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Aneurysm

# linux
wget -nc https://paddle-org.bj.bcebos.com/paddlescience/datasets/aneurysm/aneurysm_dataset.tar
# windows
# curl https://paddle-org.bj.bcebos.com/paddlescience/datasets/aneurysm/aneurysm_dataset.tar -o aneurysm_dataset.tar
# unzip it
tar -xvf aneurysm_dataset.tar
python aneurysm.py
# linux
wget -nc https://paddle-org.bj.bcebos.com/paddlescience/datasets/aneurysm/aneurysm_dataset.tar
# windows
# curl https://paddle-org.bj.bcebos.com/paddlescience/datasets/aneurysm/aneurysm_dataset.tar -o aneurysm_dataset.tar
# unzip it
tar -xvf aneurysm_dataset.tar
python aneurysm.py mode=eval EVAL.pretrained_model_path=https://paddle-org.bj.bcebos.com/paddlescience/models/aneurysm/aneurysm_pretrained.pdparams
python aneurysm.py mode=export
# linux
wget -nc https://paddle-org.bj.bcebos.com/paddlescience/datasets/aneurysm/aneurysm_dataset.tar
# windows
# curl https://paddle-org.bj.bcebos.com/paddlescience/datasets/aneurysm/aneurysm_dataset.tar -o aneurysm_dataset.tar
# unzip it
tar -xvf aneurysm_dataset.tar
python aneurysm.py mode=infer
预训练模型 指标
aneurysm_pretrained.pdparams loss(ref_u_v_w_p): 0.01488
MSE.p(ref_u_v_w_p): 0.01412
MSE.u(ref_u_v_w_p): 0.00021
MSE.v(ref_u_v_w_p): 0.00024
MSE.w(ref_u_v_w_p): 0.00032

1. 背景简介

深度学习方法可以用于处理血管瘤问题,其中包括基于物理信息的深度学习方法。这种方法可以用于脑血管瘤的压力建模,以预测和评估血管瘤破裂的风险。

针对如下血管瘤几何模型,本案例通过深度学习方式,在内部和边界施加适当的物理方程约束,以无监督学习的方式对管壁压力进行建模。

equation

2. 问题定义

假设血管瘤模型中,在入口 inlet 部分,中心点的流速为 1.5,并向四周逐渐减小;在出口 outlet 区域,压力恒为 0;在边界上无滑移,流速为 0;血管内部则符合 N-S 方程运动规律,中间段的平均流量为负(流入),出口段的平均流量为正(流出)。

3. 问题求解

接下来开始讲解如何将问题一步一步地转化为 PaddleScience 代码,用深度学习的方法求解该问题。 为了快速理解 PaddleScience,接下来仅对模型构建、方程构建、计算域构建等关键步骤进行阐述,而其余细节请参考 API文档

3.1 模型构建

在 aneurysm 问题中,每一个已知的坐标点 \((x, y, z)\) 都有对应的待求解的未知量 \((u, v, w, p)\)(速度和压力) ,在这里使用比较简单的 MLP(Multilayer Perceptron, 多层感知机) 来表示 \((x, y, z)\)\((u, v, w, p)\) 的映射函数 \(f: \mathbb{R}^3 \to \mathbb{R}^4\) ,即:

\[ (u, v, w, p) = f(x, y, z) \]

上式中 \(f\) 即为 MLP 模型本身,用 PaddleScience 代码表示如下

# set model
model = ppsci.arch.MLP(**cfg.MODEL)

为了在计算时,准确快速地访问具体变量的值,在这里指定网络模型的输入变量名是 ("x", "y", "z"),输出变量名是 ("u", "v", "w", "p"),这些命名与后续代码保持一致。

接着通过指定 MLP 的层数、神经元个数,就实例化出了一个拥有 6 层隐藏神经元,每层神经元数为 512 的神经网络模型 model,使用 silu 作为激活函数,并使用 WeightNorm 权重归一化。

3.2 方程构建

血管瘤模型涉及到 2 个方程,一是流体 N-S 方程,二是流量计算方程,因此使用 PaddleScience 内置的 NavierStokesNormalDotVec 即可。

# set equation
equation = {
    "NavierStokes": ppsci.equation.NavierStokes(
        cfg.NU * cfg.SCALE, cfg.RHO, cfg.DIM, False
    ),
    "NormalDotVec": ppsci.equation.NormalDotVec(("u", "v", "w")),
}

