ELM-DeepONets: Backpropagation-Free Training of Deep Operator Networks via Extreme Learning Machines

Hwijae Son

REVIEW 4 major objections 4 minor 25 references

ELM-DeepONets: Backpropagation-Free Training of Deep Operator Networks via Extreme Learning Machines

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read DeepONet training reduces to one least-squares solve over fixed random features, and the paper reports this beats backprop-trained DeepONets in accuracy and speed.

desk verdict A clean, simple ELM-DeepONet with a correct least-squares core, but the write-up leaves the generalized inverse and parameter counts underspecified in the very regime that produces the best numbers. read the letter →

arxiv 2501.09395 v1 pith:WYLCTJ2U submitted 2025-01-16 cs.LG cs.AIcs.NAmath.NA

Hwijae Son This is my paper · ORCID

classification cs.LGcs.AIcs.NAmath.NA MSC 68T0765F20

keywords DeepONetsExtremeLearningMachineleast-squarestrainingoperatorMoore-Penrosepseudoinversebackpropagation-freeforward-inverseproblemsfixedbasisfunctions

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 tries to establish that a DeepONet can be trained without any gradient-based optimization. By freezing the branch and trunk networks at random initial weights and inserting one extra linear layer between them, operator learning becomes a single least-squares solve for a matrix $W$, with solution $W = T^\dagger G B^\dagger$. On several benchmark operators, including nonlinear ODEs, Darcy flow, and an inverse source problem, the method is reported to reach lower or comparable relative $L^2$ error than backprop-trained DeepONets while cutting training time from hundreds of seconds to well under a second. If correct, this would make operator learning much cheaper and shift the question of DeepONet capacity onto the quality of the fixed random features rather than the trained weights.

What carries the argument

The central object is the coupling matrix $W \in \mathbb{R}^{p_2 \times p_1}$. Branch features from the frozen branch network are assembled as $B \in \mathbb{R}^{p_1 \times N}$ for $N$ input functions, and trunk features from the frozen trunk network (or from predefined bases such as sinusoids) are assembled as $T \in \mathbb{R}^{M \times p_2}$ at $M$ collocation points; the whole training objective collapses to $\|TWB - G\|_F^2$. Minimizing this gives $W = T^\dagger G B^\dagger$, where $T^\dagger$ and $B^\dagger$ are Moore-Penrose pseudoinverses, computed as left and right inverses $(T^T T)^{-1}T^T$ and $B^T(BB^T)^{-1}$ under the assumptions $p_2 \le M$ and $p_1 \le N$. This machinery turns a neural-network training problem into a linear algebra solve with only $p_1 p_2$ learnable parameters.

What would settle it

Take an operator whose true outputs lie partly outside the span of the fixed trunk features, e.g. all trunk features even while the target operator has a nonzero odd output component with $p_2=M$; then no $W$ can make $\|TWB - G\|_F$ small, and the ELM-DeepONet error would stay far above a backprop-trained DeepONet, contradicting the claim that random fixed features plus a linear map suffice. A second check is to reproduce the antiderivative experiment with $p_1=5000$, where the paper reports low error: the stated formula $B^\dagger = B^T(BB^T)^{-1}$ is undefined because $BB^T$ is singular, so any finite computed $W$ would show the implementation uses a different solve than the derivation describes.

Watch

Extended reading notes

Core claim

The paper's central claim is that the only genuinely learned part of a DeepONet can be a single matrix $W$ coupling two fixed random feature maps: trunk features $T \in \mathbb{R}^{M \times p_2}$ and branch features $B \in \mathbb{R}^{p_1 \times N}$, so the prediction over the whole training set is $T W B$. Minimizing the Frobenius error $\|TWB - G\|_F^2$ over $W$ yields $W = T^\dagger G B^\dagger$, computed with Moore-Penrose pseudoinverses under the full-rank conditions $p_2 \le M$ and $p_1 \le N$. In the reported experiments the ELM-DeepONet reaches relative $L^2$ errors of 2.12% on an antiderivative operator, 2.91% on a nonlinear ODE, 5.65% on Darcy flow with an MLP branch, and 2.77% on a parabolic inverse source problem, versus 4.12%, 4.87%, 11.77%, and 5.04% for the corresponding vanilla DeepONet baselines; the CNN-branch Darcy variant trails its baseline (6.80% versus 6.32%). Training takes roughly 0.1-3 seconds. The paper also reports that choosing $p_1$ much larger than $N$, beyond the range its derivation covers, improves accuracy, a behavior it flags for future analysis.

