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A novel approach to quantify out-of-distribution uncertainty in Neural and Universal Differential Equations.

Code ↔ Paper

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  1. [1] § Results › Numerical test cases: full reconstruction ↔ comparison_among_different_levels_of_prior_knowledge/comparison_results_on_vector_field.jl, the whole file · a weak match · score 0.52 · Wilcoxon signed rank, Lorenz system, Damped Oscillator, Lotka Volterra, CP, reconstruction

Paper

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The authors' code

Julia · 68 lines · 3.1 KB · other · 1 match

  1. cd(@__DIR__)
  3. using ComponentArrays, Lux, SciMLSensitivity, Serialization, OrdinaryDiffEq, LinearAlgebra, Random, DataFrames, CSV, Plots, Statistics
  4. using Optimization, OptimizationOptimisers, OptimizationOptimJL, StableRNGs
  5. using DiffEqFlux, Flux, Zygote, StatsPlots, LaTeXStrings, Gadfly, ColorSchemes, Dates, Distributions
  6. using HypothesisTests
  8. gr()
  10. ### vector field comparison #####
  12. test_cases = ["lotka-volterra", "damped", "lorenz"]
  13. titles = ["\nLotka-Volterra", "\nDamped oscillator", "\nLorenz system"]
  15. plots = []
  16. for test_case_index in axes(test_cases,1)
  18. test_case = test_cases[test_case_index]
  19. title_case = titles[test_case_index]
  21. cicps_vector_filed_standard_NODE = deserialize("./results_on_vector_field/full_reconstruction/" * test_case * "/cicps_distribution.jld")
  22. cicps_vector_filed_standard_UDE = deserialize("./results_on_vector_field/part_reconstruction/" * test_case * "/cicps_distribution.jld")
  23. cicps_vector_field_standard_UDE_fixed = deserialize("./results_on_vector_field/part_reconstruction_with_known_parameters/" * test_case * "/cicps_distribution.jld")
  25. #boxlot of the three distributions
  26. data = DataFrame(cicps=vcat(cicps_vector_filed_standard_NODE, cicps_vector_filed_standard_UDE, cicps_vector_field_standard_UDE_fixed),
  27. method=vcat(fill("NODE", length(cicps_vector_filed_standard_NODE)), fill("UDE", length(cicps_vector_filed_standard_UDE)), fill("UDE (fixed mech. pars)", length(cicps_vector_field_standard_UDE_fixed))))
  29. plt_tmp = @df data boxplot(:method, :cicps, legend=false, ylabel="CP", title=title_case)
  30. Plots.plot!(xrotation = 45)
  31. Plots.plot!(
  32. bottom_margin = 25Plots.mm,
  33. top_margin = 20Plots.mm,
  34. left_margin = 15Plots.mm,
  35. xtickfont = font(13)
  36. )
  38. push!(plots, plt_tmp)
  39. #save the plot
  41. #perform the wilcoxon signed-rank test between the three distributions
  42. test_NODE_UDE = HypothesisTests.MannWhitneyUTest(cicps_vector_filed_standard_NODE, cicps_vector_filed_standard_UDE)
  43. pval_NODE_UDE = pvalue(test_NODE_UDE, tail=:both)
  45. test_NODE_UDE_fixed = HypothesisTests.MannWhitneyUTest(cicps_vector_filed_standard_NODE, cicps_vector_field_standard_UDE_fixed)
  46. pval_NODE_UDE_fixed = pvalue(test_NODE_UDE_fixed, tail=:both)
  48. test_UDE_UDE_fixed = HypothesisTests.MannWhitneyUTest(cicps_vector_filed_standard_UDE, cicps_vector_field_standard_UDE_fixed)
  49. pval_UDE_UDE_fixed = pvalue(test_UDE_UDE_fixed, tail=:both)
  52. #print in a text file
  53. open("comparison_results_vector_field_standard_" * test_case * ".txt", "w") do f
  54. write(f, "Comparison of CICPS distributions on vector field reconstruction\n")
  55. write(f, "Test NODE vs UDE: p-value = " * string(pval_NODE_UDE) * "\n")
  56. write(f, "Test NODE vs UDE with known parameters: p-value = " * string(pval_NODE_UDE_fixed) * "\n")
  57. write(f, "Test UDE vs UDE with known parameters: p-value = " * string(pval_UDE_UDE_fixed) * "\n")
  58. end
  60. end
  62. #put all the plots together
  63. plt = Plots.plot(plots..., layout=(1,3), size=(1250,600), plot_title ="Comparison of the distributions of CP on vector field reconstruction")
  64. savefig("./comparison_cicps_distributions_vector_field.svg")

comparison_results_on_vector_field.jl at commit 3660854, under other · at the source

