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TopOpt

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# linux
wget -nc https://paddle-org.bj.bcebos.com/paddlescience/datasets/topopt/top_dataset.h5 -P ./datasets/
# windows
# curl https://paddle-org.bj.bcebos.com/paddlescience/datasets/topopt/top_dataset.h5 --create-dirs -o ./datasets/top_dataset.h5
python topopt.py

# linux
wget -nc https://paddle-org.bj.bcebos.com/paddlescience/datasets/topopt/top_dataset.h5 -P ./datasets/
# windows
# curl https://paddle-org.bj.bcebos.com/paddlescience/datasets/topopt/top_dataset.h5 --create-dirs -o ./datasets/top_dataset.h5
python topopt.py mode=eval 'EVAL.pretrained_model_path_dict={'Uniform': 'https://paddle-org.bj.bcebos.com/paddlescience/models/topopt/uniform_pretrained.pdparams', 'Poisson5': 'https://paddle-org.bj.bcebos.com/paddlescience/models/topopt/poisson5_pretrained.pdparams', 'Poisson10': 'https://paddle-org.bj.bcebos.com/paddlescience/models/topopt/poisson10_pretrained.pdparams', 'Poisson30': 'https://paddle-org.bj.bcebos.com/paddlescience/models/topopt/poisson30_pretrained.pdparams'}'

python topopt.py mode=export INFER.pretrained_model_name=Uniform

python topopt.py mode=export INFER.pretrained_model_name=Poisson5

python topopt.py mode=export INFER.pretrained_model_name=Poisson10

python topopt.py mode=export INFER.pretrained_model_name=Poisson30

# linux
wget -nc https://paddle-org.bj.bcebos.com/paddlescience/datasets/topopt/top_dataset.h5 -P ./datasets/
# windows
# curl https://paddle-org.bj.bcebos.com/paddlescience/datasets/topopt/top_dataset.h5 --create-dirs -o ./datasets/top_dataset.h5
python topopt.py mode=infer INFER.pretrained_model_name=Uniform INFER.img_num=3

# linux
wget -nc https://paddle-org.bj.bcebos.com/paddlescience/datasets/topopt/top_dataset.h5 -P ./datasets/
# windows
# curl https://paddle-org.bj.bcebos.com/paddlescience/datasets/topopt/top_dataset.h5 --create-dirs -o ./datasets/top_dataset.h5
python topopt.py mode=infer INFER.pretrained_model_name=Poisson5 INFER.img_num=3

# linux
wget -nc https://paddle-org.bj.bcebos.com/paddlescience/datasets/topopt/top_dataset.h5 -P ./datasets/
# windows
# curl https://paddle-org.bj.bcebos.com/paddlescience/datasets/topopt/top_dataset.h5 --create-dirs -o ./datasets/top_dataset.h5
python topopt.py mode=infer INFER.pretrained_model_name=Poisson10 INFER.img_num=3

# linux
wget -nc https://paddle-org.bj.bcebos.com/paddlescience/datasets/topopt/top_dataset.h5 -P ./datasets/
# windows
# curl https://paddle-org.bj.bcebos.com/paddlescience/datasets/topopt/top_dataset.h5 --create-dirs -o ./datasets/top_dataset.h5
python topopt.py mode=infer INFER.pretrained_model_name=Poisson30 INFER.img_num=3
预训练模型 指标
topopt_uniform_pretrained.pdparams loss(sup_validator): [0.14336, 0.10211, 0.07927, 0.06433, 0.04970, 0.04612, 0.04201, 0.03566, 0.03623, 0.03314, 0.02929, 0.02857, 0.02498, 0.02517, 0.02523, 0.02618]
metric.Binary_Acc(sup_validator): [0.9410, 0.9673, 0.9718, 0.9727, 0.9818, 0.9824, 0.9826, 0.9845, 0.9856, 0.9892, 0.9892, 0.9907, 0.9890, 0.9916, 0.9914, 0.9922]
metric.IoU(sup_validator): [0.8887, 0.9367, 0.9452, 0.9468, 0.9644, 0.9655, 0.9659, 0.9695, 0.9717, 0.9787, 0.9787, 0.9816, 0.9784, 0.9835, 0.9831, 0.9845]
topopt_poisson5_pretrained.pdparams loss(sup_validator): [0.11926, 0.09162, 0.08014, 0.06390, 0.05839, 0.05264, 0.04921, 0.04737, 0.04872, 0.04564, 0.04226, 0.04267, 0.04407, 0.04172, 0.03939, 0.03927]
metric.Binary_Acc(sup_validator): [0.9471, 0.9619, 0.9702, 0.9742, 0.9782, 0.9801, 0.9803, 0.9825, 0.9824, 0.9837, 0.9850, 0.9850, 0.9870, 0.9863, 0.9870, 0.9872]
metric.IoU(sup_validator): [0.8995, 0.9267, 0.9421, 0.9497, 0.9574, 0.9610, 0.9614, 0.9657, 0.9655, 0.9679, 0.9704, 0.9704, 0.9743, 0.9730, 0.9744, 0.9747]
topopt_poisson10_pretrained.pdparams loss(sup_validator): [0.12886, 0.07201, 0.05946, 0.04622, 0.05072, 0.04178, 0.03823, 0.03677, 0.03623, 0.03029, 0.03398, 0.02978, 0.02861, 0.02946, 0.02831, 0.02817]
metric.Binary_Acc(sup_validator): [0.9457, 0.9703, 0.9745, 0.9798, 0.9827, 0.9845, 0.9859, 0.9870, 0.9882, 0.9880, 0.9893, 0.9899, 0.9882, 0.9899, 0.9905, 0.9904]
metric.IoU(sup_validator): [0.8969, 0.9424, 0.9502, 0.9604, 0.9660, 0.9696, 0.9722, 0.9743, 0.9767, 0.9762, 0.9789, 0.9800, 0.9768, 0.9801, 0.9813, 0.9810]
topopt_poisson30_pretrained.pdparams loss(sup_validator): [0.19111, 0.10081, 0.06930, 0.04631, 0.03821, 0.03441, 0.02738, 0.03040, 0.02787, 0.02385, 0.02037, 0.02065, 0.01840, 0.01896, 0.01970, 0.01676]
metric.Binary_Acc(sup_validator): [0.9257, 0.9595, 0.9737, 0.9832, 0.9828, 0.9883, 0.9885, 0.9892, 0.9901, 0.9916, 0.9924, 0.9925, 0.9926, 0.9929, 0.9937, 0.9936]
metric.IoU(sup_validator): [0.8617, 0.9221, 0.9488, 0.9670, 0.9662, 0.9769, 0.9773, 0.9786, 0.9803, 0.9833, 0.9850, 0.9853, 0.9855, 0.9860, 0.9875, 0.9873]