3.3 计算域构建

本问题的几何区域由 stl 文件指定,按照下方命令,下载并解压到 aneurysm/ 文件夹下。

注:数据集中的 stl 文件和测试集数据(使用OpenFOAM生成)均来自 Aneurysm - NVIDIA Modulus

# linux
wget -nc https://paddle-org.bj.bcebos.com/paddlescience/datasets/aneurysm/aneurysm_dataset.tar

# windows
# curl https://paddle-org.bj.bcebos.com/paddlescience/datasets/aneurysm/aneurysm_dataset.tar -o aneurysm_dataset.tar

# unzip it
tar -xvf aneurysm_dataset.tar

解压完毕之后,aneurysm/stl 文件夹下即存放了计算域构建所需的 stl 几何文件。

注意

使用 Mesh 类之前,必须先按照1.4.2 额外依赖安装[可选]文档,安装好 open3d、pysdf、PyMesh 3 个几何依赖包。

然后通过 PaddleScience 内置的 STL 几何类 Mesh 来读取、解析这些几何文件,并且通过布尔运算,组合出各个计算域,代码如下:

# set geometry
inlet_geo = ppsci.geometry.Mesh(cfg.INLET_STL_PATH)
outlet_geo = ppsci.geometry.Mesh(cfg.OUTLET_STL_PATH)
noslip_geo = ppsci.geometry.Mesh(cfg.NOSLIP_STL_PATH)
integral_geo = ppsci.geometry.Mesh(cfg.INTEGRAL_STL_PATH)
interior_geo = ppsci.geometry.Mesh(cfg.INTERIOR_STL_PATH)

在此之后可以对几何域进行缩放和平移,以缩放输入数据的坐标范围,促进模型训练收敛。

# normalize meshes
inlet_geo = inlet_geo.translate(-np.array(cfg.CENTER)).scale(cfg.SCALE)
outlet_geo = outlet_geo.translate(-np.array(cfg.CENTER)).scale(cfg.SCALE)
noslip_geo = noslip_geo.translate(-np.array(cfg.CENTER)).scale(cfg.SCALE)
integral_geo = integral_geo.translate(-np.array(cfg.CENTER)).scale(cfg.SCALE)
interior_geo = interior_geo.translate(-np.array(cfg.CENTER)).scale(cfg.SCALE)
geom = {
    "inlet_geo": inlet_geo,
    "outlet_geo": outlet_geo,
    "noslip_geo": noslip_geo,
    "integral_geo": integral_geo,
    "interior_geo": interior_geo,
}

3.4 约束构建

本案例共涉及到 6 个约束,在具体约束构建之前,可以先构建数据读取配置,以便后续构建多个约束时复用该配置。

# set dataloader config
train_dataloader_cfg = {
    "dataset": "NamedArrayDataset",
    "iters_per_epoch": cfg.TRAIN.iters_per_epoch,
    "sampler": {
        "name": "BatchSampler",
        "drop_last": True,
        "shuffle": True,
    },
    "num_workers": 1,
}

3.4.1 内部点约束

以作用在内部点上的 InteriorConstraint 为例,代码如下:

pde = ppsci.constraint.InteriorConstraint(
    equation["NavierStokes"].equations,
    {"continuity": 0, "momentum_x": 0, "momentum_y": 0, "momentum_z": 0},
    geom["interior_geo"],
    {**train_dataloader_cfg, "batch_size": cfg.TRAIN.batch_size.pde},
    ppsci.loss.MSELoss("sum"),
    name="interior",
)

InteriorConstraint 的第一个参数是方程(组)表达式,用于描述如何计算约束目标,此处填入在 3.2 方程构建 章节中实例化好的 equation["NavierStokes"].equations

第二个参数是约束变量的目标值,在本问题中希望与 N-S 方程相关的四个值 continuity, momentum_x, momentum_y, momentum_z 均被优化至 0;

第三个参数是约束方程作用的计算域,此处填入在 3.3 计算域构建 章节实例化好的 geom["interior_geo"] 即可;

第四个参数是在计算域上的采样配置,此处设置 batch_size6000

第五个参数是损失函数,此处选用常用的 MSE 函数,且 reduction 设置为 "sum",即会将参与计算的所有数据点产生的损失项求和;

