SFO: Learning PDE Operators via Spectral Filtering

Elad Hazan; Kira Radinsky; Noam Koren; Rafael Moschopoulos

REVIEW 3 major objections 5 minor 16 references

SFO: Learning PDE Operators via Spectral Filtering

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper shows that PDE solution operators with stable, shift-invariant discretizations have Green's kernels with linear-dynamical-system structure, and that expanding the kernel in the fixed Universal Spectral Basis (eigenvectors of the H

desk verdict SFO is a solid empirical paper whose theoretical headline overshoots what is proven—push on Theorem G.1. read the letter →

arxiv 2601.17090 v2 pith:OBJHDZVD submitted 2026-01-23 cs.LG cs.AI

Noam Koren , Rafael Moschopoulos , Kira Radinsky , Elad Hazan This is my paper

classification cs.LGcs.AI

keywords neuraloperatorsPDEsolutionmapsspectralfilteringUniversalBasisHilbertmatrixGreen'sfunctionslineardynamicalsystemsoperatorlearning

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

Neural operators struggle to capture the long-range, nonlocal interactions that PDE solution maps often encode. This paper argues that for a large and representative class—stable, local, shift-invariant discretizations—the discrete Green's function decays exponentially in space, which gives it a linear-dynamical-system structure. It then shows that the Universal Spectral Basis, the fixed eigenvectors of the Hilbert matrix, approximates such kernels in their leading few modes, with a logarithmic mode-count guarantee. The resulting architecture, SFO, learns only the spectral coefficients of those modes and reports lower error than seven established baselines on all six benchmarks, using far fewer parameters. The practical claim is that a fixed, data-independent global basis can be a better inductive bias for PDE operators than learned or Fourier bases.

What carries the argument

The load-bearing object is the Universal Spectral Basis (USB): the orthonormal eigenvectors φ_l of the Hilbert matrix H_ij = 1/(i+j−1), a fixed, global, data-independent basis with a rapidly decaying spectrum. SFO parameterizes the integral kernel as κ(x,x′) = Σ_{l=1}^L Θ_l φ_l(x−x′), learns only the channel-mixing matrices Θ_l, and evaluates the resulting convolution in the frequency domain via FFT. The theory rests on two linked lemmas: a stable block three-point stencil has a Green's kernel G[t]=Σ_k Θ_k r_k^{|t|} with |r_k|<1 (exponential decay), and exponentially decaying kernels are approximated by the leading spectral-filtering/Hankel modes with error that falls exponentially, yielding

What would settle it

Construct a 1D linear diffusion problem with known Green's function, no periodic boundary conditions, and Dirichlet data; train SFO and compare boundary-band and interior L2 error to a non-periodic, boundary-aware kernel (or an exact solver). If SFO's boundary error does not shrink toward the interior error as L grows, or if a linear non-wrapped convolutional operator beats it by more than the current margins, the circular-convolution assumption—not kernel compactness—is the active limit. Separately, compute the USB coefficients of a known non-LDS kernel (e.g., a slowly decaying algebraic tail

Watch

Extended reading notes

Core claim

SFO's central discovery is a structural fact about PDE discretizations: the inverse of a stable, local, shift-invariant operator—represented by a block three-point stencil—is a convolution kernel that is a sum of geometric exponentials in the spatial index. Such kernels are impulse responses of stable linear dynamical systems, and the Universal Spectral Basis (the eigenvectors of the Hilbert matrix) compresses them exponentially, so ε accuracy is reached with L = Õ(log(1/ε)) modes. Parameterizing an integral kernel as a truncated USB expansion and evaluating the convolution with the FFT yields an operator whose only learned parameters are the spectral coefficients; across six benchmarks this

Load-bearing premise

The load-bearing premise is that the nonlinear benchmarks behave like near-linear systems in the lifted feature space, so the LDS structure proven for linear shift-invariant stencils carries over; the paper's own boundary-error analysis (Appendix F) shows the companion assumption of periodic circular convolution is already violated on Diffusion-Sorption and Maxwell, where boundary error exceeds interior error.