1. 背景简介

拓扑优化 (Topolgy Optimization) 是一种数学方法，针对给定的一组负载、边界条件和约束，在给定的设计区域内，以最大化系统性能为目标优化材料的分布。这个问题很有挑战性因为它要求解决方案是二元的，即应该说明设计区域的每个部分是否存在材料或不存在。这种优化的一个常见例子是在给定总重量和边界条件下最小化物体的弹性应变能。随着20世纪汽车和航空航天工业的发展，拓扑优化已经将应用扩展到很多其他学科：如流体、声学、电磁学、光学及其组合。SIMP (Simplied Isotropic Material with Penalization) 是目前广泛传播的一种简单而高效的拓扑优化求解方法。它通过对材料密度的中间值进行惩罚，提高了二元解的收敛性。

2. 问题定义

拓扑优化问题：

\[ \begin{aligned} & \underset{\mathbf{x}}{\text{min}} \quad && c(\mathbf{u}(\mathbf{x}), \mathbf{x}) = \sum_{j=1}^{N} E_{j}(x_{j})\mathbf{u}_{j}^{\intercal}\mathbf{k}_{0}\mathbf{u}_{j} \\ & \text{s.t.} \quad && V(\mathbf{x})/V_{0} = f_{0} \\ & \quad && \mathbf{K}\mathbf{U} = \mathbf{F} \\ & \quad && x_{j} \in \{0, 1\}, \quad j = 1,...,N \end{aligned} \]

其中：\(x_{j}\) 是材料分布 (material distribution)；\(c\) 指可塑性 (compliance)；\(\mathbf{u}_{j}\) 是 element displacement vector；\(\mathbf{k}_{0}\) 是 element stiffness matrix for an element with unit Youngs modulu；\(\mathbf{U}\), \(\mathbf{F}\) 是 global displacement and force vectors；\(\mathbf{K}\) 是 global stiffness matrix；\(V(\mathbf{x})\), \(V_{0}\) 是材料体积和设计区域的体积；\(f_{0}\) 是预先指定的体积比。

3. 问题求解

实际求解上述问题时为做简化，会把最后一个约束条件换成连续的形式：\(x_{j} \in [0, 1], \quad j = 1,...,N\)。 常见的优化算法是 SIMP 算法，它是一种基于梯度的迭代法，并对非二元解做惩罚：\(E_{j}(x_{j}) = E_{\text{min}} + x_{j}^{p}(E_{0} - E_{\text{min}})\)，这里我们不对 SIMP 算法做过多展开。由于利用 SIMP 方法, 求解器只需要进行初始的 \(N_{0}\) 次迭代就可以得到与结果的最终结果非常相近的基本视图，本案例希望通过将 SIMP 的第 \(N_{0}\) 次初始迭代结果与其对应的梯度信息作为 Unet 的输入，预测 SIMP 的100次迭代步骤后给出的优化解。

3.1 数据集准备

下载的数据集为整理过的合成数据，整理后的格式为 "iters": shape = (10000, 100, 40, 40)，"target": shape = (10000, 1, 40, 40)

  • 10000 - 随机生成问题的个数

  • 100 - SIMP 迭代次数

  • 40 - 图像高度

  • 40 - 图像宽度

数据集地址请存储于 ./datasets/top_dataset.h5

生成训练集：原始代码利用所有的10000问题生成训练数据。

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def generate_train_test(
    data_iters: np.ndarray,
    data_targets: np.ndarray,
    train_test_ratio: float,
    n_sample: int,
) -> Union[
    Tuple[np.ndarray, np.ndarray], Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray]
]:
    """Generate training and testing set

    Args:
        data_iters (np.ndarray): data with 100 channels corresponding to the results of 100 steps of SIMP algorithm
        data_targets (np.ndarray): final optimization solution given by SIMP algorithm
        train_test_ratio (float): split ratio of training and testing sets, if `train_test_ratio` = 1 then only return training data
        n_sample (int): number of total samples in training and testing sets to be sampled from the h5 dataset

    Returns:
        Union[Tuple[np.ndarray, np.ndarray], Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray]]: if `train_test_ratio` = 1, return (train_inputs, train_labels), else return (train_inputs, train_labels, test_inputs, test_labels)
    """
    n_obj = len(data_iters)
    idx = np.arange(n_obj)
    np.random.shuffle(idx)
    train_idx = idx[: int(train_test_ratio * n_sample)]
    if train_test_ratio == 1.0:
        return data_iters[train_idx], data_targets[train_idx]

    test_idx = idx[int(train_test_ratio * n_sample) :]
    train_iters = data_iters[train_idx]
    train_targets = data_targets[train_idx]
    test_iters = data_iters[test_idx]
    test_targets = data_targets[test_idx]
    return train_iters, train_targets, test_iters, test_targets

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# read h5 data
h5data = h5py.File(cfg.DATA_PATH, "r")
data_iters = np.array(h5data["iters"])
data_targets = np.array(h5data["targets"])

# generate training dataset
inputs_train, labels_train = func_module.generate_train_test(
    data_iters, data_targets, cfg.train_test_ratio, cfg.n_samples

3.2 模型构建

经过 SIMP 的 \(N_{0}\) 次初始迭代步骤得到的图像 \(I\) 可以看作是模糊了的最终结构。由于最终的优化解给出的图像 \(I^*\) 并不包含中间过程的信息，因此 \(I^*\) 可以被解释为图像 \(I\) 的掩码。于是 \(I \rightarrow I^*\) 这一优化过程可以看作是二分类的图像分割或者前景-背景分割过程，因此构建 Unet 模型进行预测，具体网络结构如图所示： Unet

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# set model

详细的模型代码在 examples/topopt/topoptmodel.py 中。

3.3 参数设定

根据论文以及原始代码给出以下训练参数：

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# other parameters
n_samples: 10000
train_test_ratio: 1.0 # use 10000 original data with different channels for training
vol_coeff: 1 # coefficient for volume fraction constraint in the loss - beta in equation (3) in paper

# training settings

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# 4 training cases parameters
LEARNING_RATE = cfg.TRAIN.learning_rate / (1 + cfg.TRAIN.epochs // 15)

3.4 data transform

根据论文以及原始代码给出以下自定义的 data transform 代码，包括随机水平或垂直翻转和随机90度旋转，对 input 和 label 同时 transform：

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def augmentation(
    input_dict: Dict[str, np.ndarray],
    label_dict: Dict[str, np.ndarray],
    weight_dict: Dict[str, np.ndarray] = None,
) -> Tuple[Dict[str, np.ndarray], Dict[str, np.ndarray], Dict[str, np.ndarray]]:
    """Apply random transformation from D4 symmetry group