Load-bearing premise

The load-bearing premise is that the fixed random branch and trunk features, together with the chosen $p_1$, $p_2$ and basis functions, are expressive enough for a single linear map $W$ to approximate the target operator; the cleanest derivation also requires $p_1 \le N$ and $p_2 \le M$, while the best reported results use $p_1 > N$, leaving the actual computation of $W$ in that regime as an unspecified modeling choice.

Editorial extensions

If this is right

Training time on the reported benchmarks drops to about 0.1-3 seconds on a single GPU, compared with hundreds of seconds for Adam-trained DeepONets.
Relative $L^2$ test errors on the antiderivative and nonlinear ODE problems are roughly half those of the smaller baseline DeepONet (2.12% versus 4.12%, and 2.91% versus 4.87%).
The trunk network can be replaced by fixed basis functions such as sinusoids, and the same least-squares solve still produces competitive errors (2.45% and 3.39% on the two ODE problems).
Using more branch features than training samples ($p_1 > N$) improves accuracy empirically, even though the paper's pseudoinverse derivation assumes $p_1 \le N$; the best-performing regime is consequently not covered by the derivation.
The method extends to forward-inverse problems: on the parabolic inverse source benchmark the ELM-DeepONet reports 2.52-2.77% relative error compared with 5.04% for the baseline DeepONet.

Reading between the lines

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

This suggests, though the paper does not state it outright, that for at least these operator-learning tasks the basis functions do not need to be learned; random fixed features plus a linear readout may carry most of the capacity.
A natural extension, not tested in the paper, is to sample the random features from a distribution matched to the operator's spectrum, since the sinusoidal trunk experiments hint that basis choice matters; this could be verified by comparing singular values of $W$ with the target operator's spectrum.
Because $W$ is fit in one shot, the method should compose with online or incremental learning via pseudoinverse updates, an avenue the paper leaves unexplored.
The reported success of $p_1 > N$ implies the training problem is underdetermined in that regime; checking whether the implementation uses the minimum-norm solution or a regularized solve would clarify how much of the gain depends on that choice, a step the paper does not take.

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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

Quick take on arXiv:2501.09395. The idea is simple and legitimate: fix random features for both trunk and branch, put a learnable matrix W between them, and fit W with one least-squares solve. The math is right in the regime p1 <= N and p2 <= M, and the reported wall-clock times are eye-catching (0.1 s vs 400-1000 s). The author is also candid in the conclusion, noting that the best results violate the stated theoretical condition.

What is actually new: I do not know of a prior ELM-DeepONet with this exact two-random-feature-plus-W formulation. It is a close cousin of random feature methods and kernel ridge regression, and the paper does not engage that literature, so the novelty is real but modest.

The problems are mostly in presentation and reproducibility. First, the derivation uses B^+ = B^T(BB^T)^{-1}, which requires p1 <= N. The sensitivity analysis and the inverse-source experiment use p1 = 5000-10000 with N = 1000 or 2000. In that regime BB^T is singular and the formula is undefined. The correct Moore-Penrose solve (B^T B)^{-1}B^T is still fine, and W = T^+ G B^+ remains the min-norm least-squares solution, so the core method survives, but the paper must say which solve is actually implemented. Second, the parameter counts do not add up: with p1 = 1000 and p2 = 100, W has 100,000 entries, not the 20K listed in Table 1. That looks like an error or a different convention; either way it blocks reconstruction of the cost. Third, comparisons are only against backprop-trained vanilla DeepONets. The paper should position itself against random-feature and kernel operator baselines and cite that literature. Finally, no code or data, and no variance bars in the headline tables, so the accuracy claims are hard to verify as written. The sensitivity tables report averages over 10 trials but no standard deviations, which is minor but easy to fix.

None of this is fatal. The central argument holds. Whoever referees it should ask for the specifics of the generalized inverse for p1 > N, corrected parameter counts, code, and a comparison to random-feature baselines. I would send it out. It is exactly the kind of paper where a referee can help sharpen a useful but under-specified idea.

Referee Report

4 major / 4 minor

Summary. The paper proposes ELM-DeepONet, a DeepONet variant in which the branch and trunk networks have fixed (random or predefined) weights and a new linear layer W ∈ R^{p2×p1} is the only trainable parameter. Training is posed as minimizing the Frobenius norm objective ||T W B − G||_F^2, and the authors derive the closed-form solution W = T† G B† under assumptions p1 ≤ N and p2 ≤ M. Experiments on an antiderivative problem, a nonlinear ODE, Darcy flow, and a parabolic inverse source problem report lower test errors and substantially shorter training times than backprop-trained DeepONets, together with a sensitivity analysis of p1 and p2 and sinusoidal trunk-basis variants.

