Writing study: Fast training of accurate physics-informed neural networks without gradient descent

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← ICLR 2026 oral writing corpus

Chinmay Datar, Taniya Kapoor, Abhishek Chandra, Qing Sun, Erik L. Bolager, Iryna Burak, Anna Veselovska, Massimo Fornasier, Felix Dietrich · Official ICLR 2026 oral record

Observed writing choices, not acceptance rules. This study examines the inspected manuscript version. It does not establish why the paper was accepted or that its presentation choices should be copied unchanged.

Figure, table and section ledger

Editorial contribution type: Learning & optimization algorithms (primary) · Inference, algorithms & systems efficiency.

Layout extraction, not a required paper template. Counts are distinct caption labels; a multi-panel figure counts once. Section spans are estimated from heading positions, including floats and whitespace. Method and limitations.

Canonical PDF

60 physical pages

Before references / appendix

≈ 10.29 page-equivalents

Main figures / tables

7 / 2

Attached appendix figures / tables

23 / 32

Attached appendix span

≈ 41.89 page-equivalents

Before references / appendix: ≈ 17.15%References / intervening material: ≈ 13.03%Attached appendix: ≈ 69.82%Estimated layout envelopes, not text/artwork area. Unresolved appendix boundaries remain within the undivided post-body material.

Section-by-section lengths and source pages

The title/abstract/front matter precede the first numbered heading. A span ends at the next detected top-level heading, not at the last sentence; end-page whitespace is included. References and statements are listed separately. These are not measurements of text-only or artwork area.

Top-level heading	Starts on PDF page	Estimated span	Share of entire PDF
1 INTRODUCTION	1	≈ 1.67 pp.	2.78%
2 SOLVING TIME-DEPENDENT PDES USING FROZEN-PINNS	3	≈ 2.33 pp.	3.88%
3 EMPIRICAL RESULTS	5	≈ 3.23 pp.	5.38%
4 CONCLUSION	8	≈ 2.31 pp.	3.85%
ACKNOWLEDGMENTS	11	≈ 0.18 pp.	0.3%
REFERENCES	11	≈ 7.81 pp.	13.02%
APPENDIX	19	≈ 2 pp.	3.33%
A EXTENDED REVIEW OF RELATED WORK	21	≈ 1.51 pp.	2.52%
B SUPPLEMENTARY METHODOLOGICAL DETAILS ON PDE SOLVERS	22	≈ 10.26 pp.	17.1%
C SUPPLEMENTARY DETAILS ON NUMERICAL EXPERIMENTS	32	≈ 28.13 pp.	46.88%

All detected figure and table locations

Main text

• Figure 1 · p. 2
• Figure 2 · p. 4
• Figure 3 · p. 5
• Figure 4 · p. 6
• Figure 5 · p. 7
• Table 1 · p. 9
• Figure 6 · p. 10
• Table 2 · p. 10
• Figure 7 · p. 10

Attached appendix

• Figure 8 · p. 24
• Table 31 · p. 24
• Figure 9 · p. 26
• Figure 10 · p. 27
• Figure 11 · p. 32
• Figure 12 · p. 34
• Figure 13 · p. 36
• Figure 14 · p. 36
• Table 3 · p. 37
• Figure 15 · p. 37
• Table 4 · p. 38
• Table 5 · p. 38
• Table 6 · p. 39
• Table 7 · p. 39
• Table 8 · p. 39
• Table 9 · p. 39
• Table 10 · p. 40
• Figure 16 · p. 40
• Figure 17 · p. 40
• Table 11 · p. 41
• Table 1 · p. 41
• Table 12 · p. 41
• Table 13 · p. 42
• Figure 18 · p. 42
• Figure 19 · p. 43
• Figure 20 · p. 44
• Figure 21 · p. 45
• Table 14 · p. 45
• Table 15 · p. 46
• Table 16 · p. 46
• Table 17 · p. 46
• Figure 22 · p. 47
• Figure 23 · p. 47
• Table 18 · p. 47
• Table 21 · p. 47
• Figure 24 · p. 48
• Figure 25 · p. 49
• Figure 26 · p. 49
• Table 19 · p. 50
• Table 20 · p. 50
• Table 22 · p. 50
• Table 24 · p. 51
• Table 23 · p. 51
• Table 25 · p. 52
• Table 26 · p. 52
• Table 27 · p. 52
• Figure 27 · p. 53
• Figure 28 · p. 54
• Table 28 · p. 54
• Table 29 · p. 56
• Figure 29 · p. 57
• Figure 30 · p. 58
• Table 30 · p. 59
• Table 32 · p. 60
• Table 33 · p. 60

Argument and organization

1. Two bottlenecks and space–time separation

   Motivates fixed spatial features with evolving output coefficients.

2. Constructing and solving the reduced system

   Adds sampling, boundary handling, SVD compression and an explicit training algorithm.

3. Experiments organized by PDE difficulty

   Tests temporal frequency, shocks, geometry, chaos and dimension before extended per-problem recipes.

Problem, claim and contribution

The introduction names optimization difficulty and non-causal time treatment separately, then maps the proposed solver components to those obstacles.

Section 1, pp. 1–3 · Main paper

Argument and section order

Experiment headings describe stressors rather than simply naming datasets, helping readers understand why each PDE is included.

Sections 3.1–3.8, pp. 6–8 · Main paper

Figures, captions and visual explanation

Color-matched spatial bases and temporal coefficients explain the factorization; separate arrows assign initial, boundary and PDE constraints to their respective solvers.

Figure 1, p. 2 · Main paper

Tables, comparisons and evidence

The summary table retains mesh-based methods, missing runtimes, GPU markers and low/high precision divisions. This is more informative than a single speedup column.

Table 1, p. 9 · Main paper

Appendix and reproducibility

A detailed contents list leads to each equation, boundary condition, parameter sweep and error map; the high-dimensional discussion admits the simple analytical structure of its benchmark.

Appendix contents, pp. 19–20; Appendix C.8, pp. 54–60 · Main paper

Clarity and writing conventions

Some supplementary numerical ratios and transcribed derivations need reconciliation: Table 22's width and timing ratios do not match its displayed operands, and the wave rearrangement changes a sign.

Appendix B.2.3, p. 29; Table 22, p. 50 · Main paper

What to reuse—and what not to infer

Transferable lesson: Organize numerical experiments around the failure modes a method is intended to solve, and compare runtime only alongside explicit accuracy targets and conventional numerical baselines.

Boundary: Do not generalize smooth, known-PDE demonstrations to arbitrary high-dimensional problems or inverse imaging. Check printed equations and ratio arithmetic against executable code before reproducing them; the observed performance is not a universal guarantee.

Apply this lesson using the argument and evidence worksheet.

Inspection record and sources

Checked

2026-09-19

Reading status

Full text + selected visuals inspected

Main text

Pages 1–10, Sections 1–4 and reproducibility/ethics statements

Appendix / supplement

Pages 19–60, contents and Appendices A–C.8: related work, all PDE-to-ODE derivations, boundary treatments, sampling discussion, every benchmark specification and ablation table

Visually inspected PDF pages

2, 9

• ICLR 2026 final proceedings · 60 pages