    Args:
        input_dict (Dict[str, np.ndarray]): input dict of np.ndarray size `(batch_size, any, height, width)`
        label_dict (Dict[str, np.ndarray]): label dict of np.ndarray size `(batch_size, 1, height, width)`
        weight_dict (Dict[str, np.ndarray]): weight dict if any
    """
    inputs = input_dict["input"]
    labels = label_dict["output"]
    assert len(inputs.shape) == 3
    assert len(labels.shape) == 3

    # random horizontal flip
    if np.random.random() > 0.5:
        inputs = np.flip(inputs, axis=2)
        labels = np.flip(labels, axis=2)
    # random vertical flip
    if np.random.random() > 0.5:
        inputs = np.flip(inputs, axis=1)
        labels = np.flip(labels, axis=1)
    # random 90* rotation
    if np.random.random() > 0.5:
        new_perm = list(range(len(inputs.shape)))
        new_perm[-2], new_perm[-1] = new_perm[-1], new_perm[-2]
        inputs = np.transpose(inputs, new_perm)
        labels = np.transpose(labels, new_perm)

3.5 约束构建

在本案例中，我们采用监督学习方式进行训练，所以使用监督约束 SupervisedConstraint，代码如下：

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# set constraints
sup_constraint = ppsci.constraint.SupervisedConstraint(
    {
        "dataset": {
            "name": "NamedArrayDataset",
            "input": {"input": inputs_train},
            "label": {"output": labels_train},
            "transforms": (
                {
                    "FunctionalTransform": {
                        "transform_func": func_module.augmentation,
                    },
                },
            ),
        },
        "batch_size": cfg.TRAIN.batch_size,
        "sampler": {
            "name": "BatchSampler",
            "drop_last": False,
            "shuffle": True,
        },
    },
    ppsci.loss.FunctionalLoss(loss_wrapper(cfg)),
    name="sup_constraint",
)

SupervisedConstraint 的第一个参数是监督约束的读取配置，配置中 "dataset" 字段表示使用的训练数据集信息，其各个字段分别表示：

  1. name： 数据集类型，此处 "NamedArrayDataset" 表示分 batch 顺序读取的 np.ndarray 类型的数据集；
  2. input： 输入变量字典：{"input_name": input_dataset}；
  3. label： 标签变量字典：{"label_name": label_dataset}；
  4. transforms： 数据集预处理配，其中 "FunctionalTransform" 为用户自定义的预处理方式。

读取配置中 "batch_size" 字段表示训练时指定的批大小，"sampler" 字段表示 dataloader 的相关采样配置。

第二个参数是损失函数，这里使用自定义损失，通过 cfg.vol_coeff 确定损失公式中 \(\beta\) 对应的值。

第三个参数是约束条件的名字，方便后续对其索引。此次命名为 "sup_constraint"。

在约束构建完毕之后，以我们刚才的命名为关键字，封装到一个字典中，方便后续访问。

3.6 采样器构建

原始数据第二维有100个通道，对应的是 SIMP 算法 100 次的迭代结果，本案例模型目标是用 SIMP 中间某一步的迭代结果直接预测 SIMP 算法100步迭代后最终的优化求解结果，这里需要构建一个通道采样器，用来将输入模型数据的第二维按一定的概率分布随机抽取某一通道或直接指定某一通道，再输入网络进行训练或推理。本案例将采样步骤放入模型的 forward 方法中。

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def uniform_sampler() -> Callable[[], int]:
    """Generate uniform sampling function from 1 to 99

    Returns:
        sampler (Callable[[], int]): uniform sampling from 1 to 99
    """
    return lambda: np.random.randint(1, 99)


def poisson_sampler(lam: int) -> Callable[[], int]:
    """Generate poisson sampling function with parameter lam with range 1 to 99

    Args:
        lam (int): poisson rate parameter

    Returns:
        sampler (Callable[[], int]): poisson sampling function with parameter lam with range 1 to 99
    """

    def func():
        iter_ = max(np.random.poisson(lam), 1)
        iter_ = min(iter_, 99)
        return iter_

    return func


def generate_sampler(sampler_type: str = "Fixed", num: int = 0) -> Callable[[], int]:
    """Generate sampler for the number of initial iteration steps

    Args:
        sampler_type (str): "Poisson" for poisson sampler; "Uniform" for uniform sampler; "Fixed" for choosing a fixed number of initial iteration steps.
        num (int): If `sampler_type` == "Poisson", `num` specifies the poisson rate parameter; If `sampler_type` == "Fixed", `num` specifies the fixed number of initial iteration steps.

    Returns:
        sampler (Callable[[], int]): sampler for the number of initial iteration steps
    """
    if sampler_type == "Poisson":
        return poisson_sampler(num)
    elif sampler_type == "Uniform":
        return uniform_sampler()
    else:
        return lambda: num

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# initialize SIMP iteration stop time sampler

3.7 优化器构建

训练过程会调用优化器来更新模型参数，此处选择 Adam 优化器。

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# set optimizer
optimizer = ppsci.optimizer.Adam(learning_rate=LEARNING_RATE, epsilon=1.0e-7)(
    model

3.8 loss和metric构建

3.8.1 loss构建

损失函数为 confidence loss + beta * volume fraction constraints:

\[ \mathcal{L} = \mathcal{L}_{\text{conf}}(X_{\text{true}}, X_{\text{pred}}) + \beta * \mathcal{L}_{\text{vol}}(X_{\text{true}}, X_{\text{pred}}) \]

confidence loss 是 binary cross-entropy:

\[ \mathcal{L}_{\text{conf}}(X_{\text{true}}, X_{\text{pred}}) = -\frac{1}{NM}\sum_{i=1}^{N}\sum_{j=1}^{M}\left[X_{\text{true}}^{ij}\log(X_{\text{pred}}^{ij}) + (1 - X_{\text{true}}^{ij})\log(1 - X_{\text{pred}}^{ij})\right] \]

volume fraction constraints:

\[ \mathcal{L}_{\text{vol}}(X_{\text{true}}, X_{\text{pred}}) = (\bar{X}_{\text{pred}} - \bar{X}_{\text{true}})^2 \]

loss 构建代码如下：

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# define loss wrapper
def loss_wrapper(cfg: DictConfig):
    def loss_expr(output_dict, label_dict, weight_dict=None):
        label_true = label_dict["output"].reshape((-1, 1))
        label_pred = output_dict["output"].reshape((-1, 1))
        conf_loss = paddle.mean(
            nn.functional.log_loss(label_pred, label_true, epsilon=1.0e-7)
        )
        vol_loss = paddle.square(paddle.mean(label_true - label_pred))
        return {"output": conf_loss + cfg.vol_coeff * vol_loss}