Significance. The core idea is simple and appealing: reformulating DeepONet training as a single least-squares solve over fixed random features is a natural extension of ELM to operator learning. The paper deserves credit for giving an explicit derivation in the p1 ≤ N, p2 ≤ M regime, for comparing several trunk choices, and for reporting large training-time reductions. However, the headline best results are obtained in the p1 > N regime, where the paper's own pseudoinverse formula is not computable as written, and the exact numerical procedure used there is not specified. The parameter-count tables also contain inconsistencies. If these gaps are fixed, the method would be a useful fast baseline for operator learning, but the current evidence is limited to four synthetic benchmarks and the central cost/accuracy claims are not yet fully reproducible.

major comments (4)

[Sec. 3, Eq. (1)–(2); Sec. 4.2, Tables 3–4; Sec. 4.4] Under p1 ≤ N and p2 ≤ M, the derivation of W = T† G B† is mathematically correct. However, the sensitivity analysis uses p1 = 5000 and 10000 with N = 1000, and the inverse source experiment uses p1 = 10000, so BB^T is singular and the stated formula B† = B^T (BB^T)^{-1} is not computable. The paper acknowledges in Section 5 that this violates the theoretical condition, but it does not specify which generalized inverse or regularization is actually used in those experiments. Because the best reported errors and the 0.1–0.15 s training times come from this p1 > N regime, please state the exact solve (e.g., left pseudoinverse (B^T B)^{-1} B^T with ridge regularization), the rank assumptions made, and whether the reported timings include that solve. Without this, the central claim is not reproducible.
[Sec. 4.1.1, Table 1] For the antiderivative example, the text sets p1 = N = 1000 and p2 = M = 100, which gives W exactly p1*p2 = 100,000 entries. Table 1 nevertheless reports 20K parameters for ELM-DeepONet. Please clarify what the reported count represents (trainable W only, fixed networks included, or a different p1/p2 setting). As written, the parameter comparison and the Section 3 claim that p1p2 is 'significantly smaller' than vanilla DeepONet cannot be evaluated.
[Sec. 4.2, Tables 3–4] The sensitivity analysis states that relative errors are averaged over 10 trials, but it does not describe a validation protocol for selecting p1 and p2. If the grid search is evaluated on the same test set used for the reported accuracies, the headline errors are optimistically biased. Please specify the train/validation/test split and state explicitly whether hyperparameter selection was performed on a validation set.
[Sec. 4.4] The inverse source experiment reports p1 = 10000 and p2 = 50 selected by grid search, but it does not state the number of training functions N or the number of collocation points M. Since the derivation's validity depends on p1 ≤ N and p2 ≤ M, these dataset sizes must be given, along with the exact computation used for W in the p1 > N regime.

minor comments (4)

[Sec. 4.1.1] The branch network architectures are written as '1-64-64-64-1', but the branch network in a DeepONet takes an m-dimensional sensor vector as input; please state the value of m and correct the input dimension notation if it is not 1.
[Sec. 3; Sec. 4.3, Table 5] The statement that p1p2 is 'significantly smaller' than vanilla DeepONet parameters is not universally true: Table 5 reports 10M parameters for ELM-DeepONet versus 181K for DeepONet-MLP. Please qualify the parameter-count claim.
[Sec. 4.3] The sinusoidal trunk basis includes expressions such as cos(nπx + π/2) and cos(nπx + π/2)sin(nπy); please verify these formulas and state whether they are exactly the functions implemented, since small typographical changes in trigonometric phases would change the basis.
[Throughout] There are formatting and typographical errors, including the title spacing ('ELM-D EEP ON ETS') and reference list errors such as 'Xin Taia'; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the least-squares derivation is self-contained; minor self-citations are not load-bearing, and the only caveats are validation-selection and p1>N reproducibility gaps.

full rationale

The central derivation in Section 3 is a standard least-squares reduction: with T, W, B, G as defined, the training loss is written as ||TWB - G||_F^2, and the solution W = T†GB† follows from the normal equations when T has full column rank and B has full row rank. This is a self-contained algebraic step, not a restatement of the conclusion. The reported test errors are held-out evaluations of the fitted W, so the accuracy claims are genuine empirical predictions rather than fitted quantities renamed as predictions. Self-citations (refs [4], [8], [9], [14]) are contextual background and are not load-bearing; the well-posedness of the inverse source problem is attributed to the independent reference [25], not to the authors' own prior work. Two caveats are worth noting but do not amount to circularity. First, hyperparameters p1 and p2 are chosen by grid search (Section 4.3) without an explicitly described validation split, and the sensitivity analysis in Section 4.2 reports test errors; if the same test errors were used as the selection criterion, the reported best errors would be optimistically biased. This is a validation-leakage risk, not a demonstrated reduction of the derivation to its inputs. Second, the paper admits in the Conclusion that large p1, such as p1 = 10000, perform well 'despite violating our theoretical insight' that the pseudoinverse should act as a proper right inverse; this means the exact solve used in the largest experiments is under-specified, creating a reproducibility gap rather than a circular step. No step in the claimed derivation chain is equivalent by construction to its inputs, so the paper is substantially self-contained and not circular in its central claim.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The method's central contribution rests on one linear algebra identity and on the untested premise that fixed random features provide an adequate basis for operator approximation. The full-rank conditions in the derivation are violated by the paper's own best configurations, creating a gap between the mathematical statement and the numerical practice.