第六个参数是约束条件的名字,需要给每一个约束条件命名,方便后续对其索引。此处命名为 "interior" 即可。

3.4.2 边界约束

接着需要对血管入口、出口、血管壁这三个表面施加约束,包括入口速度约束、出口压力约束、血管壁无滑移约束。 在 bc_inlet 约束中,入口处的流速满足从中心点开始向周围呈二次抛物线衰减,此处使用抛物线函数表示速度随着远离圆心而衰减,再将其作为 BoundaryConstraint 的第二个参数(字典)的 value。

def _compute_parabola(_in):
    centered_x = _in["x"] - cfg.INLET_CENTER[0]
    centered_y = _in["y"] - cfg.INLET_CENTER[1]
    centered_z = _in["z"] - cfg.INLET_CENTER[2]
    distance = np.sqrt(centered_x**2 + centered_y**2 + centered_z**2)
    parabola = cfg.INLET_VEL * np.maximum((1 - (distance / INLET_RADIUS) ** 2), 0)
    return parabola

def inlet_u_ref_func(_in):
    return cfg.INLET_NORMAL[0] * _compute_parabola(_in)

def inlet_v_ref_func(_in):
    return cfg.INLET_NORMAL[1] * _compute_parabola(_in)

def inlet_w_ref_func(_in):
    return cfg.INLET_NORMAL[2] * _compute_parabola(_in)

bc_inlet = ppsci.constraint.BoundaryConstraint(
    {"u": lambda d: d["u"], "v": lambda d: d["v"], "w": lambda d: d["w"]},
    {"u": inlet_u_ref_func, "v": inlet_v_ref_func, "w": inlet_w_ref_func},
    geom["inlet_geo"],
    {**train_dataloader_cfg, "batch_size": cfg.TRAIN.batch_size.bc_inlet},
    ppsci.loss.MSELoss("sum"),
    name="inlet",
)

血管出口、血管壁的无滑移约束构建方法类似,如下所示:

bc_outlet = ppsci.constraint.BoundaryConstraint(
    {"p": lambda d: d["p"]},
    {"p": 0},
    geom["outlet_geo"],
    {**train_dataloader_cfg, "batch_size": cfg.TRAIN.batch_size.bc_outlet},
    ppsci.loss.MSELoss("sum"),
    name="outlet",
)
bc_noslip = ppsci.constraint.BoundaryConstraint(
    {"u": lambda d: d["u"], "v": lambda d: d["v"], "w": lambda d: d["w"]},
    {"u": 0, "v": 0, "w": 0},
    geom["noslip_geo"],
    {**train_dataloader_cfg, "batch_size": cfg.TRAIN.batch_size.bc_noslip},
    ppsci.loss.MSELoss("sum"),
    name="no_slip",
)

3.4.3 积分边界约束

对于血管入口下方的一段区域和出口区域(面),需额外施加流入和流出的流量约束,由于流量计算涉及到具体面积,因此需要使用离散积分的方式进行计算,这些过程已经内置在了 IntegralConstraint 这一约束条件中。如下所示:

igc_outlet = ppsci.constraint.IntegralConstraint(
    equation["NormalDotVec"].equations,
    {"normal_dot_vec": 2.54},
    geom["outlet_geo"],
    {
        **train_dataloader_cfg,
        "iters_per_epoch": cfg.TRAIN.iters_integral.igc_outlet,
        "batch_size": cfg.TRAIN.batch_size.igc_outlet,
        "integral_batch_size": cfg.TRAIN.integral_batch_size.igc_outlet,
    },
    ppsci.loss.IntegralLoss("sum"),
    weight_dict=cfg.TRAIN.weight.igc_outlet,
    name="igc_outlet",
)
igc_integral = ppsci.constraint.IntegralConstraint(
    equation["NormalDotVec"].equations,
    {"normal_dot_vec": -2.54},
    geom["integral_geo"],
    {
        **train_dataloader_cfg,
        "iters_per_epoch": cfg.TRAIN.iters_integral.igc_integral,
        "batch_size": cfg.TRAIN.batch_size.igc_integral,
        "integral_batch_size": cfg.TRAIN.integral_batch_size.igc_integral,
    },
    ppsci.loss.IntegralLoss("sum"),
    weight_dict=cfg.TRAIN.weight.igc_integral,
    name="igc_integral",
)

对应的流量计算公式:

\[ flow_i = \sum_{i=1}^{M}{s_{i} (\mathbf{u_i} \cdot \mathbf{n_i})} \]

