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arxiv: 2602.14757 · v2 · submitted 2026-02-16 · 🧮 math.NA · cs.LG· cs.NA

Recognition: 1 theorem link

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Solving Inverse Parametrized Problems via Finite Elements and Extreme Learning Networks

Erik Burman , Mats G. Larson , Karl Larsson , Jonatan Vallin
Authors on Pith no claims yet

Pith reviewed 2026-05-15 21:55 UTC · model grok-4.3

classification 🧮 math.NA cs.LGcs.NA
keywords finite element methodextreme learning machineparametrized PDEinverse problemphotoacoustic tomographyerror estimatesparameter approximationuncertainty quantification
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The pith

Finite element spatial discretization paired with extreme learning machine parameter surrogates solves inverse parametrized PDE problems with explicit error estimates.

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

The paper builds a modeling framework for parameter-dependent partial differential equations that arise in inverse problems and uncertainty quantification. Spatial discretization uses standard finite elements while the dependence on a finite set of parameters is handled by a separate approximation step. Existence, uniqueness, and regularity of the parametric solution are established, and error estimates are derived that make the trade-off between mesh size and parameter approximation explicit. In low-dimensional parameter spaces classical interpolation gives algebraic rates tied to Sobolev regularity; in higher dimensions extreme learning machine surrogates replace interpolation under stated approximation and stability conditions. The method is demonstrated on quantitative photoacoustic tomography, where it produces bounded reconstruction errors for potentials and parameters at substantially lower cost than conventional approaches.

Core claim

We establish existence, uniqueness, and regularity of the parametric solution and derive rigorous error estimates that explicitly quantify the interplay between spatial discretization and parameter approximation. In low-dimensional parameter spaces, classical interpolation schemes yield algebraic convergence rates based on Sobolev regularity in the parameter variable. In higher-dimensional parameter spaces, we replace classical interpolation by extreme learning machine (ELM) surrogates and obtain error bounds under explicit approximation and stability assumptions. The proposed framework is applied to inverse problems in quantitative photoacoustic tomography, where we derive potential and 1.0

What carries the argument

The interpolation-based modeling framework that separates finite-element spatial discretization from parameter approximation performed either by classical interpolation or by extreme learning machine surrogates.

Load-bearing premise

Error bounds in higher-dimensional parameter spaces require explicit approximation and stability assumptions on the extreme learning machine surrogates.

What would settle it

A numerical test in quantitative photoacoustic tomography in which the observed reconstruction error for a chosen mesh size and ELM surrogate exceeds the theoretically derived bound for that discretization pair would falsify the error estimates.

Figures

Figures reproduced from arXiv: 2602.14757 by Erik Burman, Jonatan Vallin, Karl Larsson, Mats G. Larson.

