{"id":"554a3586-5798-41c8-b82f-31ced2042dc2","arxiv_id":"2605.24278","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"beignet replaces random Fourier feature embeddings in PINNs with a trainable multi-resolution Fourier feature pyramid, achieving higher accuracy on PDE benchmarks with fewer parameters and near machine precision residuals on the inviscid Burgers blowup using Adam.","lead":"The paper introduces beignet, a neural field model that uses a trainable multi-resolution Fourier feature pyramid instead of random features for physics-informed neural networks solving PDEs. This approach may allow more accurate and efficient solutions for physics simulations by scaling parameters in the feature grid rather than the network itself.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly isolates the optimization stability and artifact risk for the high-accuracy claim. With the full text now available, no additional load-bearing gap (e.g., in parameter counting, derivative correctness, or benchmark fairness) emerges that would alter the UNVERDICTED status; the concern remains the one already flagged.","tokens_in":1796,"tokens_out":271,"duration_ms":28124,"concrete_test":"Recompute the inviscid Burgers residual curves from the self-similar blowup experiment while holding all bandlimits fixed at the coarsest level value; if the final residual remains above 1e-12, the multi-resolution bandlimit scheduling is load-bearing for the machine-precision result.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on the trainable multi-resolution Fourier feature pyramid enabling stable high-accuracy residual minimization via Adam, with efficient spectral derivatives. The architecture description (trainable grids per resolution level, Fourier interpolation for queries, FFT-based derivative composition) is internally consistent for bandlimited functions, and the shift from random to trainable features plus explicit bandlimit control directly addresses known PINN optimization difficulties. No internal inconsistency or hidden assumption in the derivative or interpolation steps is apparent from the provided description.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript introduces beignet, a neural field architecture for physics-informed neural networks that replaces random Fourier features with a trainable multi-resolution Fourier feature pyramid. Features are queried via Fourier interpolation at each pyramid level and decoded by a fully-connected trunk; spatial derivatives are obtained by composing automatic differentiation on the trunk with FFT-based spectral derivatives on the grids. The central claims are that this yields significantly higher accuracy on PDE benchmarks using fewer parameters than prior PINN methods and that residuals on the self-similar inviscid Burgers blowup can be driven to near machine precision with Adam.","tokens_in":1867,"tokens_out":481,"duration_ms":23338,"significance":"If the experimental claims hold, the work would be significant for PINN research: the shift to trainable, explicitly band-limited features plus FFT derivatives directly targets known optimization instabilities, and the ability to scale accuracy by increasing pyramid parameters rather than trunk width offers a more efficient route to high-precision solutions. The architecture description is internally consistent for band-limited functions and avoids circularity in the derivative or interpolation steps.","major_comments":[{"comment":"The abstract and introduction assert near-machine-precision residuals on the Burgers problem and superior benchmark performance, yet the provided text supplies no quantitative details on grid resolutions, band-limit schedules, training-set sizes, error bars, or ablation studies that would allow independent verification of these load-bearing claims.","section":"Abstract and §4 (Experimental Results)"},{"comment":"The weakest assumption—that the trainable pyramid can be stably optimized without interpolation or band-limit artifacts—is not accompanied by a diagnostic (e.g., residual spectra or convergence plots under varying band limits) that would confirm the assumption holds at the reported precision.","section":"§3 (Architecture) and §4"}],"minor_comments":[{"comment":"Notation for the multi-resolution grids and the precise definition of the Fourier interpolation operator should be introduced with an equation in §3 to avoid ambiguity when readers reconstruct the derivative composition.","section":"§3"},{"comment":"Figure captions for the benchmark comparisons should explicitly state the total parameter count and optimizer settings used by each baseline method.","section":"Figure 2 and Table 1"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the constructive comments. We address each major point below and will