其中\(M\)表示离散积分点个数,\(s_i\)表示某一个点的(近似)面积,\(\mathbf{u_i}\)表示某一个点的速度矢量,\(\mathbf{n_i}\)表示某一个点的外法向矢量。

除前面章节所述的共同参数外,此处额外增加了 integral_batch_size 参数,这表示用于离散积分的采样点数量,此处使用 310 个离散点来近似积分计算;同时指定损失函数为 IntegralLoss,表示计算损失所用的最终预测值由多个离散点近似积分,再与标签值计算损失。

在微分方程约束、边界约束、初值约束构建完毕之后,以刚才的命名为关键字,封装到一个字典中,方便后续访问。

# wrap constraints together
constraint = {
    bc_inlet.name: bc_inlet,
    bc_outlet.name: bc_outlet,
    bc_noslip.name: bc_noslip,
    pde.name: pde,
    igc_outlet.name: igc_outlet,
    igc_integral.name: igc_integral,
}

3.5 超参数设定

接下来需要指定训练轮数和学习率,此处按实验经验,使用 1500 轮训练轮数,0.001 的初始学习率。

# training settings
TRAIN:
  epochs: 1500
  iters_per_epoch: 1000
  iters_integral:
    igc_outlet: 100
    igc_integral: 100
  save_freq: 20
  eval_during_train: true
  eval_freq: 20
  lr_scheduler:
    epochs: ${TRAIN.epochs}
    iters_per_epoch: ${TRAIN.iters_per_epoch}
    learning_rate: 0.001
    gamma: 0.95
    decay_steps: 15000
    by_epoch: false

3.6 优化器构建

训练过程会调用优化器来更新模型参数,此处选择较为常用的 Adam 优化器,并配合使用机器学习中常用的 ExponentialDecay 学习率调整策略。

# set optimizer
lr_scheduler = ppsci.optimizer.lr_scheduler.ExponentialDecay(
    **cfg.TRAIN.lr_scheduler
)()
optimizer = ppsci.optimizer.Adam(lr_scheduler)(model)

3.7 评估器构建

在训练过程中通常会按一定轮数间隔,用验证集(测试集)评估当前模型的训练情况,因此使用 ppsci.validate.GeometryValidator 构建评估器。

# set validator
eval_data_dict = reader.load_csv_file(
    cfg.EVAL_CSV_PATH,
    ("x", "y", "z", "u", "v", "w", "p"),
    {
        "x": "Points:0",
        "y": "Points:1",
        "z": "Points:2",
        "u": "U:0",
        "v": "U:1",
        "w": "U:2",
        "p": "p",
    },
)
input_dict = {
    "x": (eval_data_dict["x"] - cfg.CENTER[0]) * cfg.SCALE,
    "y": (eval_data_dict["y"] - cfg.CENTER[1]) * cfg.SCALE,
    "z": (eval_data_dict["z"] - cfg.CENTER[2]) * cfg.SCALE,
}
if "area" in input_dict.keys():
    input_dict["area"] *= cfg.SCALE ** (equation["NavierStokes"].dim)

label_dict = {
    "p": eval_data_dict["p"],
    "u": eval_data_dict["u"],
    "v": eval_data_dict["v"],
    "w": eval_data_dict["w"],
}
eval_dataloader_cfg = {
    "dataset": {
        "name": "NamedArrayDataset",
        "input": input_dict,
        "label": label_dict,
    },
    "sampler": {"name": "BatchSampler"},
    "num_workers": 1,
}
sup_validator = ppsci.validate.SupervisedValidator(
    {**eval_dataloader_cfg, "batch_size": cfg.EVAL.batch_size.sup_validator},
    ppsci.loss.MSELoss("mean"),
    {
        "p": lambda out: out["p"],
        "u": lambda out: out["u"],
        "v": lambda out: out["v"],
        "w": lambda out: out["w"],
    },
    metric={"MSE": ppsci.metric.MSE()},
    name="ref_u_v_w_p",
)
validator = {sup_validator.name: sup_validator}

# set visualizer(optional)
visualizer = {
    "visualize_u_v_w_p": ppsci.visualize.VisualizerVtu(
        input_dict,
        {
            "p": lambda out: out["p"],
            "u": lambda out: out["u"],
            "v": lambda out: out["v"],
            "w": lambda out: out["w"],
        },
        batch_size=cfg.EVAL.batch_size.sup_validator,
        prefix="result_u_v_w_p",
    ),
}

