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REVIEW 3 major objections 4 minor 37 references

Regularized Random Fourier Features and Finite Element Reconstruction for Operator Learning in Sobolev Space

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Frequency-weighted Tikhonov regularization makes random Fourier feature matrices well-conditioned at N ~ m log m features, and RRFF-FEM then learns noisy PDE operators more accurately and faster than unregularized random features.

desk verdict A solid, incremental random-feature method for noisy operator learning; the main theorem's condition (6) appears violated in the tested regime, but the empirical claims and Student-t generalization are worth referee time. read the letter →

arxiv 2512.17884 v1 pith:DR6JABGH submitted 2025-12-19 cs.LG cs.NAmath.NAstat.ML

Xinyue Yu , Hayden Schaeffer This is my paper · ORCID
classification cs.LGcs.NAmath.NAstat.ML MSC 65N3068T05
keywords operatorlearningrandomFourierfeaturesTikhonovregularizationStudent'stdistributionMatérnkernelconditionnumberfiniteelementinterpolationnoisyPDEdata
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

The paper is trying to establish that a random Fourier feature model for operator learning can be made stable and accurate in the presence of noisy data by adding a frequency-weighted Tikhonov penalty, without sacrificing the speed that made random features attractive. It draws the random frequencies from multivariate Student's t distributions, a family that includes both Cauchy and Gaussian limits, and pairs the learned map with a finite element interpolant so predictions are functions on the output domain. The main theoretical result is a high-probability bound on the extreme singular values of the random feature matrix: if the number of features N grows like m log m in the number of training samples m, and a scale condition on the Matérn kernel is met, the matrix is well conditioned. Across advection, Burgers', Darcy, Helmholtz, Navier-Stokes, and structural mechanics benchmarks, the regularized method reports lower test error and faster training than the unregularized random feature baseline, and accuracy comparable to kernel and neural operator methods on the tested cases.

What carries the argument

The load-bearing object is the random feature matrix A∈C^{m×N} with entries exp(i⟨ω_k, x_j⟩). Its normalized Gram matrix (1/N)AA* concentrates around the kernel matrix whose entries are the Matérn characteristic function ϕ_ω(x_j−x_k) = E[exp(i⟨ω, x_j−x_k⟩)], the Fourier transform of the Student-t sampling density. Because this characteristic function is positive, decreasing in distance, and bounded by its value at the minimum spacing κ, standard row-sum bounds for Hermitian matrices control the deviation of the expected Gram matrix from the identity; a matrix concentration inequality then converts the per-entry bounds into the high-probability spectral concentration of Theorem 3.1. The secon

What would settle it

Compute the left-hand side of condition (6) with the experiment's parameters — take σ and ν from the parameter tables, κ equal to the smallest grid spacing, and m the number of output collocation points. In several reported settings σκ is small and the Matérn characteristic function is close to 1, so the product with m is much larger than η; checking whether the empirical spectral norm ‖(1/N)AA* − I_m‖₂ nevertheless stays below 2η with probability at least 1−δ would settle whether the concentration holds despite the unverified hypothesis.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is Theorem 3.1: with feature weights drawn from the multivariate Student's t density (scale σ, degrees of freedom ν), if N ≥ C η^{-2} m log(2m/δ) and the Matérn characteristic function evaluated at the minimum separation κ of the training points is at most η/m, then with probability at least 1−δ the normalized Gram matrix satisfies ‖(1/N)AA* − I_m‖₂ ≤ 2η. This concentration makes the least-squares system well conditioned and the pseudoinverse well defined, which the paper says yields estimation and generalization bounds. The method itself solves, for each output component, a regularized least squares problem whose penalty ∑‖ω_k‖_2^p |x_k|² suppress

Load-bearing premise

The main guarantee requires the kernel's value at the shortest distance between any two training points to be at most η/m; in the paper's experiments the frequency scale is so small that this value is near 1, so the theorem's hypotheses appear to be unmet in the very regime where the numerical gains are shown, and the stated estimation/generalization bounds are inherited from earlier work rather than proved.