Figure 1
Figure 1. Figure 1: Domains. (a) The spatial domain Ωx is a bounded subset in R d for d ∈ {2, 3}. (b) The parameter domain Ωt is the hypercube [0, 1]Nt where Nt may be large. idea is to construct a fast method that allows to explore the parameter space in a global optimization step, resulting in a first approximation not far from the global minimizer. If a more accurate solution to the inverse problem is required Newton itera… view at source ↗
Figure 2
Figure 2. Figure 2: Let Vhx ⊂ H1 0 (Ωx) be the associated finite element space. For each parameter value t ∈ Ωt , the finite element approximation uhx (·, t) ∈ Vhx to the solution u(·, t) is defined by (∇uhx (·, t), ∇v)Ωx + (µ(·, t) uhx (·, t), v)Ωx = ⟨f(·, t), v⟩H−1(Ωx)×H1 0 (Ωx) ∀v ∈ Vhx (3.1) where f = f(x, t) ∈ H−1 (Ωx) may depend on the parameter t. Let {φk} K k=1 denote the basis of Vhx . The discrete solution can then … view at source ↗
Figure 2
Figure 2. Figure 2: Spatial domain discretization. The spatial domain Ωx is dis￾cretized using a finite element mesh Thx of mesh size hx. integer k ≥ 0. More precisely, for every multi-index α with |α| ≤ k, ∂ α t uhx ∈ L 2 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Low-dimensional interpolation in Ωt. For Nt ≤ 4, the param￾eter domain can be partitioned into simplices with interpolation points at the vertices. This partition constitutes a mesh Tht of the parameter domain with mesh size ht on which we may employ standard piecewise-linear inter￾polation. where σ is a fixed activation (e.g., sigmoid or ReLU) and the parameters (bm, cm) ∈ S Nt−1 × R are drawn once at ran… view at source ↗
Figure 4
Figure 4. Figure 4: High-dimensional interpolation in Ωt. For Nt > 4, an ELM surrogate is used for interpolation. (a) J quasi-random (Sobol’) points in Ωt are used as interpolation points. (b) Random ReLU feature hyperplanes for the ELM surrogate in Ωt ; each hyperplane marks where one unit switches between active and inactive regions. At the same time, the approach has several limitations. The use of random, non￾adaptive fea… view at source ↗
Figure 5
Figure 5. Figure 5: Convergence in low-dimensional setting. (a) The heatmap shows approximation error for different values of the pair (hx, ht). (b) The figure shows the approximation error as a function of hx for different choices of ht including a reference line to indicate the theoretical convergence rate of 2.0 with respect to hx. (c) The figure shows the approximation error as a function of ht for different choices of hx… view at source ↗
Figure 6
Figure 6. Figure 6: Convergence in high-dimensional setting. (a) The heatmap shows approximation error for the ELMs for different values of the pair (hx, M). (b) The figure shows the approximation error as a function of hx for dif￾ferent choices of M. A reference line is included to indicate the theoretical convergence rate of 2.0 with respect to hx. The errors for the fully trained networks with M = 25600 ReLU units are also… view at source ↗
Figure 7
Figure 7. Figure 7: Measurement operator. (a) The absorbed energy is given by µu depicted in the figure. (b) By averaging over pixels we get the measurement Q[µu] which is a piecewise-constant function on Ωx subordinate to the pixel partition. as P1, P2, . . . , PNm and with equal area |P|. We then set Mm = span{1Pi : i = 1, 2, . . . , Nm} (5.6) Thus, Mm is defined to be the span of the indicator functions associated to the p… view at source ↗
Figure 8
Figure 8. Figure 8: Potential reconstruction. Illustration of how the potential µ is successively reconstructed at different stages during the optimization. 0 50 100 150 200 i (iteration) 10-2 10-1 100 101 102 L ( t (i)) (a) 0 50 100 150 200 i (iteration) 0.5 1 1.5 2 2.5 k t (i) ! t y k (b) (c) [PITH_FULL_IMAGE:figures/full_fig_p035_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Loss and error during optimization. (a) The loss over the iter￾ations, (b) the parameter reconstruction error (measured in the Euclidean norm) over the iterations and (c) the potential reconstruction error over the iterations. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Measurement operators with different resolutions. Four exam￾ples of the information over Ωx seen by the measurement operator Q using different pixel resolutions. 0 50 100 150 200 i (iteration) 10-2 10-1 100 k t (i) ! t y k Nm = 3 # 3 Nm = 5 # 5 Nm = 7 # 7 Nm = 10 # 10 Nm = 15 # 15 Nm = 20 # 20 Nm = 25 # 25 (a) 0 100 200 300 400 500 600 700 Nm 10-2 10-1 100 k ^t ! t y k (b) [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 11
Figure 11. Figure 11: Effect of pixel resolution. Errors for the potential reconstruction when varying the pixel resolution of the measurement operator. (a) The reconstruction error over the iterations for different pixel resolutions. (b) The final reconstruction error (measured in the Euclidean norm) as a function of the number of pixels. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Measurement operators with different domain coverage. Four examples of partial pixel coverage for the measurement operator Q, where only the given percentage of the pixels in Ωx is seen by the operator [PITH_FULL_IMAGE:figures/full_fig_p037_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Effect of domain coverage. Errors for the potential reconstruc￾tion when varying the percentage of the domain seen by the measurement operator Q. (a) The reconstruction error (measured in the Euclidean norm) over the iterations for different levels of coverage. (b) The final reconstruc￾tion error as a function of the coverage. that we can rewrite the loss in (4.15) in vector form as L(t) = |P| 2 ∥qobs − q… view at source ↗
Figure 14
Figure 14. Figure 14: Effect of noise level. The mean reconstruction error, measured in the Euclidean norm, after 200 iterations as a function of noise amplitude for the original loss functional L(t) and the weighted loss functional L˜(t). To assess the robustness of the reconstruction with respect to noise, we fixed qobs and generated multiple realizations of ˜qobs for different choices of ρ. For each noise level, 20 independ… view at source ↗
read the original abstract

