Thermoacoustic tomography with variable sound speed.Inverse Problems, 25(7):075011, 2009

Plamen Stefanov and Gunther Uhlmann. Thermoacoustic tomography with variable sound speed.Inverse Problems, 25(7):075011, 2009

2009

[8] [8]

Inverse scattering.Sect

NN Bojarski. Inverse scattering.Sect. II, Third Quarterly Company Report, Naval Air Systems Command Contract N00019–73-C-0312, pages 19–73, 1973

1973