Editorial extensions

If this is right

  • With N ~ m log m features and the scale condition met, the feature matrix is spectrally concentrated around the identity, so the regularized least-squares problem is stable and the pseudoinverse is well defined.
  • The frequency-weighted penalty suppresses high-frequency noise, which the experiments indicate lowers test error on all benchmark PDEs compared to the unregularized random feature model.
  • Because RRFF-FEM's output is a finite element interpolant, the learned operator can be evaluated at arbitrary points of the output domain, not just the training grid.
  • The experiments indicate that RRFF retains the speed advantage of random features, with training times below those reported for Matérn and RBF kernel methods on the same problems.

Reading between the lines

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

  • A natural parameter-selection rule follows from condition (6): pick σ and ν so that the Matérn characteristic function at the grid spacing falls below η/m; tuning σ this way could bring the experiments into the regime where the theorem applies.
  • Because ν=1 gives Cauchy and ν→∞ gives Gaussian weights, the Student-t family offers a one-parameter interpolation between heavy-tailed and light-tailed sampling, which could be matched to the regularity of a given PDE's solution operator.
  • The frequency penalty ∑‖ω_k‖^p |x_k|² can be read as an RKHS Sobolev-norm penalty; this connection suggests a testable comparison against direct Sobolev-space kernel methods on the same noisy benchmarks.
  • The FEM recovery map decouples the number of training samples from the output grid resolution; a competing implementation could replace it with learned recovery maps and measure how much of the reported gain comes from the interpolation step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a regularized random Fourier feature method (RRFF) and a finite-element variant (RRFF-FEM) for learning PDE solution operators from noisy data. Random feature weights are drawn from multivariate Student's t distributions, and a frequency-weighted Tikhonov term is added to the least-squares objective. The main theoretical result, Theorem 3.1, gives a high-probability bound on the deviation of the normalised random feature Gram matrix from the identity under conditions (5)–(6), and the paper states that this yields estimation and generalization bounds. Extensive experiments on advection, Burgers, Darcy, Helmholtz, Navier–Stokes, and structural mechanics benchmark problems compare RRFF/RRFF-FEM with unregularized RFF, kernel methods, and neural operators.

Significance. If the theoretical guarantee held in the parameter regime used in the experiments, the paper would be a useful contribution: it unifies and extends prior random-feature conditioning results to Student-t features, gives the advertised N = O(m log m) scaling, and the benchmark suite is unusually broad and accompanied by code. The proof of Theorem 3.1 appears internally consistent under its stated hypotheses. However, the main load-bearing condition is not verified and is implausible for the reported experimental parameters, and the estimation/generalization consequences are only asserted. The numerical study is substantial and the empirical improvements are interesting, but the advertised provable well-conditioning is not actually connected to the experiments in the present form.

