Feature importance-informed symbolic drag modelling, from discovery to interpretation.
Research · Model performance · Web app · Local setup · PCE & sensitivity · Related publication · Cite
Predicting drag across diverse rough surfaces calls for a model that is both accurate and interpretable. This work identifies influential geometric features, learns a structured analytical expression for equivalent sandgrain roughness, and uses polynomial chaos expansion (PCE) to interpret the model's sensitivity.
Ao Cheng, Zhideng Zhou, Xiaolei Yang and Guowei He
A Drag Model for Rough Surfaces Learned Using Feature Importance-Informed Symbolic Regression
Journal of Fluids Engineering 148(4), 041302 (April 2026). Read the article ↗
| Training surfaces | Unseen test surfaces | Overall mean relative error | Overall maximum relative error |
|---|---|---|---|
| 96 | 8 | 9.60% | 30.79% |
The paper reports these results across the 104 supplied cases. Evaluating the public expression on the repository data reproduces 9.5979% mean and 30.7924% maximum absolute relative error.
Five geometric inputs form a product of univariate subfunctions:
The multiplicative template makes each feature's contribution available for inspection. Sections 4.4–4.5 and Appendix C of the article describe the sensitivity analysis and PCE formulation.
Each point evaluates the public analytical expression. DNSLC diamonds mark the eight held-out cases; the second panel summarizes errors within each data family. The plots are newly drawn from the repository data.
The application also displays the two archived symbolic-search fronts in results/csvFiles/, separately from newly executed searches.
Explore the model in your browser, run a symbolic search, and watch a polynomial surrogate take shape.
| Workspace | What you can explore |
|---|---|
| Overview | Rotate the pairwise subfunction product and inspect model predictions across five data families. |
| Symbolic Lab | Restore the paper configuration or set operators, nesting limits, weighted complexity and search budget; inspect equations, logs and held-out predictions. |
| PCE Studio | Configure independent uniform/normal input distributions, polynomial degree and LHS size; fit PolyChaos, play the expansion animation and inspect Sobol indices. |
| Model Explorer | Adjust the geometric statistics and examine univariate subfunctions, pairwise products and source data. |
| Research & Cite | Read the publications, copy BibTeX and review the experiment settings. |
A completed PySR experiment is included as a saved run, process log and result screenshot. Select View verified PySR demo to inspect it without starting another search.
Python 3.12 is the recommended version and was used for local verification.
python -m venv .venv
# Windows PowerShell:
.venv\Scripts\Activate.ps1
# macOS / Linux: source .venv/bin/activate
python -m pip install -r requirements.txt
python -m streamlit run streamlit_app.py --server.port 8507Open http://localhost:8507. On Windows, the launcher handles environment creation and dependency installation:
.\start_streamlit.ps1See the setup and demonstration guide for a walkthrough, or the Streamlit deployment guide to deploy your own instance.
Python GP provides a lightweight genetic-programming search for immediate exploration without Julia. It enforces the selected nested-operator limits and constant-complexity weight, with an optional product template and protected numerical operators.
PySR reads the original results/SR_train.py configuration, including the five-factor template, nested-operator constraints, weighted complexity, float64 precision and L1 objective. To enable live PySR searches locally:
python -m pip install -r requirements-pysr.txtThen select PySR in the app. The first run initializes Julia and SymbolicRegression.jl. Searches run in separate processes, expose their logs and can be cancelled.
Select Paper settings to restore 1500 iterations, 90 populations, population size 80, 1200 cycles per iteration, maxsize 60 and constant complexity 3, with the original multithreading mode and nesting limits. Quick run reduces the search budget and uses serial execution while retaining the same expression constraints. See the symbolic-regression configuration guide for the exact nesting table and control definitions.
The left panel shows g(Ex) × f(Sk), the product of two learned subfunctions. All other subfunctions have unit contribution, following the paper's pairwise visualization. The right panel shows successive degrees of the PCE projected onto this two-factor product. The animation illustrates polynomial approximation rather than fluid evolution in time.
This experiment uses the paper's Table 5 mixed distributions, degree 6, 462 basis terms, 2500 LHS fitting samples and seed 42. Validation uses 1200 independent LHS points, including distribution tails beyond the empirical feature bounds. Exact settings and results are available in verification-results.json.
The app also offers a bounded uniform preset and an Ishigami benchmark for checking sensitivity-index convergence. Statistical moments, validation metrics and Sobol indices use the complete model. For the pairwise animation, orthogonality removes terms involving the other inputs and their mean subfunction contribution is divided out. Calculations call the repository's PolyChaos implementation; Sobol indices assume independent inputs.
Helpful examples and documentation:
- PySR examples and API reference: symbolic regression, expression templates, operators and search settings.
- UQpy PCE gallery: surrogate construction and sensitivity-analysis examples.
- Chaospy polynomial introduction: orthogonal polynomial expansions and uncertainty quantification.
streamlit_app.py Streamlit application entry point
roughlab/ Model adapters, visualizations and experiment workers
code_files/ Original PCE research implementation
pce_core.py PolyChaos: LHS, least squares, moments and Sobol indices
pce_utils.py Polynomial and quadrature utilities
run_analysis.py Original model and analysis example
results/
SR_train.py Original PySR training configuration
csvFiles/ 104 roughness cases and archived search fronts
subFunc_plot.ipynb Factor and interaction notebook
loss_plot.ipynb Search-front notebook
assets/ Figures, animation and application screenshots
docs/ Setup, deployment, saved runs and citation metadata
tools/ Figure generation, browser checks and release packaging
tests/ Numerical and application verification
CITATION.bib / CITATION.cff Publication metadata
requirements*.txt Base dependencies and optional PySR backend
start_streamlit.ps1 Windows launcher
A physics-embedded symbolic wall model for large-eddy simulation of separated flows
Ao Cheng, Xin-Lei Zhang, Zhideng Zhou, Xiaolei Yang and Guowei He.
Journal of Computational Physics 567, 115311 (2026).
A related publication on interpretable, physics-embedded symbolic wall modelling for separated-flow large-eddy simulation.
@article{cheng2026wall,
title = {A physics-embedded symbolic wall model for large-eddy simulation of separated flows},
author = {Cheng, Ao and Zhang, Xin-Lei and Zhou, Zhideng and Yang, Xiaolei and He, Guowei},
journal = {Journal of Computational Physics},
volume = {567},
pages = {115311},
year = {2026},
doi = {10.1016/j.jcp.2026.115311},
url = {https://doi.org/10.1016/j.jcp.2026.115311}
}If you use this code or model, please cite the JFE article:
@article{cheng2026drag,
title = {A Drag Model for Rough Surfaces Learned Using Feature Importance-Informed Symbolic Regression},
author = {Cheng, Ao and Zhou, Zhideng and Yang, Xiaolei and He, Guowei},
journal = {Journal of Fluids Engineering},
volume = {148},
number = {4},
pages = {041302},
year = {2026},
month = apr,
doi = {10.1115/1.4070838},
url = {https://doi.org/10.1115/1.4070838}
}Both entries are available in CITATION.bib. CITATION.cff supports GitHub's citation interface. Publication metadata is documented in citation verification.
Research, original code and scientific interpretation: Ao Cheng and coauthors.
OpenAI Codex assisted with the Streamlit interface, code integration, figure generation, documentation and verification.
Released under the MIT License.