Prandtl

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CFD surrogate modeling toolkit. Train fast surrogates for aerodynamic predictions — scikit-learn-like API.

📖 Full Documentation — install guide, user guide, API reference, examples

import prandtl as pr

# Sample parameter space + analytical truth
X, Y = pr.sample(pr.analytical.cl_flat_plate, bounds=[(-5, 15), (0.01, 0.1)], n=100)

# Train Gaussian Process surrogate
surrogate = pr.Surrogate(params=["alpha", "camber"], outputs=["CL"], method="gp")
surrogate.fit(X, Y)

# Predict + validate
Y_pred = surrogate.predict(X_test)
report = pr.metrics(Y_test, Y_pred)
print(report)  # {"CL": {"r2": 0.9998, "rmse": 0.0012, "mae": 0.0010}}

# Export for deployment
surrogate.export("model.onnx")  # one .onnx file per output

The problem

Question: How much lift does your drone's rotor generate? → Run a CFD simulation: 40 minutes.

You actually want 100 different RPM–angle-of-attack combos → That's 40×100 = 66 hours.

Prandtl's approach: Learn the pattern from 100 sampled points → predict the remaining 10,000 combos in milliseconds, error < 0.2%.

Plain English: CFD is an expensive calculator — each button press costs 30 minutes. Prandtl clones that calculator — the clone returns instant results that are almost indistinguishable from the original.

This is ML at its most practical: not images, not chat, not recommendations — just learning one function to replace another that's too slow.

Install

pip install prandtl-cfd             # Base (numpy, scipy, torch)
pip install prandtl-cfd[gp]         # Gaussian Process backend (GPyTorch)
pip install prandtl-cfd[export]     # ONNX export support
pip install prandtl-cfd[all]        # Everything (gp + export)

v0.5.0 highlights

New model backends: Random Forest & Gradient Boosting

# Random Forest — no PyTorch/GPyTorch needed
surrogate = pr.Surrogate(params=["alpha", "mach"], outputs=["CL", "CD"], method="rf")
surrogate.fit(X, Y)
Y_pred = surrogate.predict(X_test)

# RF uncertainty: standard deviation across tree ensemble
Y_mu, Y_std = surrogate.predict_with_uncertainty(X_test)

Active learning — "where to sample next?"

from prandtl import ActiveLearner

learner = ActiveLearner(surrogate, X_pool, strategy="max_std")
X_next = learner.query(n=10)          # pick the 10 most uncertain points
surrogate.fit(X_next, Y_new)          # label them and retrain

Co-Kriging: multi-fidelity surrogate

from prandtl import CoKriging

ck = CoKriging(params=["alpha"], outputs=["CL"])
ck.fit(X_cheap, Y_cheap, X_expensive, Y_expensive)
Y_pred = ck.predict(X_test)

GPU/CUDA support

surrogate = pr.Surrogate(params=["alpha"], outputs=["CL"], method="mlp", device="cuda")
surrogate.fit(X, Y)  # trains on GPU

More

• Sobolev training — GradientConstraint for physics-informed gradient matching
• Uncertainty quantification — predict_with_uncertainty() for GP and RF
• Analytical benchmarks — NACA0012, RAE2822 added to prandtl.analytical

v0.4.0 highlights

Sobol sampling (new) + Matern kernels

# Low-discrepancy Sobol sequences — deterministic, reproducible
X, Y = pr.sample(func, bounds=[(0, 1), (-2, 2)], n=128, method="sobol")

# GP with Matern kernel variants for different smoothness assumptions
surrogate = pr.Surrogate(params=["alpha"], outputs=["CL"], method="gp",
                          gp_kernel="matern52")  # ν=2.5 (smooth)
# Also: "matern15" (ν=1.5), "matern25" (ν=0.5, rough), "rbf" (default)

Cross-validation & metrics

# K-fold cross-validation — one line
scores = pr.cross_validate(surrogate, X, Y, cv=5)
# → {"CL": {"mae_mean": 0.012, "mae_std": 0.003, "r2_mean": 0.999, ...}}

# Extended metrics beyond RMSE/R²
metrics = pr.metrics(Y, Y_pred)
# → {"CL": {"r2": 0.9996, "rmse": 0.0010, "mae": 0.0008,
#            "max_re": 0.0034, "explained_variance": 0.9996}}

# Residual diagnostics
res = pr.residual_analysis(Y, Y_pred)
# → {"CL": {"shapiro_stat": 0.987, "shapiro_p": 0.42,  # p>0.05 → normal ✓
#            "skewness": -0.15, "kurtosis": 2.91, "max_residual_idx": 7,
#            "residuals": array([...])}}

# Learning curve — performance vs training size
curve = pr.learning_curve(surrogate, X, Y, sizes=[20, 40, 60, 80, 100])
# → {"train_sizes": [20, 40, 60, 80, 100],
#     "train_mae": [0.005, 0.008, 0.010, 0.011, 0.012],
#     "val_mae":   [0.018, 0.014, 0.013, 0.012, 0.012]}

Physics constraints (v0.2.0+)

from prandtl import Monotonicity, Convexity, BoundaryValue

surrogate.fit(X, Y, physics=[
    Monotonicity(param_idx=0, sign=1, weight=0.1),          # CL ↑ monotonically with α
    BoundaryValue({"alpha": 0.0}, {"CL": 0.0}, weight=10.0), # CL=0 at α=0
    Convexity(param_idx=0, sign=-1, weight=0.05),            # concave drag polar
], n_iter=500, lr=0.01)

CFD data I/O

from prandtl import read_foam_forces, read_su2_history

X, Y = read_foam_forces("postProcessing/forces/0/coefficient.dat")
surrogate.fit(X, Y)

What it does

Prandtl lets you replace expensive CFD simulations with fast ML surrogates — without writing any ML boilerplate.