3.8.2 metric构建

本案例原始代码选择 Binary Accuracy 和 IoU 进行评估:

\[ \text{Bin. Acc.} = \frac{w_{00}+w_{11}}{n_{0}+n_{1}} \]
\[ \text{IoU} = \frac{1}{2}\left[\frac{w_{00}}{n_{0}+w_{10}} + \frac{w_{11}}{n_{1}+w_{01}}\right] \]

其中 \(n_{0} = w_{00} + w_{01}\) ， \(n_{1} = w_{10} + w_{11}\) ，\(w_{tp}\) 表示实际是 \(t\) 类且被预测为 \(p\) 类的像素点的数量 metric 构建代码如下：

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# define metric
def val_metric(output_dict, label_dict, weight_dict=None):
    label_pred = output_dict["output"]
    label_true = label_dict["output"]
    accurates = paddle.equal(paddle.round(label_true), paddle.round(label_pred))
    acc = paddle.mean(paddle.cast(accurates, dtype=paddle.get_default_dtype()))
    true_negative = paddle.sum(
        paddle.multiply(
            paddle.equal(paddle.round(label_pred), 0.0),
            paddle.equal(paddle.round(label_true), 0.0),
        ),
        dtype=paddle.get_default_dtype(),
    )
    true_positive = paddle.sum(
        paddle.multiply(
            paddle.equal(paddle.round(label_pred), 1.0),
            paddle.equal(paddle.round(label_true), 1.0),
        ),
        dtype=paddle.get_default_dtype(),
    )
    false_negative = paddle.sum(
        paddle.multiply(
            paddle.equal(paddle.round(label_pred), 1.0),
            paddle.equal(paddle.round(label_true), 0.0),
        ),
        dtype=paddle.get_default_dtype(),
    )
    false_positive = paddle.sum(
        paddle.multiply(
            paddle.equal(paddle.round(label_pred), 0.0),
            paddle.equal(paddle.round(label_true), 1.0),
        ),
        dtype=paddle.get_default_dtype(),
    )
    n_negative = paddle.add(false_negative, true_negative)
    n_positive = paddle.add(true_positive, false_positive)
    iou = 0.5 * paddle.add(
        paddle.divide(true_negative, paddle.add(n_negative, false_positive)),
        paddle.divide(true_positive, paddle.add(n_positive, false_negative)),
    )

3.9 模型训练

本案例根据采样器的不同选择共有四组子案例，案例参数如下：

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# general settings
mode: train # running mode: train/eval
seed: 42

训练代码如下：

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# train models for 4 cases
for sampler_key, num in cfg.CASE_PARAM:

    # initialize SIMP iteration stop time sampler
    SIMP_stop_point_sampler = func_module.generate_sampler(sampler_key, num)

    # initialize logger for training
    sampler_name = sampler_key + str(num) if num else sampler_key
    OUTPUT_DIR = osp.join(
        cfg.output_dir, f"{sampler_name}_vol_coeff{cfg.vol_coeff}"
    )
    logger.init_logger("ppsci", osp.join(OUTPUT_DIR, "train.log"), "info")

    # set model
    model = TopOptNN(**cfg.MODEL, channel_sampler=SIMP_stop_point_sampler)

    # set optimizer
    optimizer = ppsci.optimizer.Adam(learning_rate=LEARNING_RATE, epsilon=1.0e-7)(
        model
    )

    # initialize solver
    solver = ppsci.solver.Solver(
        model,
        constraint,
        OUTPUT_DIR,
        optimizer,
        epochs=cfg.TRAIN.epochs,
        iters_per_epoch=ITERS_PER_EPOCH,
        eval_during_train=cfg.TRAIN.eval_during_train,
        seed=cfg.seed,
    )

    # train model

3.10 评估模型

对四个训练好的模型，分别使用不同的通道采样器 (原始数据的第二维对应表示的是 SIMP 算法的 100 步输出结果，统一取原始数据第二维的第 5，10，15，20，...，80 通道以及对应的梯度信息作为新的输入构建评估数据集) 进行评估，每次评估时只取 cfg.EVAL.num_val_step 个 bacth 的数据，计算它们的平均 Binary Accuracy 和 IoU 指标；同时评估结果需要与输入数据本身的阈值判定结果 (0.5作为阈值) 作比较。具体代码请参考完整代码

3.10.1 评估器构建

为应用 PaddleScience API，此处在每一次评估时构建一个评估器 SupervisedValidator 进行评估：

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sup_validator = ppsci.validate.SupervisedValidator(
    {
        "dataset": {
            "name": "NamedArrayDataset",
            "input": {"input": inputs_eval},
            "label": {"output": labels_eval},
            "transforms": (
                {
                    "FunctionalTransform": {
                        "transform_func": func_module.augmentation,
                    },
                },
            ),
        },
        "batch_size": cfg.EVAL.batch_size,
        "sampler": {
            "name": "BatchSampler",
            "drop_last": False,
            "shuffle": True,
        },
        "num_workers": 0,
    },
    ppsci.loss.FunctionalLoss(loss_wrapper(cfg)),
    {"output": lambda out: out["output"]},
    {"metric": ppsci.metric.FunctionalMetric(val_metric)},
    name="sup_validator",
)

评估器配置与 约束构建 的设置类似，读取配置中 "num_workers"：0 表示单线程读取；评价指标 "metric" 为自定义评估指标，包含 Binary Accuracy 和 IoU。

3.11 评估结果可视化

使用 ppsci.utils.misc.plot_curve() 方法直接绘制 Binary Accuracy 和 IoU 的结果：

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ppsci.utils.misc.plot_curve(
    acc_results_summary,
    xlabel="iteration",
    ylabel="accuracy",
    output_dir=cfg.output_dir,
)
ppsci.utils.misc.plot_curve(
    iou_results_summary, xlabel="iteration", ylabel="iou", output_dir=cfg.output_dir

4. 完整代码

topopt.py
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# Copyright (c) 2023 PaddlePaddle Authors. All Rights Reserved.

# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at

#     http://www.apache.org/licenses/LICENSE-2.0

# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.

from os import path as osp
from typing import Dict

import functions as func_module
import h5py
import hydra
import numpy as np
import paddle
from omegaconf import DictConfig
from paddle import nn
from topoptmodel import TopOptNN

import ppsci
from ppsci.utils import logger


def train(cfg: DictConfig):
    # set random seed for reproducibility
    ppsci.utils.misc.set_random_seed(cfg.seed)
    # initialize logger
    logger.init_logger("ppsci", osp.join(cfg.output_dir, f"{cfg.mode}.log"), "info")

    # 4 training cases parameters
    LEARNING_RATE = cfg.TRAIN.learning_rate / (1 + cfg.TRAIN.epochs // 15)
    ITERS_PER_EPOCH = int(cfg.n_samples * cfg.train_test_ratio / cfg.TRAIN.batch_size)

    # read h5 data
    h5data = h5py.File(cfg.DATA_PATH, "r")
    data_iters = np.array(h5data["iters"])
    data_targets = np.array(h5data["targets"])

    # generate training dataset
    inputs_train, labels_train = func_module.generate_train_test(
        data_iters, data_targets, cfg.train_test_ratio, cfg.n_samples
    )

    # set constraints
    sup_constraint = ppsci.constraint.SupervisedConstraint(
        {
            "dataset": {
                "name": "NamedArrayDataset",
                "input": {"input": inputs_train},
                "label": {"output": labels_train},
                "transforms": (
                    {
                        "FunctionalTransform": {
                            "transform_func": func_module.augmentation,
                        },
                    },
                ),
            },
            "batch_size": cfg.TRAIN.batch_size,
            "sampler": {
                "name": "BatchSampler",
                "drop_last": False,
                "shuffle": True,
            },
        },
        ppsci.loss.FunctionalLoss(loss_wrapper(cfg)),
        name="sup_constraint",
    )
    constraint = {sup_constraint.name: sup_constraint}

    # train models for 4 cases
    for sampler_key, num in cfg.CASE_PARAM:

        # initialize SIMP iteration stop time sampler
        SIMP_stop_point_sampler = func_module.generate_sampler(sampler_key, num)

        # initialize logger for training
        sampler_name = sampler_key + str(num) if num else sampler_key
        OUTPUT_DIR = osp.join(
            cfg.output_dir, f"{sampler_name}_vol_coeff{cfg.vol_coeff}"
        )
        logger.init_logger("ppsci", osp.join(OUTPUT_DIR, "train.log"), "info")

        # set model
        model = TopOptNN(**cfg.MODEL, channel_sampler=SIMP_stop_point_sampler)

        # set optimizer
        optimizer = ppsci.optimizer.Adam(learning_rate=LEARNING_RATE, epsilon=1.0e-7)(
            model
        )

        # initialize solver
        solver = ppsci.solver.Solver(
            model,
            constraint,
            OUTPUT_DIR,
            optimizer,
            epochs=cfg.TRAIN.epochs,
            iters_per_epoch=ITERS_PER_EPOCH,
            eval_during_train=cfg.TRAIN.eval_during_train,
            seed=cfg.seed,
        )

        # train model
        solver.train()


# evaluate 4 models
def evaluate(cfg: DictConfig):
    # set random seed for reproducibility
    ppsci.utils.misc.set_random_seed(cfg.seed)
    # initialize logger
    logger.init_logger("ppsci", osp.join(cfg.output_dir, f"{cfg.mode}.log"), "info")

    # fixed iteration stop times for evaluation
    iterations_stop_times = range(5, 85, 5)
    model = TopOptNN(**cfg.MODEL)

    # evaluation for 4 cases
    acc_results_summary = {}
    iou_results_summary = {}

    # read h5 data
    h5data = h5py.File(cfg.DATA_PATH, "r")
    data_iters = np.array(h5data["iters"])
    data_targets = np.array(h5data["targets"])

    for case_name, model_path in cfg.EVAL.pretrained_model_path_dict.items():
        acc_results, iou_results = evaluate_model(
            cfg, model, model_path, data_iters, data_targets, iterations_stop_times
        )

        acc_results_summary[case_name] = acc_results
        iou_results_summary[case_name] = iou_results

    # calculate thresholding results
    th_acc_results = []
    th_iou_results = []
    for stop_iter in iterations_stop_times:
        SIMP_stop_point_sampler = func_module.generate_sampler("Fixed", stop_iter)

        current_acc_results = []
        current_iou_results = []

        # only calculate for NUM_VAL_STEP times of iteration
        for _ in range(cfg.EVAL.num_val_step):
            input_full_channel, label = func_module.generate_train_test(
                data_iters, data_targets, 1.0, cfg.EVAL.batch_size
            )
            # thresholding
            SIMP_initial_iter_time = SIMP_stop_point_sampler()  # channel k
            input_channel_k = paddle.to_tensor(
                input_full_channel, dtype=paddle.get_default_dtype()
            )[:, SIMP_initial_iter_time, :, :]
            input_channel_k_minus_1 = paddle.to_tensor(
                input_full_channel, dtype=paddle.get_default_dtype()
            )[:, SIMP_initial_iter_time - 1, :, :]
            input = paddle.stack(
                (input_channel_k, input_channel_k - input_channel_k_minus_1), axis=1
            )
            out = paddle.cast(
                paddle.to_tensor(input)[:, 0:1, :, :] > 0.5,
                dtype=paddle.get_default_dtype(),
            )
            th_result = val_metric(
                {"output": out},
                {"output": paddle.to_tensor(label, dtype=paddle.get_default_dtype())},
            )
            acc_results, iou_results = th_result["Binary_Acc"], th_result["IoU"]
            current_acc_results.append(acc_results)
            current_iou_results.append(iou_results)

        th_acc_results.append(np.mean(current_acc_results))
        th_iou_results.append(np.mean(current_iou_results))

    acc_results_summary["thresholding"] = th_acc_results
    iou_results_summary["thresholding"] = th_iou_results

    ppsci.utils.misc.plot_curve(
        acc_results_summary,
        xlabel="iteration",
        ylabel="accuracy",
        output_dir=cfg.output_dir,
    )
    ppsci.utils.misc.plot_curve(
        iou_results_summary, xlabel="iteration", ylabel="iou", output_dir=cfg.output_dir
    )


def evaluate_model(
    cfg, model, pretrained_model_path, data_iters, data_targets, iterations_stop_times
):
    # load model parameters
    solver = ppsci.solver.Solver(
        model,
        epochs=1,
        iters_per_epoch=cfg.EVAL.num_val_step,
        eval_with_no_grad=True,
        pretrained_model_path=pretrained_model_path,
    )

    acc_results = []
    iou_results = []

    # evaluation for different fixed iteration stop times
    for stop_iter in iterations_stop_times:
        # only evaluate for NUM_VAL_STEP times of iteration
        inputs_eval, labels_eval = func_module.generate_train_test(
            data_iters, data_targets, 1.0, cfg.EVAL.batch_size * cfg.EVAL.num_val_step
        )

        sup_validator = ppsci.validate.SupervisedValidator(
            {
                "dataset": {
                    "name": "NamedArrayDataset",
                    "input": {"input": inputs_eval},
                    "label": {"output": labels_eval},
                    "transforms": (
                        {
                            "FunctionalTransform": {
                                "transform_func": func_module.augmentation,
                            },
                        },
                    ),
                },
                "batch_size": cfg.EVAL.batch_size,
                "sampler": {
                    "name": "BatchSampler",
                    "drop_last": False,
                    "shuffle": True,
                },
                "num_workers": 0,
            },
            ppsci.loss.FunctionalLoss(loss_wrapper(cfg)),
            {"output": lambda out: out["output"]},
            {"metric": ppsci.metric.FunctionalMetric(val_metric)},
            name="sup_validator",
        )
        validator = {sup_validator.name: sup_validator}
        solver.validator = validator