major comments (3)
  1. [§3.2, Theorem 3.1, Eq. (6)] Condition (6) is load-bearing but never verified, and it is not satisfied by the reported experimental parameter values. Since φ(t) is the characteristic function of the feature distribution, φ(0)=1 and φ decreases with ∥t∥; therefore (6) requires the minimum separation κ among the rows of A — the training input functions, not the spatial grid — to be large enough that φ(κ) ≤ η/m. For the Gaussian case and the reported Darcy/Helmholtz scale σ=√2×10^{-5}, (6) reads exp(-σ²κ²/2) ≤ η/m, so for m=10^3–10^4 one needs κ of order 10^5; the discretized PDE input vectors have Euclidean norms of order 10–10^3, so this is not attainable. The paper never computes κ or checks (6). Even if κ could be chosen large, the very small η forced by (6) would make condition (5) require far more than the reported N. Consequently Theorem 3.1 does not apply in the regime where RRFF is tested, and the statement th
  2. [§3.2, final paragraph] The paper asserts that the concentration property can be used to derive estimation and generalization bounds 'similar to [18]', but no statement or proof is given. This is not a routine corollary: the estimator in (3) solves a regularized, frequency-weighted least-squares problem with Student-t features, whereas [18] analyzes unregularized minimum-norm interpolation with Cauchy features. The abstract and introduction advertise these guarantees, so the missing derivation is load-bearing. Either state the precise bounds and prove them, or explicitly limit the theoretical claim to the conditioning result.
  3. [§4.2–§4.3, Tables 7 and 10] The regularization parameter α is selected by minimizing the reported average relative test error ('we tested a range of values for α to find the optimal α such that the RRFF method minimizes the test error', Figures 8 and 12), and Tables 7 and 10 report the corresponding errors. This is a test-set leak: the reported 'test' errors no longer measure genuine generalization. The observed improvement of RRFF over RFF may therefore be inflated. Use a held-out validation split or nested cross-validation to choose α, and report test errors only for the final model.
minor comments (4)
  1. [§3, definition of ρ(ω)] The normalization factor appears to be σ π^{d/2} ν^{d/2} Γ(ν/2), but a d-dimensional scale parameter σ should give a factor σ^d π^{d/2} ν^{d/2} Γ(ν/2). Please correct the density or clarify the notation for σ.
  2. [§2 vs §3.2] The symbol m is overloaded: in Section 2 it denotes the output dimension, while in Assumption 1 and Theorem 3.1 it denotes the number of training samples. This makes the statement of condition (6) confusing and should be fixed by renaming one of the two quantities.
  3. [§2, Lipschitz assumption] The displayed Lipschitz condition contains a typo: ∥u2−u2∥ should presumably be ∥u1−u2∥.
  4. [§4.7, Table 7] The phrase 'consistently outperform' is stronger than the data support for Navier–Stokes with ν=2,3, where the reported differences are only 0.04–0.06 percentage points. Reporting standard deviations or confidence intervals over the 20 trials would help assess these differences.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity: Theorem 3.1 is independently proven from stated assumptions; only the generalization-bound corollary is imported from a same-group preprint.

  1. self citation load bearing [Section 1 (Contributions bullet 2) and end of Section 3]
    "We show that if the complexity ratio N/m scales like log m, where N is the number of random features and m is the number of training samples, then the random feature matrix is well-conditioned with high probability. Estimation and generalization bounds follow from [18]. ... We can use the concentration property of the random feature matrix A∈C^{m×N} to derive estimation and generalization bounds similar to [18]."

    The paper's advertised theoretical payoff—'which yields estimation and generalization guarantees'—is not derived in this paper. It is deferred to [18], an arXiv preprint by Liao, Needell, and Schaeffer, which shares the current author Hayden Schaeffer and is not reproduced or machine-checked here. No proof is given that the regularized Student-t estimator in (3) satisfies the assumptions or inherits the Cauchy-feature bounds of [18]. Thus the generalization-guarantee component of the central claim reduces to an unverified self-citation. This is partial and minor, because the well-conditioning theorem itself is proven independently through the Matérn characteristic function, Gershgorin, and Matrix Bernstein arguments.

full rationale

The main theoretical result, Theorem 3.1, is not circular: it is a genuine derivation from Assumption 1 and the explicit conditions (5)-(6). The proof identifies the Student-t characteristic function with the Matérn kernel, uses the minimum-separation assumption κ, and applies Gershgorin plus Matrix Bernstein. Condition (6) is an explicit sufficient condition on the kernel off-diagonal entries, not a fitted parameter, and the theorem does not claim it holds automatically. The experimental RRFF/RRFF-FEM results are evaluated on external benchmark PDEs from [24,7], so the reported improvements over RFF are not forced by construction. The main correctness caveat—that condition (6) appears not to be verified in the experimental regime, since σ√νκ is small and φ(κ)≈1—is an applicability gap, not a circularity. The only circularity-adjacent item is the import of estimation/generalization bounds from [18], a same-group preprint with no derivation supplied; this is a minor self-citation rather than a reduction of the central claim to its inputs. Overall score 2.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The method introduces no new physical or mathematical entities; the frequency-weighted penalty is a regularizer, not a new object. The central claim depends instead on many per-benchmark hyperparameters (σ, α, p, N, ν, FEM grids), with α selected by test-error minimization. The theory requires a well-separated grid and condition (6), which is not checked for the experimental parameter values.