incorporate revisions to enhance the clarity and verifiability of the experimental claims.","responses":[{"response":"We agree that the abstract and introduction summarize results at a high level. Section 4 of the manuscript contains the experimental configurations, but to facilitate independent verification we will expand the revised manuscript with a dedicated experimental setup subsection that explicitly lists grid resolutions, band-limit schedules, training-set sizes, and training hyperparameters. We will also add error bars from multiple independent runs and ablation studies on pyramid depth and band limits.","revision_made":"yes","referee_comment":"[Abstract and §4 (Experimental Results)] The abstract and introduction assert near-machine-precision residuals on the Burgers problem and superior benchmark performance, yet the provided text supplies no quantitative details on grid resolutions, band-limit schedules, training-set sizes, error bars, or ablation studies that would allow independent verification of these load-bearing claims."},{"response":"The empirical evidence for stable optimization is the successful minimization of residuals to near machine precision on the inviscid Burgers blowup using Adam, a regime not previously reported with first-order methods. Nevertheless, we concur that explicit diagnostics would strengthen the claim. In the revision we will add residual spectra and training convergence curves for multiple band-limit schedules to directly demonstrate the absence of interpolation or band-limit artifacts at the reported precision levels.","revision_made":"yes","referee_comment":"[§3 (Architecture) and §4] The weakest assumption—that the trainable pyramid can be stably optimized without interpolation or band-limit artifacts—is not accompanied by a diagnostic (e.g., residual spectra or convergence plots under varying band limits) that would confirm the assumption holds at the reported precision."}],"tokens_in":1451,"tokens_out":386,"duration_ms":24678,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing here is beignet, a PINN architecture that uses a trainable multi-resolution Fourier feature pyramid instead of random features, with Fourier interpolation for continuous queries and FFT-based derivatives composed with automatic differentiation.\n\nThis is new in the explicit shift to trainable grids at multiple resolutions plus direct bandlimit control. It lets parameter count scale in the feature pyramid rather than the MLP trunk, and it computes spatial derivatives spectrally for efficiency. The architecture is internally consistent for bandlimited functions and targets real PINN pain points around optimization stability and poor scaling.\n\nThe paper shows gains on PDE benchmarks with fewer parameters than prior methods and reaches near machine precision on the self-similar inviscid Burgers blowup using only Adam. That last result is notable if it holds, since high accuracy on that problem has usually required more expensive optimizers.\n\nThe soft spot is the lack of any experimental specifics in the abstract: no dataset sizes, error bars, ablation controls on pyramid levels or interpolation, or training curves. Without those it is hard to judge whether the accuracy improvements are robust or sensitive to choices in the trainable grids. The assumption that the features can be stably optimized without artifacts from interpolation or bandlimit selection is plausible but untested in the given description.\n\nThis is for researchers building or tuning PINNs for forward and inverse PDEs who want a concrete architectural option. It deserves peer review because the proposal is grounded and the claims are falsifiable, even if the current writeup needs more validation to be convincing.","headline":"Beignet swaps random Fourier features for a trainable multi-resolution pyramid with spectral derivatives, claiming better PINN accuracy but with thin experimental detail.","tokens_in":2368,"tokens_out":378,"would_cite":false,"duration_ms":37496,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A trainable multi-resolution Fourier feature pyramid in PINNs yields higher accuracy with fewer parameters than prior methods and reaches near machine precision on difficult problems using Adam.","keywords":["physics-informed neural networks","Fourier features","partial differential equations","neural fields","multi-resolution embeddings","bandlimited representations","Burgers equation","neural PDE solvers"],"falsifier":"A side-by-side evaluation on the PDE benchmarks or Burgers blowup problem showing that beignet does not produce lower residuals than prior PINN methods at matched parameter counts, or that Adam optimization fails to reach residuals near machine precision, would falsify the central performance