3.8 可视化器构建

在模型评估时,如果评估结果是可以可视化的数据,可以选择合适的可视化器来对输出结果进行可视化。

本文中的输出数据是一个区域内的三维点集,因此只需要将评估的输出数据保存成 vtu格式 文件,最后用可视化软件打开查看即可。代码如下:

# set visualizer(optional)
visualizer = {
    "visualize_u_v_w_p": ppsci.visualize.VisualizerVtu(
        input_dict,
        {
            "p": lambda out: out["p"],
            "u": lambda out: out["u"],
            "v": lambda out: out["v"],
            "w": lambda out: out["w"],
        },
        batch_size=cfg.EVAL.batch_size.sup_validator,
        prefix="result_u_v_w_p",
    ),
}

3.9 模型训练、评估与可视化

完成上述设置之后,只需要将上述实例化的对象按顺序传递给 ppsci.solver.Solver,然后启动训练、评估、可视化。

# initialize solver
solver = ppsci.solver.Solver(
    model,
    constraint,
    cfg.output_dir,
    optimizer,
    lr_scheduler,
    cfg.TRAIN.epochs,
    cfg.TRAIN.iters_per_epoch,
    save_freq=cfg.TRAIN.save_freq,
    log_freq=cfg.log_freq,
    eval_during_train=True,
    eval_freq=cfg.TRAIN.eval_freq,
    seed=cfg.seed,
    equation=equation,
    geom=geom,
    validator=validator,
    visualizer=visualizer,
    pretrained_model_path=cfg.TRAIN.pretrained_model_path,
    checkpoint_path=cfg.TRAIN.checkpoint_path,
    eval_with_no_grad=cfg.EVAL.eval_with_no_grad,
)
# train model
solver.train()
# evaluate after finished training
solver.eval()
# visualize prediction after finished training
solver.visualize()

4. 完整代码

aneurysm.py
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"""
Reference: https://docs.nvidia.com/deeplearning/modulus/modulus-v2209/user_guide/intermediate/adding_stl_files.html
"""

import hydra
import numpy as np
from omegaconf import DictConfig

import ppsci
from ppsci.utils import reader


def train(cfg: DictConfig):
    # set model
    model = ppsci.arch.MLP(**cfg.MODEL)

    # set equation
    equation = {
        "NavierStokes": ppsci.equation.NavierStokes(
            cfg.NU * cfg.SCALE, cfg.RHO, cfg.DIM, False
        ),
        "NormalDotVec": ppsci.equation.NormalDotVec(("u", "v", "w")),
    }

    # set geometry
    inlet_geo = ppsci.geometry.Mesh(cfg.INLET_STL_PATH)
    outlet_geo = ppsci.geometry.Mesh(cfg.OUTLET_STL_PATH)
    noslip_geo = ppsci.geometry.Mesh(cfg.NOSLIP_STL_PATH)
    integral_geo = ppsci.geometry.Mesh(cfg.INTEGRAL_STL_PATH)
    interior_geo = ppsci.geometry.Mesh(cfg.INTERIOR_STL_PATH)

    # normalize meshes
    inlet_geo = inlet_geo.translate(-np.array(cfg.CENTER)).scale(cfg.SCALE)
    outlet_geo = outlet_geo.translate(-np.array(cfg.CENTER)).scale(cfg.SCALE)
    noslip_geo = noslip_geo.translate(-np.array(cfg.CENTER)).scale(cfg.SCALE)
    integral_geo = integral_geo.translate(-np.array(cfg.CENTER)).scale(cfg.SCALE)
    interior_geo = interior_geo.translate(-np.array(cfg.CENTER)).scale(cfg.SCALE)
    geom = {
        "inlet_geo": inlet_geo,
        "outlet_geo": outlet_geo,
        "noslip_geo": noslip_geo,
        "integral_geo": integral_geo,
        "interior_geo": interior_geo,
    }

    # set dataloader config
    train_dataloader_cfg = {
        "dataset": "NamedArrayDataset",
        "iters_per_epoch": cfg.TRAIN.iters_per_epoch,
        "sampler": {
            "name": "BatchSampler",
            "drop_last": True,
            "shuffle": True,
        },
        "num_workers": 1,
    }