Feature	Description
Four backends	GP (gp), MLP (mlp), Random Forest (rf), Gradient Boosting (gb) — no PyTorch needed for tree models
Uncertainty	predict_with_uncertainty() — GP analytic variance, RF tree-ensemble variance
Zero CFD required	Validate your surrogate pipeline with built-in analytical truth functions (thin airfoil theory, cylinder drag, propeller thrust, NACA 0012, RAE 2822)
Active learning	ActiveLearner — Bayesian optimization for smart sampling
Multi-fidelity	CoKriging — combine cheap + expensive simulation data
GPU/CUDA	device='cuda' flag for MLP backend
Multi-output	One surrogate predicts CL, CD, CM simultaneously
Validation suite	Cross-validation, learning curves, residual analysis, and extended metrics (R², RMSE, MAE, MaxRE, Explained Variance)
Physics constraints	Monotonicity, convexity, boundary value, and Sobolev gradient constraints during training
ONNX export	Export trained MLP surrogates for deployment in any ONNX runtime
Sci-kit learn style	.fit(), .predict(), .validate() — if you know sklearn, you know Prandtl

Quick Tour

1. Validate with analytical truth (zero CFD)

import prandtl as pr

# Thin airfoil lift coefficient: CL = 2π(α + 2camber)
X, Y = pr.sample(
    pr.analytical.cl_flat_plate,
    bounds=[(-5, 15),    # alpha: -5° to 15°
            (0.01, 0.1)],  # camber: 1% to 10%
    n=100,
    method="lhs",
    seed=42,
)

surrogate = pr.Surrogate(params=["alpha", "camber"], outputs=["CL"], method="gp")
surrogate.fit(X, Y)  # learns the analytical function

# Test on new points
X_test, Y_test = pr.sample(pr.analytical.cl_flat_plate, bounds=[(-5, 15), (0.01, 0.1)], n=30, seed=99)
Y_pred = surrogate.predict(X_test)
report = pr.metrics(Y_test, Y_pred)
print(report)  # R² > 0.999 on smooth analytical functions

2. Multiple outputs

def my_airfoil(alpha, mach):
    cl = 2 * np.pi * (np.radians(alpha) + 0.04)
    cd = 0.01 + 0.1 * cl**2  # quadratic drag polar
    return {"CL": cl, "CD": cd}

X, Y = pr.sample(my_airfoil, bounds=[(-5, 15), (0.15, 0.85)], n=200)

surrogate = pr.Surrogate(
    params=["alpha", "mach"], outputs=["CL", "CD"], method="mlp"
)
surrogate.fit(X, Y, n_iter=3000)

# Single call validates all outputs
Y_pred = surrogate.predict(X_test)
report = pr.metrics(Y_test, Y_pred)
# {"CL": {"r2": 0.9995, "rmse": ..., "mae": ...},
#  "CD": {"r2": 0.9987, "rmse": ..., "mae": ...}}

3. Export to ONNX

# MLP surrogates can be exported for deployment
surrogate.export("airfoil_model.onnx")
# Creates: airfoil_model__CL.onnx, airfoil_model__CD.onnx

# Load with onnxruntime
import onnxruntime as ort
session = ort.InferenceSession("airfoil_model__CL.onnx")
cl = session.run(None, {"X": x_new.astype(np.float32)})[0]

4. Sampling methods

# Latin Hypercube Sampling (default) — space-filling
X, Y = pr.sample(func, bounds=[(0, 1), (-2, 2)], n=100, method="lhs")

# Uniform random
X, Y = pr.sample(func, bounds=[(0, 1), (-2, 2)], n=100, method="uniform")

# Sobol sequences — low-discrepancy, reproducible
X, Y = pr.sample(func, bounds=[(0, 1), (-2, 2)], n=128, method="sobol")

# From existing data
surrogate = pr.Surrogate(params=["alpha", "mach"], outputs=["CL"], method="gp")
surrogate.fit(X, Y)  # X: (n_points, n_params), Y: (n_points, n_outputs)