Editorial extensions

If this is right

With L≈16–20 Hilbert modes, SFO concentrates essentially all learned kernel energy in a few global modes, so parameter counts stay small: 119K–406K parameters across 1D/2D/3D benchmarks versus orders of magnitude more for several baselines.
Per-layer cost is O(d n log n + L d n + L d² n); experiments place SFO's training time on par with the fastest Fourier-style baselines.
On all six benchmarks, SFO reports lower mean L2 relative error than every baseline, with relative improvements over the best baseline between 0.9% and 38.5%, and the largest gains on reaction–diffusion and Cahn–Hilliard problems.
The same trained coefficients transfer to coarser grids by reconstructing the USB basis at test resolution, with error degrading gracefully (except boundary-layer benchmarks where coarsening can artificially reduce measured error).
Theoretical result: for block three-point stencil operators satisfying stability condition a_j > 2|b_j|, the inverse Green's kernel is a finite sum of geometric exponentials, so an SFO operator reaches ε accuracy with L = Õ(log(1/ε)) modes.

Reading between the lines

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

An implicit consequence is resolution transfer: because the USB is defined by index structure rather than by learned spatial features, the same learned coefficients can be reused on finer or coarser grids by recomputing the basis; the paper's subsampling experiments support this, but full grid-invariance would require a more systematic test.
A testable extension is to check the near-linearity assumption directly: record latent feature trajectories during inference and compute how much of the variance a best-fit linear dynamical model explains. If those residuals are large, the compactness observed on nonlinear benchmarks would need a different explanation than the LDS proof.
Boundary sensitivity is the clearest next frontier. Since Appendix F shows boundary error exceeding interior error for Diffusion-Sorption and Maxwell, one could augment the FFT kernel with a small learned boundary-correcting term or switch to non-uniform convolutions; the current architecture deliberately trades this for speed.
The tied-index tensor product couples all spatial axes through the same mode index; a natural alternative is learned mode-mixing across axes or data-dependent mode selection, which the multi-index ablation suggests is not automatically better.

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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

SFO is a solid empirical paper whose theoretical headline overshoots what is proven. If you work on neural operators, worth your time; if you serve as a referee, push on Theorem G.1.

What is genuinely new: the transfer of the Universal Spectral Basis / STU spectral-filtering machinery from sequence modeling to spatial PDE kernels, with an explicit theoretical motivation (exponential decay of Green's functions for stable shift-invariant stencils) and a careful six-benchmark evaluation. The empirical reporting is a cut above the usual: confidence intervals, ablations against Fourier/Chebyshev/random bases, parameter counts, runtime, and a boundary-vs-interior error diagnostic. The coefficient decay plots and the rank L / lifting-dimension sweeps are convincing. The core USB-basis choice is plausible and appears to beat the alternatives in the ablation.

The soft spot is real, and the stress-test note is correct. Theorem G.1 states an operator-norm bound, but the proof only shows ℓ2 approximation of the Green's kernel. For a convolution operator, operator norm is controlled by the ℓ1 norm of the kernel (or the sup norm of its Fourier symbol). An ℓ2-small kernel difference can be large in ℓ1—spiky or high-frequency errors. So the logarithmic-mode claim is not established by the given argument. This is a genuine gap in the central theoretical claim, and it is internal, not a matter of taste. It may be repairable by proving an ℓ1/H∞ decay bound for the truncated USB expansion, but that is absent.

The nonlinear extension is also asserted rather than shown: 'the pre-SFO projection is a Koopman lift' (Section 4.1) is a placeholder, and the theory does not cover the nonlinear benchmarks where the accuracy gains are largest. The experiments don't depend on that theory, so the empirical case stands, but the abstract's 'theoretically motivated' claim should be softened.

Minor: Appendix F discloses that FFT circular convolution causes larger boundary error on Diffusion-Sorption and Maxwell—that's honest and a reasonable scope limit. The anonymized code repo has no commit hash, so reproducibility is not fully checkable yet.

For whom: neural-operator people, especially FNO variants; also anyone interested in whether the LDS spectral-filtering ideas carry over to spatial problems. It deserves a serious referee; I'd accept for review, with the theory as the main thing to fix.

