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Learning missing physics from legacy simulators with alternating neural integrators.

Code ↔ Paper

4 matches between paragraphs of the paper and lines of its authors' code, computed by the harvester (lexical-v1). Click a colored paragraph or line to see its counterpart.

The 4 matches · 3 of them tie a paragraph to a whole file, not to given lines: weak matches, whose lines are not tinted
  1. [1] § Methods › Evaluation metrics and baselines ↔ Lorenz-stenflo/sindy_chaos_metric.m, the whole file · a weak match · score 0.74 · Lyapunov exponents, correlation dimension, approximate entropy, metrics, baselines, trajectory
  2. [2] § Results › Recovering missing coupling in a chaotic system ↔ Lorenz-stenflo/sindy_chaos_metric.m, the whole file · a weak match · score 0.69 · Lyapunov exponent, correlation dimension, approximate entropy, RK4, baselines, trajectories
  3. [3] § Methods › Implementation and non-intrusive coupling ↔ Morris/dataset/morris_lecar_ode.m, the whole file · a weak match · score 0.59 · Morris Lecar, activations, fractional, simulation
  4. [4] § Results › From discrepancy to closed-loop refinement ↔ Lorenz-stenflo/reproduce_symbolic_writeback.py, lines 212–239 · score 0.54 · closed loop, Lorenz Stenflo, refinement, distillation, frozen, predictive

Paper

Loaded from Europe PMC by your browser, not stored by OSCR: doi.org · Europe PMC

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The authors' code

MATLAB · 79 lines · 2.4 KB · MIT · 2 matches

  1. % metric_trajectories_for_chaos.m
  3. here = fileparts(mfilename('fullpath'));
  4. matfile = fullfile(here, 'trajectories_for_chaos.mat');
  6. if exist(matfile, 'file') ~= 2
  7. error('Missing %s', matfile);
  8. end
  10. S = load(matfile);
  11. dt = double(S.dt(1));
  12. fs = 1 / dt;
  14. % Order: row 1 = reference truth; rows 2–3 = models (vs your true/base/4th; no 2nd in this .mat)
  15. vars = {'truth', 'rk4_prior_plus_B_baseline', 'ani4_strang_B_ani4'};
  17. nVar = numel(vars);
  18. nTraj = size(S.truth, 1);
  19. T = size(S.truth, 2);
  21. K = 2000;
  22. K = min(K, T);
  23. if K < 500
  24. warning('metric_trajectories_for_chaos: T=%d < 500, using K=%d.', T, K);
  25. end
  27. % 1-based trajectory indices (default: all trajectories in the file)
  28. traj_indices = 1:nTraj;
  30. % Optional: use only (x,y,z) for metrics [1 2 3]; [] = all 4 states
  31. state_cols = [];
  33. get_metrics = @(d) [correlationDimension(d), ...
  34. approximateEntropy(d), ...
  35. lyapunovExponent(d, fs)];
  37. all_metrics = zeros(nVar, 3, numel(traj_indices));
  39. for idx = 1:numel(traj_indices)
  40. t_idx = traj_indices(idx);
  41. for i = 1:nVar
  42. data = squeeze(S.(vars{i})(t_idx, 1:K, :));
  43. if ~isempty(state_cols)
  44. data = data(:, state_cols);
  45. end
  46. if size(data, 1) ~= K
  47. data = reshape(data, K, []);
  48. end
  49. all_metrics(i, :, idx) = get_metrics(data);
  50. end
  51. end
  53. avg_metrics = mean(all_metrics, 3);
  55. m_true = avg_metrics(1, :);
  56. den = abs(m_true);
  57. den(den < eps) = NaN;
  58. rel_errors = abs(avg_metrics(2:end, :) - m_true) ./ den;
  60. methods = {'Baseline (RK4+SINDy)', 'ANI4 (Strang)'};
  61. metrics_names = {'Corr Dimension', 'App Entropy', 'Lyapunov Exp'};
  63. fprintf('\n%s\n', matfile);
  64. fprintf('fs = 1/dt = %.6g (dt = %.6g)\n', fs, dt);
  65. fprintf('K = %d time steps, nTraj averaged = %d\n\n', K, numel(traj_indices));
  67. fprintf('%-22s | %-12s | %-12s | %-12s\n', 'Metric', 'Ground Truth', 'Method Value', 'Rel. Error');
  68. fprintf('------------------------------------------------------------------------------\n');
  70. for m = 1:3
  71. fprintf('--- %s ---\n', metrics_names{m});
  72. for i = 1:size(rel_errors, 1)
  73. val = avg_metrics(i + 1, m);
  74. err = rel_errors(i, m) * 100;
  75. fprintf('%-22s | %-12.5f | %-12.5f | %-7.2f%%\n', ...
  76. methods{i}, m_true(m), val, err);
  77. end
  78. fprintf('------------------------------------------------------------------------------\n');
  79. end

