Training

Choosing a backend

Backend	Method	Strengths	Limitations
Gaussian Process	method="gp"	Uncertainty estimates, excellent with small data	O(n³) scaling, no ONNX export
MLP	method="mlp"	Scales to large data, ONNX export	Needs tuning, no built-in uncertainty
Random Forest	method="rf"	No PyTorch needed, uncertainty via ensemble	Large memory for many trees
Gradient Boosting	method="gb"	High accuracy, no PyTorch	No built-in uncertainty, needs quantile regression

Gaussian Process

Good for small datasets (< 1000 points) where you want prediction uncertainty.

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

GP models use GPyTorch's ExactGP. They automatically optimize kernel hyperparameters.

MLP

Good for larger datasets or when you need ONNX deployment.

surrogate = pr.Surrogate(
    params=["alpha", "mach"],
    outputs=["CL", "CD"],
    method="mlp"
)
surrogate.fit(X, Y, n_iter=3000, lr=0.001)

Parameter	Default	Description
n_iter	1000	Number of training iterations
lr	0.001	Learning rate
physics	None	List of physics constraints

树模型 (RF/GB)

Random Forest 和 Gradient Boosting 基于 scikit-learn，无需 PyTorch/GPyTorch。适用于中等数据量，训练速度快，内存友好。

# Random Forest
surrogate = pr.Surrogate(
    params=["alpha", "mach"],
    outputs=["CL", "CD"],
    method="rf"
)
surrogate.fit(X, Y)
Y_pred = surrogate.predict(X_test)

# Gradient Boosting
surrogate = pr.Surrogate(
    params=["alpha", "mach"],
    outputs=["CL", "CD"],
    method="gb"
)
surrogate.fit(X, Y)
Y_pred = surrogate.predict(X_test)

Parameter	Default	Description
n_estimators	100	树的数量
max_depth	None	树的最大深度

RF 不确定性

Random Forest 支持 predict_with_uncertainty()：

Y_mu, Y_std = surrogate.predict_with_uncertainty(X_test)

Y_std 是树集成中预测值的标准差，可作为不确定性估计。

Multi-output

One surrogate predicts multiple outputs simultaneously:

def my_airfoil(alpha, mach):
    return {"CL": ..., "CD": ..., "CM": ...}

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

Each output gets its own independent GP or MLP head.