Referee Report

3 major / 5 minor

Summary. This paper introduces the Spectral Filtering Operator (SFO), a neural operator that parameterizes integral kernels as truncated expansions in the fixed Universal Spectral Basis (USB) derived from Hilbert-matrix eigenvectors, and evaluates the resulting convolution via FFT. The paper claims state-of-the-art accuracy on six PDE benchmarks, with up to 40% relative error reduction and substantially fewer parameters than strong baselines, and a theoretical result (Theorem G.1) asserting that for stable block three-point stencil operators, an SFO-parameterized operator approximates the inverse to accuracy ε with L = Õ(log(1/ε)) modes. The empirical section reports mean L2 relative errors with 95% confidence intervals, ablations on basis choice, rank, lifting dimension, and tied vs. multi-index modes, and an appendix with boundary-vs-interior error diagnostics.

Significance. The empirical contribution is solid: the evaluation protocol is standard, the confidence intervals are reported, the ablations are informative, and the parameter/compute complexity analysis is useful. The fixed Hilbert basis with FFT evaluation is a clean and simple idea, and the empirical gains on several benchmarks are credible. If the logarithmic-mode theorem were proved in operator norm, it would be a valuable bridge between LDS spectral filtering and PDE operator learning. However, the theoretical result as stated is not established: there is a norm gap between the kernel approximation lemma and the operator-norm theorem, and the theorem is proved on an infinite grid while the implementation uses finite-grid circular convolution. The extension to nonlinear benchmarks rests on an unproven Koopman-lift assertion. The paper would be strengthened by either repairing these gaps or explicitly downgrading the theoretical claims to heuristics.

major comments (3)

[Appendix G, Lemma G.3 and Theorem G.1] Theorem G.1 asserts an operator-norm bound ||A^{-1}-M_L|| ≤ ε, but the proof derives only an ℓ2 bound on the kernel, ||g-g_L||_{ℓ2} ≤ ε. Since A^{-1} acts by convolution with g (Eq. 15), the ℓ2→ℓ2 operator norm of the error is the H∞ norm of the Fourier symbol difference; an ℓ2-small kernel perturbation need not be symbol-small, e.g., a high-frequency oscillatory error. To justify L=Õ(log(1/ε)) as an operator-norm statement, the paper needs an ℓ1 or H∞ bound for the truncated USB expansion, which Lemma G.3 does not provide. No norm is specified in the theorem statement; if the intended norm is operator norm, the proof is incomplete.
[§5.1, Eq. (13); App. F; App. G] The theorem is proved on the infinite grid ℓ2(Z) after 'ignoring tails and boundary effects,' whereas the deployed operator uses a finite uniform grid and FFT-based circular convolution. Appendix F itself documents nontrivial boundary-vs-interior error for Diffusion-Sorption and Maxwell (Table 10), and App. G offers no quantitative transfer from infinite-grid Green's functions to finite periodic convolution. The paper should either prove Theorem G.1 for the finite periodic operator actually implemented, or explicitly state that the theorem is an idealized motivation rather than a guarantee for the evaluated setup.
[§4.1, §6.3, Conclusion] The only justification for applying the theory to the nonlinear benchmarks is the sentence 'the pre-SFO projection is a Koopman lift, making feature-space dynamics near-linear.' No Koopman operator, observable embedding, or approximation error is defined. Since the strongest empirical gains occur on nonlinear reaction–diffusion problems, this is a load-bearing claim for the paper's narrative that the theory explains the benchmark results. It should be reworded as an explicit heuristic hypothesis, or a formal Koopman analysis must be supplied.

minor comments (5)