        # modify the channel_sampler in model
        SIMP_stop_point_sampler = func_module.generate_sampler("Fixed", stop_iter)
        solver.model.channel_sampler = SIMP_stop_point_sampler

        _, eval_result = solver.eval()

        current_acc_results = eval_result["metric"]["Binary_Acc"]
        current_iou_results = eval_result["metric"]["IoU"]

        acc_results.append(current_acc_results)
        iou_results.append(current_iou_results)

    return acc_results, iou_results


# define loss wrapper
def loss_wrapper(cfg: DictConfig):
    def loss_expr(output_dict, label_dict, weight_dict=None):
        label_true = label_dict["output"].reshape((-1, 1))
        label_pred = output_dict["output"].reshape((-1, 1))
        conf_loss = paddle.mean(
            nn.functional.log_loss(label_pred, label_true, epsilon=1.0e-7)
        )
        vol_loss = paddle.square(paddle.mean(label_true - label_pred))
        return {"output": conf_loss + cfg.vol_coeff * vol_loss}

    return loss_expr


# define metric
def val_metric(output_dict, label_dict, weight_dict=None):
    label_pred = output_dict["output"]
    label_true = label_dict["output"]
    accurates = paddle.equal(paddle.round(label_true), paddle.round(label_pred))
    acc = paddle.mean(paddle.cast(accurates, dtype=paddle.get_default_dtype()))
    true_negative = paddle.sum(
        paddle.multiply(
            paddle.equal(paddle.round(label_pred), 0.0),
            paddle.equal(paddle.round(label_true), 0.0),
        ),
        dtype=paddle.get_default_dtype(),
    )
    true_positive = paddle.sum(
        paddle.multiply(
            paddle.equal(paddle.round(label_pred), 1.0),
            paddle.equal(paddle.round(label_true), 1.0),
        ),
        dtype=paddle.get_default_dtype(),
    )
    false_negative = paddle.sum(
        paddle.multiply(
            paddle.equal(paddle.round(label_pred), 1.0),
            paddle.equal(paddle.round(label_true), 0.0),
        ),
        dtype=paddle.get_default_dtype(),
    )
    false_positive = paddle.sum(
        paddle.multiply(
            paddle.equal(paddle.round(label_pred), 0.0),
            paddle.equal(paddle.round(label_true), 1.0),
        ),
        dtype=paddle.get_default_dtype(),
    )
    n_negative = paddle.add(false_negative, true_negative)
    n_positive = paddle.add(true_positive, false_positive)
    iou = 0.5 * paddle.add(
        paddle.divide(true_negative, paddle.add(n_negative, false_positive)),
        paddle.divide(true_positive, paddle.add(n_positive, false_negative)),
    )
    return {"Binary_Acc": acc, "IoU": iou}


# export model
def export(cfg: DictConfig):
    # set model
    model = TopOptNN(**cfg.MODEL)

    # initialize solver
    solver = ppsci.solver.Solver(
        model,
        eval_with_no_grad=True,
        pretrained_model_path=cfg.INFER.pretrained_model_path_dict[
            cfg.INFER.pretrained_model_name
        ],
    )

    # export model
    from paddle.static import InputSpec

    input_spec = [{"input": InputSpec([None, 2, 40, 40], "float32", name="input")}]

    solver.export(input_spec, cfg.INFER.export_path)


def inference(cfg: DictConfig):
    # read h5 data
    h5data = h5py.File(cfg.DATA_PATH, "r")
    data_iters = np.array(h5data["iters"])
    data_targets = np.array(h5data["targets"])
    idx = np.random.choice(len(data_iters), cfg.INFER.img_num, False)
    data_iters = data_iters[idx]
    data_targets = data_targets[idx]

    sampler = func_module.generate_sampler(cfg.INFER.sampler_key, cfg.INFER.sampler_num)
    data_iters = channel_sampling(sampler, data_iters)

    from deploy.python_infer import pinn_predictor

    predictor = pinn_predictor.PINNPredictor(cfg)

    input_dict = {"input": data_iters}
    output_dict = predictor.predict(input_dict, cfg.INFER.batch_size)

    # mapping data to output_key
    output_dict = {
        store_key: output_dict[infer_key]
        for store_key, infer_key in zip({"output"}, output_dict.keys())
    }

    save_topopt_img(
        input_dict,
        output_dict,
        data_targets,
        cfg.INFER.save_res_path,
        cfg.INFER.res_img_figsize,
        cfg.INFER.save_npy,
    )


# used for inference
def channel_sampling(sampler, input):
    SIMP_initial_iter_time = sampler()
    input_channel_k = input[:, SIMP_initial_iter_time, :, :]
    input_channel_k_minus_1 = input[:, SIMP_initial_iter_time - 1, :, :]
    input = np.stack(
        (input_channel_k, input_channel_k - input_channel_k_minus_1), axis=1
    )
    return input


# used for inference
def save_topopt_img(
    input_dict: Dict[str, np.ndarray],
    output_dict: Dict[str, np.ndarray],
    ground_truth: np.ndarray,
    save_dir: str,
    figsize: tuple = None,
    save_npy: bool = False,
):

    input = input_dict["input"]
    output = output_dict["output"]
    import os

    import matplotlib.pyplot as plt

    os.makedirs(save_dir, exist_ok=True)
    for i in range(len(input)):
        plt.figure(figsize=figsize)
        plt.subplot(1, 4, 1)
        plt.axis("off")
        plt.imshow(input[i][0], cmap="gray")
        plt.title("Input Image")
        plt.subplot(1, 4, 2)
        plt.axis("off")
        plt.imshow(input[i][1], cmap="gray")
        plt.title("Input Gradient")
        plt.subplot(1, 4, 3)
        plt.axis("off")
        plt.imshow(np.round(output[i][0]), cmap="gray")
        plt.title("Prediction")
        plt.subplot(1, 4, 4)
        plt.axis("off")
        plt.imshow(np.round(ground_truth[i][0]), cmap="gray")
        plt.title("Ground Truth")
        plt.show()
        plt.savefig(osp.join(save_dir, f"Prediction_{i}.png"))
        plt.close()
        if save_npy:
            with open(osp(save_dir, f"Prediction_{i}.npy"), "wb") as f:
                np.save(f, output[i])


@hydra.main(version_base=None, config_path="./conf", config_name="topopt.yaml")
def main(cfg: DictConfig):
    if cfg.mode == "train":
        train(cfg)
    elif cfg.mode == "eval":
        evaluate(cfg)
    elif cfg.mode == "export":
        export(cfg)
    elif cfg.mode == "infer":
        inference(cfg)
    else:
        raise ValueError(
            f"cfg.mode should in ['train', 'eval', 'export', 'infer'], but got '{cfg.mode}'"
        )


if __name__ == "__main__":
    main()
functions.py
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# Copyright (c) 2023 PaddlePaddle Authors. All Rights Reserved.

# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at

#     http://www.apache.org/licenses/LICENSE-2.0

# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.

from typing import Callable
from typing import Dict
from typing import Tuple
from typing import Union

import numpy as np


def uniform_sampler() -> Callable[[], int]:
    """Generate uniform sampling function from 1 to 99

    Returns:
        sampler (Callable[[], int]): uniform sampling from 1 to 99
    """
    return lambda: np.random.randint(1, 99)


def poisson_sampler(lam: int) -> Callable[[], int]:
    """Generate poisson sampling function with parameter lam with range 1 to 99

    Args:
        lam (int): poisson rate parameter

    Returns:
        sampler (Callable[[], int]): poisson sampling function with parameter lam with range 1 to 99
    """

    def func():
        iter_ = max(np.random.poisson(lam), 1)
        iter_ = min(iter_, 99)
        return iter_

    return func


def generate_sampler(sampler_type: str = "Fixed", num: int = 0) -> Callable[[], int]:
    """Generate sampler for the number of initial iteration steps

    Args:
        sampler_type (str): "Poisson" for poisson sampler; "Uniform" for uniform sampler; "Fixed" for choosing a fixed number of initial iteration steps.
        num (int): If `sampler_type` == "Poisson", `num` specifies the poisson rate parameter; If `sampler_type` == "Fixed", `num` specifies the fixed number of initial iteration steps.

    Returns:
        sampler (Callable[[], int]): sampler for the number of initial iteration steps
    """
    if sampler_type == "Poisson":
        return poisson_sampler(num)
    elif sampler_type == "Uniform":
        return uniform_sampler()
    else:
        return lambda: num


def generate_train_test(
    data_iters: np.ndarray,
    data_targets: np.ndarray,
    train_test_ratio: float,
    n_sample: int,
) -> Union[
    Tuple[np.ndarray, np.ndarray], Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray]
]:
    """Generate training and testing set

    Args:
        data_iters (np.ndarray): data with 100 channels corresponding to the results of 100 steps of SIMP algorithm
        data_targets (np.ndarray): final optimization solution given by SIMP algorithm
        train_test_ratio (float): split ratio of training and testing sets, if `train_test_ratio` = 1 then only return training data
        n_sample (int): number of total samples in training and testing sets to be sampled from the h5 dataset

    Returns:
        Union[Tuple[np.ndarray, np.ndarray], Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray]]: if `train_test_ratio` = 1, return (train_inputs, train_labels), else return (train_inputs, train_labels, test_inputs, test_labels)
    """
    n_obj = len(data_iters)
    idx = np.arange(n_obj)
    np.random.shuffle(idx)
    train_idx = idx[: int(train_test_ratio * n_sample)]
    if train_test_ratio == 1.0:
        return data_iters[train_idx], data_targets[train_idx]

    test_idx = idx[int(train_test_ratio * n_sample) :]
    train_iters = data_iters[train_idx]
    train_targets = data_targets[train_idx]
    test_iters = data_iters[test_idx]
    test_targets = data_targets[test_idx]
    return train_iters, train_targets, test_iters, test_targets


def augmentation(
    input_dict: Dict[str, np.ndarray],
    label_dict: Dict[str, np.ndarray],
    weight_dict: Dict[str, np.ndarray] = None,
) -> Tuple[Dict[str, np.ndarray], Dict[str, np.ndarray], Dict[str, np.ndarray]]:
    """Apply random transformation from D4 symmetry group

    Args:
        input_dict (Dict[str, np.ndarray]): input dict of np.ndarray size `(batch_size, any, height, width)`
        label_dict (Dict[str, np.ndarray]): label dict of np.ndarray size `(batch_size, 1, height, width)`
        weight_dict (Dict[str, np.ndarray]): weight dict if any
    """
    inputs = input_dict["input"]
    labels = label_dict["output"]
    assert len(inputs.shape) == 3
    assert len(labels.shape) == 3

    # random horizontal flip
    if np.random.random() > 0.5:
        inputs = np.flip(inputs, axis=2)
        labels = np.flip(labels, axis=2)
    # random vertical flip
    if np.random.random() > 0.5:
        inputs = np.flip(inputs, axis=1)
        labels = np.flip(labels, axis=1)
    # random 90* rotation
    if np.random.random() > 0.5:
        new_perm = list(range(len(inputs.shape)))
        new_perm[-2], new_perm[-1] = new_perm[-1], new_perm[-2]
        inputs = np.transpose(inputs, new_perm)
        labels = np.transpose(labels, new_perm)

    return {"input": inputs}, {"output": labels}, weight_dict
topoptmodel.py
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# Copyright (c) 2023 PaddlePaddle Authors. All Rights Reserved.

# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at

#     http://www.apache.org/licenses/LICENSE-2.0

# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.

import paddle
from paddle import nn

import ppsci


# NCHW data format
class TopOptNN(ppsci.arch.UNetEx):
    """Neural network for Topology Optimization, inherited from `ppsci.arch.UNetEx`

    [Sosnovik, I., & Oseledets, I. (2019). Neural networks for topology optimization. Russian Journal of Numerical Analysis and Mathematical Modelling, 34(4), 215-223.](https://arxiv.org/pdf/1709.09578)

    Args:
        input_key (str): Name of function data for input.
        output_key (str): Name of function data for output.
        in_channel (int): Number of channels of input.
        out_channel (int): Number of channels of output.
        kernel_size (int, optional): Size of kernel of convolution layer. Defaults to 3.
        filters (Tuple[int, ...], optional): Number of filters. Defaults to (16, 32, 64).
        layers (int, optional): Number of encoders or decoders. Defaults to 3.
        channel_sampler (callable, optional): The sampling function for the initial iteration time
                (corresponding to the channel number of the input) of SIMP algorithm. The default value
                is None, when it is None, input for the forward method should be sampled and prepared
                with the shape of [batch, 2, height, width] before passing to forward method.
        weight_norm (bool, optional): Whether use weight normalization layer. Defaults to True.
        batch_norm (bool, optional): Whether add batch normalization layer. Defaults to True.
        activation (Type[nn.Layer], optional): Name of activation function. Defaults to nn.ReLU.