claims.","tokens_in":2701,"feed_emoji":"🧮","tokens_out":737,"duration_ms":36695,"temperature":0.7,"pith_summary":"The paper presents beignet, an architecture for physics-informed neural networks that solves partial differential equations more accurately. It replaces random Fourier feature embeddings with a trainable multi-resolution Fourier feature pyramid queried via Fourier interpolation. This design enables efficient spectral computation of derivatives using the FFT and allows scaling accuracy by increasing pyramid parameters rather than network size. The result is higher accuracy on benchmarks with fewer total parameters and the ability to reach near machine precision residuals on the inviscid Burgers blowup problem using only the Adam optimizer.","feed_headline":"Trainable Fourier pyramids boost PINN accuracy with fewer parameters","feed_subtitle":"Replacing random features with a multi-resolution pyramid scales accuracy efficiently and reaches near machine precision on hard PDEs using","key_machinery":"Trainable multi-resolution Fourier feature pyramid queried by Fourier interpolation, with derivatives obtained via FFT on the grids.","core_discovery":"beignet replaces the random Fourier feature embedding used by existing PINN models with a trainable multi-resolution Fourier feature pyramid. To query beignet at a continuous coordinate, Fourier interpolation at each level of the pyramid returns features at the input coordinate, and then decodes this vector with a fully-connected neural network trunk. This enables efficient derivative computation by composing automatic differentiation with spectral FFT derivatives, efficient accuracy scaling by increasing pyramid parameters, and direct bandlimit control for stable optimization on difficult PDEs.","pith_inferences":["The multi-resolution pyramid structure may transfer to other coordinate-based neural representations where controlling frequency content across scales is useful.","Bandlimit control could reduce reliance on specialized optimizers in a broader class of physics-constrained learning tasks.","The same interpolation-plus-FFT derivative mechanism might be applied to time-dependent or higher-dimensional PDEs to handle multi-scale behavior.","Parameter efficiency gains could translate to reduced memory requirements when deploying these models for large-scale or real-time simulations."],"forward_implications":["Spatial derivatives are computed efficiently by composing automatic differentiation on the trunk network with spectral derivatives of the feature grids via the FFT.","Accuracy scales by increasing the parameter count of the Fourier feature pyramid rather than the size of the neural network trunk.","Direct control over the representation bandlimit produces more stable optimization on difficult PDEs.","PDE benchmark solutions achieve significantly higher accuracy using fewer total parameters than state-of-the-art PINN methods.","Residuals on the self-similar inviscid Burgers blowup problem reach near machine precision using only the Adam optimizer."],"fun_headline_variants":["Fourier pyramids replace random features in physics-informed networks","Trainable Fourier feature pyramids scale accuracy in PINNs","Multi-resolution pyramids enable efficient derivative computation in PINNs","Beignet controls bandlimit for stable PINN optimization on PDEs"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The trainable multi-resolution Fourier feature pyramid can be stably optimized to produce features that satisfy the PDE residual at the required accuracy without introducing artifacts from the interpolation or bandlimit choices.","fun_headline_variants_meta":{"raw":{"variants":["Fourier pyramids replace random features in physics-informed networks","Trainable Fourier feature pyramids scale accuracy in PINNs","Multi-resolution pyramids enable efficient derivative computation in PINNs","Beignet controls bandlimit for stable PINN optimization on PDEs"]},"model":"grok-4.3","cost_usd":0.006848,"raw_usage":{"total_tokens":3143,"prompt_tokens":754,"num_sources_used":0,"completion_tokens":64,"cost_in_usd_ticks":68478000,"prompt_tokens_details":{"text_tokens":754,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2325,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":754,"tokens_out":64,"duration_ms":17429,"temperature":1.0,"reasoning_tokens":2325,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T15:39:32.315880+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A side-by-side evaluation on the PDE benchmarks or Burgers blowup problem showing that beignet does not produce lower residuals than prior PINN methods at matched parameter counts, or that Adam optimization fails to reach residuals near machine precision, would falsify the central performance claims.","supporting_citations":[],"review_version":1}