[G.5, proof of Lemma G.2] The symbol B0 is used but never defined; it should be A1 throughout that proof.
[G.6, proof sketch] 'Therefore only approximate G[0 : N]' is incomplete; it should read 'We therefore only approximate G[0 : N]'.
[Introduction, references] The parenthetical citation (Herman) is incomplete; it needs a year, venue, and URL.
[Appendix E] The description of zero-shot resolution transfer should specify how the Hilbert basis is reconstructed at the target resolution and why the learned coefficients remain meaningful at a different grid size.
[Table 4] The tied-index vs. multi-index ablation uses L=6 rather than the tuned L=20, so the comparison does not match the main configuration; the conclusion that multi-index modes are not worth their parameter cost should be qualified accordingly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the empirical results are independent benchmark evaluations, and the theoretical claim is an application of prior spectral-filtering results rather than a definitional identity.

full rationale

The paper's central empirical claim (Table 1) comes from training SFO on standard PDE benchmarks with a fixed, data-independent Hilbert/USB basis and evaluating on held-out data; no fitted parameter is renamed as a prediction of a closely related quantity. The theoretical chain (Theorem G.1) proves a stable block three-point stencil has an exponentially decaying Green's kernel (Lemma G.2) and then invokes a spectral-filtering approximation lemma (Lemma G.3) for exponentially decaying kernels. The last step is a citation to prior work with overlapping authorship, but it is a stated theorem with a proof sketch, not an identity, and the exponential-decay step is derived rather than assumed. The paper itself flags real limitations: App. F reports larger boundary than interior error for Diffusion-Sorption and Maxwell under circular convolution, Sec. 4.1's Koopman-lift extension to nonlinear benchmarks is presented as intuition, and the step from Lemma G.3's l2 kernel bound to Theorem G.1's operator-norm statement omits the required l1/symbol-norm argument. These are correctness or evidence gaps, not circular reductions; no derivation step is equivalent to its input by construction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on prior spectral-filtering theory (partly co-authored by Hazan), on the completeness and decay of Hilbert modes, on periodic FFT convolution, and on an asserted Koopman linearization for nonlinear PDEs. The per-benchmark hyperparameters L and d are tuned and should be treated as fitted choices.

free parameters (2)

L (number of USB modes) = 16 or 20 per dataset
Grid-searched over {16,20}; central to the compactness and accuracy claims.
d (lifting dimension) = 32, 64, or 128 per dataset
Grid-searched; affects parameter count and accuracy; selected per benchmark.

assumptions (5)

domain assumption Hilbert/USB modes form a complete orthonormal system and give exponentially accurate approximations of exponentially decaying LDS impulse responses.
Invoked in Lemma G.3 and Section 4.1 to conclude L=Õ(log(1/ε)) modes suffice; the result is cited to Hazan et al. 2017, a co-author, and not re-derived.
standard math Stable local shift-invariant discretizations have exponentially decaying Green's functions.
Used in Lemma G.2 via Demko et al. 1984 and Benzi 2007; accepted numerical linear algebra.
ad hoc to paper For nonlinear PDEs, the lifting map approximates a Koopman lift, making feature-space dynamics near-linear.
Section 4.1 says 'intuitively'; no derivation or empirical verification is provided, yet it is used to extend the linear stencil theory to nonlinear benchmarks.
domain assumption FFT-based circular convolution faithfully realizes the integral kernel on the evaluated domains.
Assumed in Eq. (8) and Section 5.1; Appendix F shows boundary-vs-interior error gaps for Diffusion-Sorption and Maxwell, so this premise is partially violated.
ad hoc to paper Finite-grid behavior can be approximated by an infinite-grid analysis with tails and boundaries ignored.
Appendix G.4 states: 'For computational feasibility, we considered a centered finite window on the grid, ignoring tails and boundary effects.' This is not fully justified for finite benchmarks.

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Cite this review

Pith. "Pith review of SFO: Learning PDE Operators via Spectral Filtering." pith.science (2026). https://pith.science/paper/OBJHDZVD