free parameters (6)
  • Feature scale σ = 0.001–0.2 (Tables 1–6)
    Chosen by hand per benchmark and per distribution; controls the frequency spread of the random features.
  • Regularization strength α = 10^-5–0.5 (Tables 1–6)
    Selected per benchmark by minimizing the test error (§4.2, §4.3); tuning on the test set makes reported errors optimistic.
  • Regularization exponent p = 2 or 4
    Set per benchmark: p=4 for Helmholtz and Navier-Stokes, p=2 otherwise.
  • Number of random features N = 5,000–100,000
    Chosen per benchmark; roughly consistent with O(m log m) scaling but not derived from the theory or validated against it.
  • Student-t degrees of freedom ν = 2, 3, ∞
    Three variants compared; ν is a modeling choice and changes the effective Sobolev smoothness.
  • FEM mesh / grid size = 27–6,800 grid points
    Coarse nonuniform grids chosen per problem for the FEM recovery map; affects interpolation error and the minimum separation κ.
assumptions (6)
  • domain assumption Training noise is i.i.d., isotropic Gaussian, with relative magnitude fixed per problem and test inputs also corrupted.
    Section 2 and Appendix A define the noise model; the method and theory are not shown to cover other noise types.
  • domain assumption The target operator G is Lipschitz and each component f_j is Lipschitz.
    Used in the problem formulation (Section 2); not verified for the benchmark operators but plausible.
  • domain assumption Assumption 1: training points are separated by ||x_j − x_j'|| ≥ κ > 0 and feature weights follow a Student-t distribution.
    Central to the Gershgorin and Bernstein bounds in Theorem 3.1; small uniform or nonuniform grid spacings make κ small.
  • standard math Student-t random features generate a Matérn kernel whose RKHS equals the Sobolev space H^{(ν+d)/2}.
    Invoked in Section 3.1 via [31] and Matérn-Sobolev equivalence; standard but imported from the literature.
  • domain assumption Estimation and generalization bounds from [18] apply to the regularized Student-t estimator.
    Section 3.2 states 'Estimation and generalization bounds follow from [18]' without proof; [18] analyzes unregularized Cauchy features.
  • standard math Matrix Bernstein inequality, Gershgorin's theorem, Sobolev embedding, and Bessel function monotonicity are valid and applicable.
    Standard tools used in the proof of Theorem 3.1.

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Cite this review

Pith. "Pith review of Regularized Random Fourier Features and Finite Element Reconstruction for Operator Learning in Sobolev Space." pith.science (2026). https://pith.science/paper/DR6JABGH

@misc{pith2026251217884,
  author       = {Pith},
  title        = {Pith review of: Regularized Random Fourier Features and Finite Element Reconstruction for Operator Learning in Sobolev Space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DR6JABGH}},
  note         = {Machine review of arXiv:2512.17884}
}
abstract

Operator learning is a data-driven approximation of mappings between infinite-dimensional function spaces, such as the solution operators of partial differential equations. Kernel-based operator learning can offer accurate, theoretically justified approximations that require less training than standard methods. However, they can become computationally prohibitive for large training sets and can be sensitive to noise. We propose a regularized random Fourier feature (RRFF) approach, coupled with a finite element reconstruction map (RRFF-FEM), for learning operators from noisy data. The method uses random features drawn from multivariate Student's $t$ distributions, together with frequency-weighted Tikhonov regularization that suppresses high-frequency noise. We establish high-probability bounds on the extreme singular values of the associated random feature matrix and show that when the number of features $N$ scales like $m \log m$ with the number of training samples $m$, the system is well-conditioned, which yields estimation and generalization guarantees. Detailed numerical experiments on benchmark PDE problems, including advection, Burgers', Darcy flow, Helmholtz, Navier-Stokes, and structural mechanics, demonstrate that RRFF and RRFF-FEM are robust to noise and achieve improved performance with reduced training time compared to the unregularized random feature model, while maintaining competitive accuracy relative to kernel and neural operator tests.

Figures

Figures reproduced from arXiv: 2512.17884 by the authors.