Overview

Authors: Stefano Giampiccolo1,2, Giovanni Iacca2, Luca Marchetti1,3
  1. Fondazione The Microsoft Research-University of Trento Centre for Computational and Systems Biology (COSBI), Rovereto,Trento, Italy
  2. Department of Information Engineering and Computer Science (DISI), University of Trento, Povo,Trento, Italy
  3. Department of Cellular, Computational and Integrative Biology (CIBIO), University of Trento, Povo,Trento, Italy
Institutions: The Microsoft Research - University of Trento Centre for Computational and Systems Biology (Italy); University of Trento (Italy)
Journal: NPJ systems biology and applications, volume 12, issue 1, article 125
Dates: received 22 October 2025; accepted 14 May 2026; published online 27 May 2026
Type: Research article · Language: English
License: CC BY-NC-ND
Identifiers: DOI 10.1038/s41540-026-00749-5 · PMID 42204152 · PMCID PMC13494032 · OpenAlex W4414778742
Open access: gold, a free copy (OpenAlex)
Preprint: doi:10.1101/2025.10.01.679743
Status: code verified
Categories: computational modeling (no new data) (modality), human (organism), methods / tools (subfield)
Methods: Statistics, Machine learning
Keywords: Computational biology and bioinformatics, Mathematics and computing, Physics
MeSH: Systems Biology*, Algorithms, Computer Simulation, Ensemble Learning, Humans, Models, Biological, Reproducibility of Results, Uncertainty (* major topic)
Topic: Control Systems and Identification (Control and Systems Engineering, Engineering), according to OpenAlex
Citations: not cited yet (Europe PMC); 60 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

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cosbi-research/UQNU

License: other
State: the link answers, verified on 28 September 2026
Evidence: files inventoried
Commit: 36608549685e0d5e8ca4cc422af0c2d7956b35d1, 16 April 2026
Languages: Julia (613), Shell (48)
Size: 3,434 files, 661 scripts
Software Heritage: archived
Found in: “Code availability”
Holds: README, license file, environment (Manifest.toml, Project.toml, analysis_NODE_Laplace/Manifest.toml, analysis_NODE_Laplace/Project.toml, analysis_NODE_maximized/Manifest.toml, analysis_NODE_maximized/Project.toml, analysis_NODE_maximized_Lorenz_1500_epochs/Manifest.toml, analysis_NODE_maximized_Lorenz_1500_epochs/Project.toml, analysis_NODE_maximized_multiple_training_set/Manifest.toml, analysis_NODE_maximized_multiple_training_set/Project.toml, analysis_NODE_maximized_thre_e-4/Manifest.toml, analysis_NODE_maximized_thre_e-4/Project.toml)
Not found: CITATION.cff, tests, continuous integration, documentation
Tools: Plots.jl (542 files), DataFrames.jl (521 files), DifferentialEquations.jl (482 files), Flux.jl (446 files), Distributions.jl (402 files), Makie (9 files)
Availability: 1 check, the latest on 28 September 2026: the link answers
  • 28 September 2026: the link answers
663 files
At the source: github.com/cosbi-research/UQNU

Code availability statement

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Read it in the paper: doi.org/10.1038/s41540-026-00749-5.

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Data

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Code and data availability statement

The paper has a code and 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/s41540-026-00749-5.

Versions

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Version 1, 28 September 2026: the first record

Recorded: type, language, journal, volume, issue, pages, dates, 3 authors, 3 keywords, 8 MeSH terms, 36 references.

Cite

This paper

Giampiccolo, S., Iacca, G., & Marchetti, L. (2026). A novel approach to quantify out-of-distribution uncertainty in Neural and Universal Differential Equations. NPJ systems biology and applications, 12(1), 125. https://doi.org/10.1038/s41540-026-00749-5

BibTeX

@article{giampiccolo2026novel,
author = {Giampiccolo, Stefano and Iacca, Giovanni and Marchetti, Luca},
title = {{A novel approach to quantify out-of-distribution uncertainty in Neural and Universal Differential Equations}},
journal = {NPJ systems biology and applications},
year = {2026},
month = may,
volume = {12},
number = {1},
pages = {125},
publisher = {Nature Publishing Group},
issn = {2056-7189},
doi = {10.1038/s41540-026-00749-5},
url = {https://doi.org/10.1038/s41540-026-00749-5},
pmid = {42204152},
pmcid = {PMC13494032}
}

RIS

TY - JOUR
AU - Giampiccolo, Stefano
AU - Iacca, Giovanni
AU - Marchetti, Luca
TI - A novel approach to quantify out-of-distribution uncertainty in Neural and Universal Differential Equations
T2 - NPJ systems biology and applications
J2 - NPJ Syst Biol Appl
PY - 2026
DA - 2026/05/27
VL - 12
IS - 1
SP - 125
SN - 2056-7189
PB - Nature Publishing Group
DO - 10.1038/s41540-026-00749-5
UR - https://doi.org/10.1038/s41540-026-00749-5
LA - en
ER -

CSL-JSON

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"author": [
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"family": "Giampiccolo",
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