We develop an interpolation-based modeling framework for parameter-dependent partial differential equations arising in control, inverse problems, and uncertainty quantification. The solution is discretized in the physical domain using finite element methods, while the dependence on a finite-dimensional parameter is approximated separately. We establish existence, uniqueness, and regularity of the parametric solution and derive rigorous error estimates that explicitly quantify the interplay between spatial discretization and parameter approximation. In low-dimensional parameter spaces, classical interpolation schemes yield algebraic convergence rates based on Sobolev regularity in the parameter variable. In higher-dimensional parameter spaces, we replace classical interpolation by extreme learning machine (ELM) surrogates and obtain error bounds under explicit approximation and stability assumptions. The proposed framework is applied to inverse problems in quantitative photoacoustic tomography, where we derive potential and parameter reconstruction error estimates and demonstrate substantial computational savings compared to standard approaches, without sacrificing 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

1 major / 0 minor

Summary. The paper develops an interpolation-based modeling framework for parameter-dependent PDEs, using finite element discretization in the physical domain and separate approximation of the parameter dependence. It establishes existence, uniqueness, and regularity of the parametric solution, derives rigorous error estimates quantifying the interplay between spatial discretization and parameter approximation, and applies the framework to inverse problems in quantitative photoacoustic tomography, claiming computational savings without loss of accuracy. Low-dimensional cases use classical interpolation with Sobolev rates; high-dimensional cases employ ELM surrogates under explicit approximation and stability assumptions.

Significance. If the central claims hold, the work offers a structured approach to parametric inverse problems with explicit error control that separates spatial and parameter errors, potentially enabling scalable computations in high-dimensional settings via ELM while preserving rigor in applications such as photoacoustic tomography.

major comments (1)
  1. [Abstract / high-dimensional parameter approximation] The error estimates for high-dimensional parameter spaces (as stated in the abstract) are derived only under explicit approximation and stability assumptions on the ELM surrogates. These assumptions are not shown to follow from the PDE structure or verified numerically within the manuscript, rendering the claimed rigor for the high-dimensional regime conditional rather than self-contained.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. We address the major comment below and will incorporate revisions to strengthen the presentation of the high-dimensional results.

read point-by-point responses

  1. Referee: [Abstract / high-dimensional parameter approximation] The error estimates for high-dimensional parameter spaces (as stated in the abstract) are derived only under explicit approximation and stability assumptions on the ELM surrogates. These assumptions are not shown to follow from the PDE structure or verified numerically within the manuscript, rendering the claimed rigor for the high-dimensional regime conditional rather than self-contained.

    Authors: We appreciate the referee highlighting this point. The manuscript explicitly conditions the high-dimensional error estimates on approximation and stability assumptions for the ELM surrogates, as stated in the abstract and elaborated in the relevant theoretical sections. This is by design: the framework separates spatial FEM discretization from parameter-space approximation to remain modular and applicable to general parameter-dependent PDEs, with ELM serving as one possible surrogate in high dimensions. The assumptions are not derived from the specific PDE structure because they concern the general approximation theory of ELM networks, which is supported by prior literature on neural network surrogates. We agree, however, that the conditional nature merits clearer emphasis to prevent any perception of unconditional rigor. In the revised version we will (i) update the abstract to foreground the assumptions, (ii) add a concise discussion paragraph (with references to ELM approximation results) explaining when the assumptions are expected to hold for the photoacoustic tomography problem, and (iii) include targeted numerical checks of the stability and approximation quality on the concrete example. These changes will make the high-dimensional regime more self-contained while preserving the modular character of the framework. revision: yes

Circularity Check

0 steps flagged

Error estimates derived from discretization theory and stated ELM assumptions, no reduction to fitted inputs

full rationale

The paper's core claims rest on standard existence/uniqueness results for parametric PDEs, classical Sobolev interpolation rates for low-dimensional parameter spaces, and finite-element spatial error bounds. High-dimensional cases invoke ELM surrogates only under explicitly stated approximation and stability assumptions that are external to the derivation; these are not obtained by fitting parameters inside the paper's own equations and then relabeling them as predictions. No self-definitional loops, load-bearing self-citations, or ansatz smuggling appear in the derivation chain. The overall argument therefore remains self-contained against external benchmarks, warranting only a minor score for the conditional nature of the high-dimensional bounds.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework relies on standard PDE existence and regularity results plus specific ELM approximation and stability assumptions to obtain error bounds; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)
  • domain assumption Existence, uniqueness, and regularity of the parametric solution
    Stated as established for the modeling framework.
  • ad hoc to paper Explicit approximation and stability assumptions for ELM surrogates
    Invoked to derive error bounds in high-dimensional parameter spaces.

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