Figure 1
Figure 1. Commutative diagram of the operator learning framework following [25, 1, 2, 20, 18]. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Advection I: (left to right) An example of (a) training input with 5% noise, (b) training output with 5% noise, (c) test example and predictions of RV ◦ ˆf using RFF-FEM-∞ and RRFF￾FEM-∞, (d) pointwise errors for predictions of RV ◦ ˆf using RFF-FEM-∞ and RRFF-FEM-∞ [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Advection I: (left to right) (a) test example and predictions of RV ◦ ˆf using RFF￾FEM-2 and RRFF-FEM-2, (b) pointwise errors for predictions of RV ◦ ˆf using RFF-FEM-2 and RRFF-FEM-2, (c) test example and predictions of RV ◦ ˆf using RFF-FEM-3 and RRFF-FEM-3, (d) pointwise errors for predictions of RV ◦ ˆf using RFF-FEM-3 and RRFF-FEM-3. A more complex initial condition was also considered in [24], which takes the … view at source ↗
Figures from the paper (42 more)
Figure 4
Figure 4. Figure 4: Advection II: (left to right) An example of (a) training input with 5% noise, (b) training output with 5% noise, (c) test example and predictions of RV ◦ ˆf using RFF-FEM-∞ and RRFF￾FEM-∞, (d) pointwise errors for predictions of RV ◦ ˆf using RFF-FEM-∞ and RRFF-FEM-∞ …
Figure 5
Figure 5. Figure 5: Advection II: (left to right) (a) test example and predictions of RV ◦ ˆf using RFF￾FEM-2 and RRFF-FEM-2, (b) pointwise errors for predictions of RV ◦ ˆf using RFF-FEM-2 and RRFF-FEM-2, (c) test example and predictions of RV ◦ ˆf using RFF-FEM-3 and RRFF-FEM-3, (d) poi…
Figure 6
Figure 6. Figure 6: Advection III: (left to right) An example of (a) training input with 5% noise, (b) training output with 5% noise, (c) test example and predictions of RV ◦ ˆf using RFF-FEM-∞ and RRFF-FEM-∞, (d) pointwise errors for predictions of RV ◦ ˆf using RFF-FEM-∞ and RRFF-FEM- ∞…
Figure 7
Figure 7. Figure 7: Advection III: (left to right) (a) test example and predictions of RV ◦ ˆf using RFF￾FEM-2 and RRFF-FEM-2, (b) pointwise errors for predictions of RV ◦ ˆf using RFF-FEM-2 and RRFF-FEM-2, (c) test example and predictions of RV ◦ ˆf using RFF-FEM-3 and RRFF-FEM-3, (d) po…
Figure 8
Figure 8. Figure 8: Burgers’ Equation: (left to right) An example of (a) training input with 5% noise, (b) training output with 5% noise, (c) log10(α) versus average relative test error for RRFF-∞ over 20 trials, (d) log10(α) versus average relative test error for RRFF-2 and RRFF-3 over 2…
Figure 9
Figure 9. Figure 9: Burgers’ Equation: (left to right) (a) test example and RFF-∞ and RRFF-∞ ap￾proximations of ˆf, (b) pointwise errors for RFF-∞ and RRFF-∞ approximations of ˆf, (c) test example and RFF-2 and RRFF-2 approximations of ˆf, (d) pointwise errors for RFF-2 and RRFF-2 approxi…
Figure 10
Figure 10. Figure 10: Burgers’ Equation: (left to right) (a) test example and RFF-3 and RRFF-3 approxi￾mations of ˆf, (b) pointwise errors for RFF-3 and RRFF-3 approximations of ˆf, (c) test example and predictions of RV ◦ ˆf using RFF-FEM-∞ and RRFF-FEM-∞, (d) pointwise errors for predict…
Figure 11
Figure 11. Figure 11: Burgers’ Equation: (left to right) (a) test example and predictions of RV ◦ ˆf using RFF-FEM-2 and RRFF-FEM-2, (b) pointwise errors for predictions of RV ◦ ˆf using RFF-FEM-2 and RRFF-FEM-2, (c) test example and predictions of RV ◦ ˆf using RFF-FEM-3 and RRFF-FEM￾3, (…