sindy_chaos_metric.m at commit 6b23f21, under MIT · at the source

Overview

Authors: Hao Wang1, Qinghe Wang2,3, Caiyou Yuan2,3, Kailiang Wu2,3,4
ORCID iDs: Hao Wang, Qinghe Wang, Caiyou Yuan, Kailiang Wu
  1. School of Mathematical Sciences, Zhejiang University,Hangzhou, China
  2. Department of Mathematics, Southern University of Science and Technology,Shenzhen, China
  3. Shenzhen International Center for Mathematics, Southern University of Science and Technology,Shenzhen, China
  4. Guangdong Provincial Key Laboratory of Computational Science and Material Design, Southern University of Science and Technology,Shenzhen, China
Institutions: Zhejiang University (China); Southern University of Science and Technology (China)
Journal: Nature communications, volume 17, issue 1, article 7877
Dates: received 17 November 2025; accepted 26 May 2026; published online 23 June 2026
Type: Research article · Language: English
License: CC BY-NC-ND
Identifiers: DOI 10.1038/s41467-026-74002-2 · PMID 42337241 · PMCID PMC13444323 · OpenAlex W7165570580
Open access: gold, a free copy (OpenAlex)
Status: code verified
Methods: Complexity, Statistics, Machine learning
Keywords: Applied mathematics, Computational science
Topic: Model Reduction and Neural Networks (Statistical and Nonlinear Physics, Physics and Astronomy), according to OpenAlex
Funding: National Natural Science Foundation of China (National Science Foundation of China) (92370108); Shenzhen Science and Technology Innovation Commission (JCYJ20250604144300001, RCJC20221008092757098); Science Challenge Project (No. TZ2025007)
Citations: not cited yet (Europe PMC); 24 references in the paper

Abstract

The abstract is not reproduced here: the paper's license (CC BY-NC-ND) does not allow it. Read it in the paper, at the publisher or on Europe PMC.

Repository

Its files are read in the Code ↔ Paper reader above, with 4 matches between paragraphs and lines of code.

shanxue-w/ANI

License: MIT
State: the link answers, verified on 27 September 2026
Evidence: files inventoried
Commit: 6b23f2183cd89ec9779d2a51a81b99b0bdfe5d16, 21 June 2026
Languages: Python (183), MATLAB (24), Shell (8), Jupyter (3), C (2), C++ (2), C/C++ (1)
Size: 911 files, 223 scripts
Software Heritage: not checked
Found in: “Code availability”
Holds: README, license file, environment (requirements.txt, setup.py), tests, 3 notebooks
Not found: CITATION.cff, continuous integration, documentation
Tools: NumPy (168 files), Matplotlib (140 files), PyTorch (127 files), seaborn (43 files), SciPy (23 files), pandas (16 files), SymPy (2 files), scikit-learn (1 file)
Availability: 1 check, the latest on 27 September 2026: the link answers
  • 27 September 2026: the link answers
225 files
At the source: github.com/shanxue-w/ANI

Code availability statement

The paper has a code availability statement. Its license (CC BY-NC-ND) does not allow reproducing it here; in short, from what the harvester recognized in it:

Read it in the paper: doi.org/10.1038/s41467-026-74002-2.

Tracing map

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What the map holds:

  • 1 repository of the authors' code, each at its verified commit, with its license and how the link was found in the paper;
  • 223 scripts, each with its path and the digest of its content;
  • 4 matches between paragraphs of the paper and lines of the code (method lexical-v1);
  • neither the text of the paper nor the code itself.

Its JSON (tracing-map.json) is deposited on Zenodo with its DOI once the map is validated.

Data

Datasets cited

Data availability statement

The paper has a data availability statement. Its license (CC BY-NC-ND) does not allow reproducing it here; in short, from what the harvester recognized in it:

Read it in the paper: doi.org/10.1038/s41467-026-74002-2.

Versions

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Version 1, 27 September 2026: the first record

Recorded: type, language, journal, volume, issue, pages, dates, 4 authors, 2 keywords, 3 funders, 19 references.

Cite

This paper

Wang, H., Wang, Q., Yuan, C., & Wu, K. (2026). Learning missing physics from legacy simulators with alternating neural integrators. Nature communications, 17(1), 7877. https://doi.org/10.1038/s41467-026-74002-2

BibTeX

@article{wang2026learning,
author = {Wang, Hao and Wang, Qinghe and Yuan, Caiyou and Wu, Kailiang},
title = {{Learning missing physics from legacy simulators with alternating neural integrators}},
journal = {Nature communications},
year = {2026},
month = jun,
volume = {17},
number = {1},
pages = {7877},
publisher = {Nature Publishing Group},
issn = {2041-1723},
doi = {10.1038/s41467-026-74002-2},
url = {https://doi.org/10.1038/s41467-026-74002-2},
pmid = {42337241},
pmcid = {PMC13444323}
}

RIS

TY - JOUR
AU - Wang, Hao
AU - Wang, Qinghe
AU - Yuan, Caiyou
AU - Wu, Kailiang
TI - Learning missing physics from legacy simulators with alternating neural integrators
T2 - Nature communications
J2 - Nat Commun
PY - 2026
DA - 2026/06/23
VL - 17
IS - 1
SP - 7877
SN - 2041-1723
PB - Nature Publishing Group
DO - 10.1038/s41467-026-74002-2
UR - https://doi.org/10.1038/s41467-026-74002-2
LA - en
ER -

CSL-JSON

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"container-title": "Nature communications",
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"family": "Wang",
"given": "Hao"
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"given": "Caiyou"
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"given": "Kailiang"
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"container-title-short": "Nat Commun",
"volume": "17",
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"DOI": "10.1038/s41467-026-74002-2",
"PMID": "42337241",
"PMCID": "PMC13444323",
"ISSN": "2041-1723",
"publisher": "Nature Publishing Group",
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"language": "en",
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"date-parts": [
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]
}
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