    Examples:
        >>> import ppsci
        >>> model = ppsci.arch.ppsci.arch.TopOptNN("input", "output", 2, 1, 3, (16, 32, 64), 2, lambda: 1, Flase, False)
    """

    def __init__(
        self,
        input_key="input",
        output_key="output",
        in_channel=2,
        out_channel=1,
        kernel_size=3,
        filters=(16, 32, 64),
        layers=2,
        channel_sampler=None,
        weight_norm=False,
        batch_norm=False,
        activation=nn.ReLU,
    ):
        super().__init__(
            input_key=input_key,
            output_key=output_key,
            in_channel=in_channel,
            out_channel=out_channel,
            kernel_size=kernel_size,
            filters=filters,
            layers=layers,
            weight_norm=weight_norm,
            batch_norm=batch_norm,
            activation=activation,
        )
        self.in_channel = in_channel
        self.out_channel = out_channel
        self.filters = filters
        self.channel_sampler = channel_sampler
        self.activation = activation

        # Modify Layers
        self.encoder[1] = nn.Sequential(
            nn.MaxPool2D(self.in_channel, padding="SAME"),
            self.encoder[1][0],
            nn.Dropout2D(0.1),
            self.encoder[1][1],
        )
        self.encoder[2] = nn.Sequential(
            nn.MaxPool2D(2, padding="SAME"), self.encoder[2]
        )
        # Conv2D used in reference code in decoder
        self.decoders[0] = nn.Sequential(
            nn.Conv2D(
                self.filters[-1], self.filters[-1], kernel_size=3, padding="SAME"
            ),
            self.activation(),
            nn.Conv2D(
                self.filters[-1], self.filters[-1], kernel_size=3, padding="SAME"
            ),
            self.activation(),
        )
        self.decoders[1] = nn.Sequential(
            nn.Conv2D(
                sum(self.filters[-2:]), self.filters[-2], kernel_size=3, padding="SAME"
            ),
            self.activation(),
            nn.Dropout2D(0.1),
            nn.Conv2D(
                self.filters[-2], self.filters[-2], kernel_size=3, padding="SAME"
            ),
            self.activation(),
        )
        self.decoders[2] = nn.Sequential(
            nn.Conv2D(
                sum(self.filters[:-1]), self.filters[-3], kernel_size=3, padding="SAME"
            ),
            self.activation(),
            nn.Conv2D(
                self.filters[-3], self.filters[-3], kernel_size=3, padding="SAME"
            ),
            self.activation(),
        )
        self.output = nn.Sequential(
            nn.Conv2D(
                self.filters[-3], self.out_channel, kernel_size=3, padding="SAME"
            ),
            nn.Sigmoid(),
        )

    def forward(self, x):
        if self.channel_sampler is not None:
            SIMP_initial_iter_time = self.channel_sampler()  # channel k
            input_channel_k = x[self.input_keys[0]][:, SIMP_initial_iter_time, :, :]
            input_channel_k_minus_1 = x[self.input_keys[0]][
                :, SIMP_initial_iter_time - 1, :, :
            ]
            x = paddle.stack(
                (input_channel_k, input_channel_k - input_channel_k_minus_1), axis=1
            )
        else:
            x = x[self.input_keys[0]]
        # encode
        upsampling_size = []
        skip_connection = []
        n_encoder = len(self.encoder)
        for i in range(n_encoder):
            x = self.encoder[i](x)
            if i is not (n_encoder - 1):
                upsampling_size.append(x.shape[-2:])
                skip_connection.append(x)

        # decode
        n_decoder = len(self.decoders)
        for i in range(n_decoder):
            x = self.decoders[i](x)
            if i is not (n_decoder - 1):
                up_size = upsampling_size.pop()
                x = nn.UpsamplingNearest2D(up_size)(x)
                skip_output = skip_connection.pop()
                x = paddle.concat((skip_output, x), axis=1)

        out = self.output(x)
        return {self.output_keys[0]: out}

5. 结果展示

下图展示了4个模型分别在16组不同的评估数据集上的表现，包括 Binary Accuracy 以及 IoU 这两种指标。其中横坐标代表不同的评估数据集，例如：横坐标 \(i\) 表示由原始数据第二维的第 \(5\cdot(i+1)\) 个通道及其对应梯度信息构建的评估数据集；纵坐标为评估指标。thresholding 对应的指标可以理解为 benchmark。

bin_acc

Binary Accuracy结果

iou

IoU结果

用表格表示上图指标为：

bin_acc eval_dataset_ch_5 eval_dataset_ch_10 eval_dataset_ch_15 eval_dataset_ch_20 eval_dataset_ch_25 eval_dataset_ch_30 eval_dataset_ch_35 eval_dataset_ch_40 eval_dataset_ch_45 eval_dataset_ch_50 eval_dataset_ch_55 eval_dataset_ch_60 eval_dataset_ch_65 eval_dataset_ch_70 eval_dataset_ch_75 eval_dataset_ch_80
Poisson5 0.9471 0.9619 0.9702 0.9742 0.9782 0.9801 0.9803 0.9825 0.9824 0.9837 0.9850 0.9850 0.9870 0.9863 0.9870 0.9872
Poisson10 0.9457 0.9703 0.9745 0.9798 0.9827 0.9845 0.9859 0.9870 0.9882 0.9880 0.9893 0.9899 0.9882 0.9899 0.9905 0.9904
Poisson30 0.9257 0.9595 0.9737 0.9832 0.9828 0.9883 0.9885 0.9892 0.9901 0.9916 0.9924 0.9925 0.9926 0.9929 0.9937 0.9936
Uniform 0.9410 0.9673 0.9718 0.9727 0.9818 0.9824 0.9826 0.9845 0.9856 0.9892 0.9892 0.9907 0.9890 0.9916 0.9914 0.9922
iou eval_dataset_ch_5 eval_dataset_ch_10 eval_dataset_ch_15 eval_dataset_ch_20 eval_dataset_ch_25 eval_dataset_ch_30 eval_dataset_ch_35 eval_dataset_ch_40 eval_dataset_ch_45 eval_dataset_ch_50 eval_dataset_ch_55 eval_dataset_ch_60 eval_dataset_ch_65 eval_dataset_ch_70 eval_dataset_ch_75 eval_dataset_ch_80
Poisson5 0.8995 0.9267 0.9421 0.9497 0.9574 0.9610 0.9614 0.9657 0.9655 0.9679 0.9704 0.9704 0.9743 0.9730 0.9744 0.9747
Poisson10 0.8969 0.9424 0.9502 0.9604 0.9660 0.9696 0.9722 0.9743 0.9767 0.9762 0.9789 0.9800 0.9768 0.9801 0.9813 0.9810
Poisson30 0.8617 0.9221 0.9488 0.9670 0.9662 0.9769 0.9773 0.9786 0.9803 0.9833 0.9850 0.9853 0.9855 0.9860 0.9875 0.9873
Uniform 0.8887 0.9367 0.9452 0.9468 0.9644 0.9655 0.9659 0.9695 0.9717 0.9787 0.9787 0.9816 0.9784 0.9835 0.9831 0.9845

6. 参考文献