Figure 12
Figure 12. Figure 12: Darcy Flow: (left to right) An example of (a) training input with 5% noise, (b) training output with 5% noise, (c) log10(α) versus average relative test error for RRFF-∞ over 20 trials, (d) log10(α) versus average relative test error for RRFF-2 and RRFF-3 over 20 tria…
Figure 13
Figure 13. Figure 13: Darcy Flow: (left to right) (a) test example, (b) RFF-∞ approximation of ˆf, (c) RRFF-∞ approximation of ˆf, (d) pointwise error for RFF-∞ approximation of ˆf, (e) pointwise error for RRFF-∞ approximation of ˆf. 4.4 Helmholtz Equation Consider the Helmholtz equation o…
Figure 14
Figure 14. Figure 14: Darcy Flow: (left to right) (a) test example, (b) RFF-2 approximation of ˆf, (c) RRFF￾2 approximation of ˆf, (d) pointwise error for RFF-2 approximation of ˆf, (e) pointwise error for RRFF-2 approximation of ˆf [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]
Figure 15
Figure 15. Figure 15: Darcy Flow: (left to right) (a) test example, (b) RFF-3 approximation of ˆf, (c) RRFF￾3 approximation of ˆf, (d) pointwise error for RFF-3 approximation of ˆf, (e) pointwise error for RRFF-3 approximation of ˆf [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]
Figure 16
Figure 16. Figure 16: Darcy Flow: (left to right) (a) test example, (b) prediction of RV ◦ ˆf using RFF-FEM- ∞, (c) prediction of RV ◦ ˆf using RRFF-FEM-∞, (d) pointwise error for prediction of RV ◦ ˆf using RFF-FEM-∞, (e) pointwise error for prediction of RV ◦ ˆf using RRFF-FEM-∞ [PITH_F…
Figure 17
Figure 17. Figure 17: Darcy Flow: (left to right) (a) test example, (b) prediction of RV ◦ ˆf using RFF-FEM￾2, (c) prediction of RV ◦ ˆf using RRFF-FEM-2, (d) pointwise error for prediction of RV ◦ ˆf using RFF-FEM-2, (e) pointwise error for prediction of RV ◦ ˆf using RRFF-FEM-2. 19 [PIT…
Figure 18
Figure 18. Figure 18: Darcy Flow: (left to right) (a) test example, (b) prediction of RV ◦ ˆf using RFF-FEM￾3, (c) prediction of RV ◦ ˆf using RRFF-FEM-3, (d) pointwise error for prediction of RV ◦ ˆf using RFF-FEM-3, (e) pointwise error for prediction of RV ◦ ˆf using RRFF-FEM-3. where ∂D…
Figure 19
Figure 19. Figure 19: Helmholtz Equation: (left to right) An example of (a) training input with 1% noise, (b) training output with 1% noise. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_19.png]
Figure 20
Figure 20. Figure 20: Helmholtz Equation: (left to right) (a) test example, (b) prediction of RV ◦ ˆf using RFF-FEM-∞, (c) prediction of RV ◦ ˆf using RRFF-FEM-∞, (d) pointwise error for prediction of RV ◦ ˆf using RFF-FEM-∞, (e) pointwise error for prediction of RV ◦ ˆf using RRFF-FEM-∞ …
Figure 21
Figure 21. Figure 21: Helmholtz Equation: (left to right) (a) test example, (b) prediction of RV ◦ ˆf using RFF-FEM-2, (c) prediction of RV ◦ ˆf using RRFF-FEM-2, (d) pointwise error for prediction of RV ◦ ˆf using RFF-FEM-2, (e) pointwise error for prediction of RV ◦ ˆf using RRFF-FEM-2 …
Figure 22
Figure 22. Figure 22: Helmholtz Equation: (left to right) (a) test example, (b) prediction of RV ◦ ˆf using RFF-FEM-3, (c) prediction of RV ◦ ˆf using RRFF-FEM-3, (d) pointwise error for prediction of RV ◦ ˆf using RFF-FEM-3, (e) pointwise error for prediction of RV ◦ ˆf using RRFF-FEM-3. …
Figure 23