5. Physics-informed training

from prandtl import Monotonicity, BoundaryValue, Convexity

constraints = [
    Monotonicity(param_idx=0, sign=1, weight=0.1),
    # CL must increase with alpha (param_idx=0). sign=+1 enforces monotonic increase.
    BoundaryValue({"alpha": 0.0}, {"CL": 0.0}, weight=10.0),
    # At alpha=0°, CL must be 0. High weight = strict constraint.
    Convexity(param_idx=0, sign=-1, weight=0.05),
    # Concave relationship (sign=-1) — e.g., drag polar curvature.
]

surrogate = pr.Surrogate(params=["alpha", "mach"], outputs=["CL", "CD"], method="mlp")
surrogate.fit(X, Y, physics=constraints, n_iter=500, lr=0.01)

6. Cross-validation

# 5-fold CV: train on 80%, test on 20%, repeat 5 times
scores = pr.cross_validate(surrogate, X, Y, cv=5, verbose=True)
print(f"MAE: {scores['CL']['mae_mean']:.4f} ± {scores['CL']['mae_std']:.4f}")
print(f"R²:  {scores['CL']['r2_mean']:.4f} ± {scores['CL']['r2_std']:.4f}")

# All outputs scored automatically
# {'CL': {'mae_mean': ..., 'mae_std': ..., 'rmse_mean': ..., 'r2_mean': ..., ...},
#  'CD': {'mae_mean': ..., ...}}

7. Learning curve

# See how performance scales with training data
curve = pr.learning_curve(surrogate, X, Y, sizes=[10, 20, 50, 100, 150])

# Interpret: if val_mae plateaus, you have enough data.
# If train_mae ≪ val_mae, you're overfitting — try simpler model or fewer iterations.
print(f"Final train MAE: {curve['train_mae'][-1]:.4f}")
print(f"Final val MAE:   {curve['val_mae'][-1]:.4f}")

8. Residual analysis

res = pr.residual_analysis(Y_test, Y_pred)

# Shapiro-Wilk normality test: p > 0.05 → residuals are normally distributed ✓
for output in res:
    r = res[output]
    print(f"{output}:")
    print(f"  Shapiro-Wilk p={r['shapiro_p']:.3f} {'✓' if r['shapiro_p'] > 0.05 else '✗'}")
    print(f"  Skewness={r['skewness']:.3f}, Kurtosis={r['kurtosis']:.3f}")
    print(f"  Max residual at index {r['max_residual_idx']}")

# High skewness → systematic bias. High kurtosis → outliers.
# Non-normal residuals → model is missing physics or needs more data.

Built-in analytical functions

All return exact mathematical values — perfect for framework validation with zero CFD.

Function	Formula	Parameters
cl_flat_plate(alpha, camber)	CL = 2π(α + 2camber)	α: angle of attack [°], camber: ratio
cd_cylinder(reynolds)	Piecewise Re-dependent CD	Re: Reynolds number
thrust_propeller(rpm, diameter, pitch)	T = CT·ρ·n²·D⁴	rpm, diameter [m], pitch [m]

Architecture

prandtl/
├── __init__.py          # Public API: Surrogate, sample(), cross_validate(), metrics(), ...
├── _surrogate.py        # Core Surrogate class (fit/predict/validate/export)
├── _gaussian.py         # GPyTorch ExactGP wrapper
├── _neural.py           # PyTorch MLP wrapper
├── _tree.py             # Random Forest & Gradient Boosting (scikit-learn)
├── _active.py           # Active learning / Bayesian optimization
├── _co_kriging.py       # Multi-fidelity Co-Kriging
├── _sobolev.py          # Sobolev gradient constraints
├── _validate.py         # Cross-validation, learning curves, residual analysis, metrics
├── _physics.py          # Physics-informed constraints (Monotonicity, Convexity, BoundaryValue)
├── _sampling.py         # LHS, uniform, Sobol samplers
├── _io.py               # CFD data I/O (OpenFOAM forces, SU2 history)
├── _analytical.py       # Analytical truth functions (NACA0012, RAE2822, flat plate, cylinder, propeller)
└── analytical.py        # Public re-export

Limitations

• GP ONNX export: GP models are non-parametric and cannot be exported to ONNX. Use method='mlp' if you need exportable surrogates.
• Tree model export: RF/GB models (scikit-learn) cannot be exported to ONNX. Use method='mlp' for export.
• GB uncertainty: Gradient Boosting uncertainty requires quantile regression. Use GradientBoosting.fit_with_uncertainty() directly.
• Co-Kriging scale: Limited to two fidelity levels in this release.

Roadmap

Done:

• GP + MLP + RF + GB quad backends
• Physics-informed constraints (Monotonicity, Convexity, BoundaryValue, Sobolev gradients)
• Validation suite (cross-validation, learning curves, residual analysis)
• CFD data I/O (OpenFOAM, SU2)
• ONNX export (MLP)
• GPU/CUDA support
• Uncertainty quantification API
• Active learning / Bayesian optimization
• Analytical benchmark functions (NACA 0012, RAE 2822, flat plate, cylinder, propeller)
• Multi-fidelity surrogates (Co-Kriging)

Mid-term (v0.6+):

• Multi-level Co-Kriging (3+ fidelity levels)
• Adaptive sampling strategies (expected improvement, UCB)
• Model interpretability tools (SHAP, partial dependence)
• Distributed training for large-scale datasets

License

MIT