@misc{pith2026260117090,
  author       = {Pith},
  title        = {Pith review of: SFO: Learning PDE Operators via Spectral Filtering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OBJHDZVD}},
  note         = {Machine review of arXiv:2601.17090}
}

read the original abstract

Partial differential equations (PDEs) govern complex systems, yet neural operators often struggle to efficiently capture the long-range, nonlocal interactions inherent in their solution maps. We introduce Spectral Filtering Operator (SFO), a neural operator that parameterizes integral kernels using the Universal Spectral Basis (USB), a fixed, global orthonormal basis derived from the eigenmodes of the Hilbert matrix in spectral filtering theory. Motivated by our theoretical finding that the discrete Green's functions of shift-invariant PDE discretizations exhibit spatial Linear Dynamical System (LDS) structure, we prove that these kernels admit compact approximations in the USB. By learning only the spectral coefficients of rapidly decaying eigenvalues, SFO achieves a highly efficient representation. Across six benchmarks, including reaction-diffusion, fluid dynamics, and 3D electromagnetics, SFO achieves state-of-the-art accuracy, reducing error by up to 40% relative to strong baselines while using substantially fewer parameters.

Figures

Figures reproduced from arXiv: 2601.17090 by the authors.

**Figure 1.** SFO architecture. The input field (a(x), x), is lifted by a map P to a latent representation v 0 . The latent field is updated by T stacked STU layers, each applying an operator K. Finally, a projection Q maps the last latent state v T back to the solution field u(x) [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 3.** Magnitude of learned Hilbert spectral coefficients vs. mode index l. We plot the Frobenius norms ∥Θl∥F averaged across STU layers. Coefficients decay with l, indicating SFO concentrates most of its capacity on a few low-index global modes. PDEs are nonlinear, the corresponding lifted operators exhibit geometrically decaying Green’s kernels (App. G). 7.1. Ablation Studies To evaluate the contribution of the key arch… view at source ↗

**Figure 4.** Test loss vs. STU rank L. The test loss decreases rapidly for small L, indicating that the leading modes capture most of the structure, and then flattens as additional modes give only marginal gains. All curves are min–max normalized to a common scale. Effect of lifting dimension d. We sweep the latent width d with all other hyperparameters fixed [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: Average magnitude of learned Hilbert spectral coefficients across benchmarks. For each STU layer, we compute the Frobenius norm ∥Θl∥F of the coefficient matrix associated with mode l and average across layers. The learned coefficients emphasize low-index modes, support…

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Reference graph

Works this paper leans on

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Analysis of learned Hilbert coefficients

10 SFO: Learning PDE Solution Operators via Hilbert Spectral Modes A. Analysis of learned Hilbert coefficients. We analyze the learned Hilbert spectral coefficients across PDE benchmarks. For each STU layer, we compute the Frobenius norm ∥Θl∥F of the coefficient matrix associated with mode l and average across layers. Figure 5 shows that across 5/6 cases,...

2025

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On a finite discretized grid, FFT corresponds to circular convolution and thus implicitly assumes a periodic extension of the domain

and related variants such as UNO (Rahman et al., 2023), F- FNO (Tran et al., 2023), and U-FNO (Wen et al., 2022)). On a finite discretized grid, FFT corresponds to circular convolution and thus implicitly assumes a periodic extension of the domain. This design reflects an efficiency–expressivity trade-off: FFT- based evaluation scales favorably with grid ...

2023

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Finally, we obtain G[t] =Udiag(g 1[t],

With the above choice ofr k,α k + 2βkrk = p α2 k −4β 2 k, hence gk[t] = r|t| kp α2 k −4β 2 k (βk ̸= 0). Finally, we obtain G[t] =Udiag(g 1[t], . . . , gd[t])U ⊤ = dX k=1 Θk r|t| k ,Θ k := 1p α2 k −4β 2 k U·,kU ⊤ ·,k, G.6. Lemma 2: Exponential decay implies spectral filter learnability Lemma G.3(Spectral filtering approximation for exponentially decaying k...

2017

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Moreover, the Hankel spectrum decays exponentially (see Chapter 11 of (Hazan & Singh, 2022)), yielding the stated logarithmic mode complexity

shows that such responses admit exponentially accurate approximation by projection onto the top eigenvectors of an associated Hankel matrix, with error controlled by the tail of its spectrum (Lemma 4.1 therein). Moreover, the Hankel spectrum decays exponentially (see Chapter 11 of (Hazan & Singh, 2022)), yielding the stated logarithmic mode complexity. Re...

2022