free parameters (4)

p1 (branch feature count) = 1000 for ODE examples (= N); 5000 for Darcy (grid search); 10000 for inverse source (grid search)
Controls the capacity of the random branch feature map. Larger p1 improves test error in the sensitivity analysis but violates the derived p1 <= N condition and increases pseudoinverse cost.
p2 (trunk feature count) = 100 for ODE examples (= M); 50 for inverse source; grid-searched for Darcy
Controls the resolution of the output basis. The derivation requires p2 <= M for the left pseudoinverse of T to be valid.
Fixed trunk/branch network sizes = Not fixed; varies by experiment (e.g., 3-layer MLP with p2 hidden nodes for trunk)
Architecture choices affect feature quality and are selected by hand without a stated protocol.
Sinusoidal basis frequency scale (the '32' factor) = 32 (formula garbled in text)
Defines the harmonic basis for the Sinusoidal ELM-DeepONet variant; chosen by hand.

assumptions (4)

standard math Moore-Penrose pseudoinverse minimizes the Frobenius least-squares objective
Used to derive W = T† G B† from ||TWB - G||_F^2.
domain assumption Random fixed features are expressive enough for operator approximation
The method's accuracy relies on random branch/trunk features spanning a space that can represent the target operator; the cited ELM universal approximation theorem [18] covers functions, not operators, so this is an extrapolation.
ad hoc to paper T and B have full rank so that T† and B† are left/right inverses
Stated in Section 3 with conditions p2 <= M and p1 <= N. The paper's best experiments use p1 > N, contradicting this assumption and leaving the computation of W in that regime undefined.
domain assumption Benchmark datasets are correctly generated and representative
Dataset construction relies on GRF sampling and numerical integration, but the sensor count m is not stated, so the input representation is incomplete.

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Cite this review

Pith. "Pith review of ELM-DeepONets: Backpropagation-Free Training of Deep Operator Networks via Extreme Learning Machines." pith.science (2026). https://pith.science/paper/WYLCTJ2U

@misc{pith2026250109395,
  author       = {Pith},
  title        = {Pith review of: ELM-DeepONets: Backpropagation-Free Training of Deep Operator Networks via Extreme Learning Machines},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WYLCTJ2U}},
  note         = {Machine review of arXiv:2501.09395}
}

read the original abstract

Deep Operator Networks (DeepONets) are among the most prominent frameworks for operator learning, grounded in the universal approximation theorem for operators. However, training DeepONets typically requires significant computational resources. To address this limitation, we propose ELM-DeepONets, an Extreme Learning Machine (ELM) framework for DeepONets that leverages the backpropagation-free nature of ELM. By reformulating DeepONet training as a least-squares problem for newly introduced parameters, the ELM-DeepONet approach significantly reduces training complexity. Validation on benchmark problems, including nonlinear ODEs and PDEs, demonstrates that the proposed method not only achieves superior accuracy but also drastically reduces computational costs. This work offers a scalable and efficient alternative for operator learning in scientific computing.

Figures

Figures reproduced from arXiv: 2501.09395 by the authors.

**Figure 2.** An illustration of DeepONet, with the picture adapted from [10]. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** An illustration of the proposed ELM-DeepONets. The parameters of the branch and trunk networks are fixed [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Numerical results for the antiderivative example on a single test sample demonstrate that DeepONet fails to [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: An illustration of sinusoidal basis functions for the Darcy Flow example. Fixed basis functions are used as an [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: Results for a randomly selected test sample demonstrate that the ELM-DeepONet with an MLP branch [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: Results for two randomly selected test samples demonstrate that the proposed methods outperform the vanilla [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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

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Min Sue Park, Cheolhyeong Kim, Hwijae Son, and Hyung Ju Hwang. The deep minimizing movement scheme. Journal of Computational Physics, 494:112518, 2023

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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

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