Figure 23. Figure 23: Navier-Stokes Equation: (left to right) An example of (a) training input with 5% noise, (b) training output with 5% noise. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_23.png]
Figure 24
Figure 24. Figure 24: Navier-Stokes Equation: (left to right) (a) test example, (b) prediction of RV ◦ ˆf using RFF-FEM-∞, (c) prediction of RV ◦ ˆf using RRFF-FEM-∞, (d) pointwise error for prediction of RV ◦ ˆf using RFF-FEM-∞, (e) pointwise error for prediction of RV ◦ ˆf using RRFF-FEM…
Figure 25
Figure 25. Figure 25: Navier-Stokes Equation: (left to right) (a) test example, (b) prediction of RV ◦ ˆf using RFF-FEM-2, (c) prediction of RV ◦ ˆf using RRFF-FEM-2, (d) pointwise error for prediction of RV ◦ ˆf using RFF-FEM-2, (e) pointwise error for prediction of RV ◦ ˆf using RRFF-FEM…
Figure 26
Figure 26. Figure 26: Navier-Stokes Equation: (left to right) (a) test example, (b) prediction of RV ◦ ˆf using RFF-FEM-3, (c) prediction of RV ◦ ˆf using RRFF-FEM-3, (d) pointwise error for prediction of RV ◦ ˆf using RFF-FEM-3, (e) pointwise error for prediction of RV ◦ ˆf using RRFF-FEM…
Figure 27
Figure 27. Figure 27: Structural Mechanics: (left to right) An example of (a) training input with 5% noise, (b) training output with 5% noise. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_27.png]
Figure 28
Figure 28. Figure 28: Structural Mechanics: (left to right) (a) test example, (b) prediction of RV ◦ ˆf using RFF-FEM-∞, (c) prediction of RV ◦ ˆf using RRFF-FEM-∞, (d) pointwise error for prediction of RV ◦ ˆf using RFF-FEM-∞, (e) pointwise error for prediction of RV ◦ ˆf using RRFF-FEM-∞…
Figure 29
Figure 29. Figure 29: Structural Mechanics: (left to right) (a) test example, (b) prediction of RV ◦ ˆf using RFF-FEM-2, (c) prediction of RV ◦ ˆf using RRFF-FEM-2, (d) pointwise error for prediction of RV ◦ ˆf using RFF-FEM-2, (e) pointwise error for prediction of RV ◦ ˆf using RRFF-FEM-2…
Figure 30
Figure 30. Figure 30: Structural Mechanics: (left to right) (a) test example, (b) prediction of RV ◦ ˆf using RFF-FEM-3, (c) prediction of RV ◦ ˆf using RRFF-FEM-3, (d) pointwise error for prediction of RV ◦ ˆf using RFF-FEM-3, (e) pointwise error for prediction of RV ◦ ˆf using RRFF-FEM-3…
Figure 31
Figure 31. Figure 31: Advection I: (left to right) (a) test example and RFF-∞ and RRFF-∞ approximations of ˆf, (b) pointwise errors for RFF-∞ and RRFF-∞ approximations of ˆf, (c) test example and RFF￾2 and RRFF-2 approximations of ˆf, (d) pointwise errors for RFF-2 and RRFF-2 approximation…
Figure 32
Figure 32. Figure 32: Advection I: (left to right) (a) test example and RFF-3 and RRFF-3 approximations of ˆf, (b) pointwise errors for RFF-3 and RRFF-3 approximations of ˆf. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_32.png]
Figure 33
Figure 33. Figure 33: Advection II: (left to right) (a) test example and RFF-∞ and RRFF-∞ approximations of ˆf, (b) pointwise errors for RFF-∞ and RRFF-∞ approximations of ˆf, (c) test example and RFF￾2 and RRFF-2 approximations of ˆf, (d) pointwise errors for RFF-2 and RRFF-2 approximatio…
Figure 34
Figure 34. Figure 34: Advection II: (left to right) (a) test example and RFF-3 and RRFF-3 approximations of ˆf, (b) pointwise errors for RFF-3 and RRFF-3 approximations of ˆf [PITH_FULL_IMAGE:figures/full_fig_p032_34.png]
Figure 35
Figure 35. Figure 35: Advection III: (left to right) (a) test example and RFF-∞ and RRFF-∞ approxima￾tions of ˆf, (b) pointwise errors for RFF-∞ and RRFF-∞ approximations of ˆf, (c) test example and RFF-2 and RRFF-2 approximations of ˆf, (d) pointwise errors for RFF-2 and RRFF-2 approx￾ima…
Figure 36
Figure 36. Figure 36: Advection III: (left to right) (a) test example and RFF-3 and RRFF-3 approximations of ˆf, (b) pointwise errors for RFF-3 and RRFF-3 approximations of ˆf. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_36.png]
Figure 37
Figure 37. Figure 37: Helmholtz Equation: (left to right) (a) test example, (b) RFF-∞ approximation of ˆf, (c) RRFF-∞ approximation of ˆf, (d) pointwise error for RFF-∞ approximation of ˆf, (e) pointwise error for RRFF-∞ approximation of ˆf [PITH_FULL_IMAGE:figures/full_fig_p033_37.png]
Figure 38
Figure 38. Figure 38: Helmholtz Equation: (left to right) (a) test example, (b) RFF-2 approximation of ˆf, (c) RRFF-2 approximation of ˆf, (d) pointwise error for RFF-2 approximation of ˆf, (e) pointwise error for RRFF-2 approximation of ˆf [PITH_FULL_IMAGE:figures/full_fig_p033_38.png]
Figure 39
Figure 39. Figure 39: Helmholtz Equation: (left to right) (a) test example, (b) RFF-3 approximation of ˆf, (c) RRFF-3 approximation of ˆf, (d) pointwise error for RFF-3 approximation of ˆf, (e) pointwise error for RRFF-3 approximation of ˆf. Navier-Stokes Equation In the numerical experime…
Figure 40
Figure 40. Figure 40: Navier-Stokes Equation: (left to right) (a) test example, (b) RFF-∞ approximation of ˆf, (c) RRFF-∞ approximation of ˆf, (d) pointwise error for RFF-∞ approximation of ˆf, (e) pointwise error for RRFF-∞ approximation of ˆf [PITH_FULL_IMAGE:figures/full_fig_p034_40.png]
Figure 41
Figure 41. Figure 41: Navier-Stokes Equation: (left to right) (a) test example, (b) RFF-2 approximation of ˆf, (c) RRFF-2 approximation of ˆf, (d) pointwise error for RFF-2 approximation of ˆf, (e) pointwise error for RRFF-2 approximation of ˆf [PITH_FULL_IMAGE:figures/full_fig_p034_41.png]
Figure 42
Figure 42. Figure 42: Navier-Stokes Equation: (left to right) (a) test example, (b) RFF-3 approximation of ˆf, (c) RRFF-3 approximation of ˆf, (d) pointwise error for RFF-3 approximation of ˆf, (e) pointwise error for RRFF-3 approximation of ˆf. Structural Mechanics The one-dimensional loa…
Figure 43
Figure 43. Figure 43: Structural Mechanics: (left to right) (a) test example, (b) RFF-∞ approximation of ˆf, (c) RRFF-∞ approximation of ˆf, (d) pointwise error for RFF-∞ approximation of ˆf, (e) pointwise error for RRFF-∞ approximation of ˆf [PITH_FULL_IMAGE:figures/full_fig_p035_43.png]
Figure 44
Figure 44. Figure 44: Structural Mechanics: (left to right) (a) test example, (b) RFF-2 approximation of ˆf, (c) RRFF-2 approximation of ˆf, (d) pointwise error for RFF-2 approximation of ˆf, (e) pointwise error for RRFF-2 approximation of ˆf [PITH_FULL_IMAGE:figures/full_fig_p035_44.png]
Figure 45
Figure 45. Figure 45: Structural Mechanics: (left to right) (a) test example, (b) RFF-3 approximation of ˆf, (c) RRFF-3 approximation of ˆf, (d) pointwise error for RFF-3 approximation of ˆf, (e) pointwise error for RRFF-3 approximation of ˆf. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_45.png]

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