    # set constraint
    INLET_AREA = 21.1284 * (cfg.SCALE**2)
    INLET_RADIUS = np.sqrt(INLET_AREA / np.pi)

    def _compute_parabola(_in):
        centered_x = _in["x"] - cfg.INLET_CENTER[0]
        centered_y = _in["y"] - cfg.INLET_CENTER[1]
        centered_z = _in["z"] - cfg.INLET_CENTER[2]
        distance = np.sqrt(centered_x**2 + centered_y**2 + centered_z**2)
        parabola = cfg.INLET_VEL * np.maximum((1 - (distance / INLET_RADIUS) ** 2), 0)
        return parabola

    def inlet_u_ref_func(_in):
        return cfg.INLET_NORMAL[0] * _compute_parabola(_in)

    def inlet_v_ref_func(_in):
        return cfg.INLET_NORMAL[1] * _compute_parabola(_in)

    def inlet_w_ref_func(_in):
        return cfg.INLET_NORMAL[2] * _compute_parabola(_in)

    bc_inlet = ppsci.constraint.BoundaryConstraint(
        {"u": lambda d: d["u"], "v": lambda d: d["v"], "w": lambda d: d["w"]},
        {"u": inlet_u_ref_func, "v": inlet_v_ref_func, "w": inlet_w_ref_func},
        geom["inlet_geo"],
        {**train_dataloader_cfg, "batch_size": cfg.TRAIN.batch_size.bc_inlet},
        ppsci.loss.MSELoss("sum"),
        name="inlet",
    )
    bc_outlet = ppsci.constraint.BoundaryConstraint(
        {"p": lambda d: d["p"]},
        {"p": 0},
        geom["outlet_geo"],
        {**train_dataloader_cfg, "batch_size": cfg.TRAIN.batch_size.bc_outlet},
        ppsci.loss.MSELoss("sum"),
        name="outlet",
    )
    bc_noslip = ppsci.constraint.BoundaryConstraint(
        {"u": lambda d: d["u"], "v": lambda d: d["v"], "w": lambda d: d["w"]},
        {"u": 0, "v": 0, "w": 0},
        geom["noslip_geo"],
        {**train_dataloader_cfg, "batch_size": cfg.TRAIN.batch_size.bc_noslip},
        ppsci.loss.MSELoss("sum"),
        name="no_slip",
    )
    pde = ppsci.constraint.InteriorConstraint(
        equation["NavierStokes"].equations,
        {"continuity": 0, "momentum_x": 0, "momentum_y": 0, "momentum_z": 0},
        geom["interior_geo"],
        {**train_dataloader_cfg, "batch_size": cfg.TRAIN.batch_size.pde},
        ppsci.loss.MSELoss("sum"),
        name="interior",
    )
    igc_outlet = ppsci.constraint.IntegralConstraint(
        equation["NormalDotVec"].equations,
        {"normal_dot_vec": 2.54},
        geom["outlet_geo"],
        {
            **train_dataloader_cfg,
            "iters_per_epoch": cfg.TRAIN.iters_integral.igc_outlet,
            "batch_size": cfg.TRAIN.batch_size.igc_outlet,
            "integral_batch_size": cfg.TRAIN.integral_batch_size.igc_outlet,
        },
        ppsci.loss.IntegralLoss("sum"),
        weight_dict=cfg.TRAIN.weight.igc_outlet,
        name="igc_outlet",
    )
    igc_integral = ppsci.constraint.IntegralConstraint(
        equation["NormalDotVec"].equations,
        {"normal_dot_vec": -2.54},
        geom["integral_geo"],
        {
            **train_dataloader_cfg,
            "iters_per_epoch": cfg.TRAIN.iters_integral.igc_integral,
            "batch_size": cfg.TRAIN.batch_size.igc_integral,
            "integral_batch_size": cfg.TRAIN.integral_batch_size.igc_integral,
        },
        ppsci.loss.IntegralLoss("sum"),
        weight_dict=cfg.TRAIN.weight.igc_integral,
        name="igc_integral",
    )
    # wrap constraints together
    constraint = {
        bc_inlet.name: bc_inlet,
        bc_outlet.name: bc_outlet,
        bc_noslip.name: bc_noslip,
        pde.name: pde,
        igc_outlet.name: igc_outlet,
        igc_integral.name: igc_integral,
    }

    # set optimizer
    lr_scheduler = ppsci.optimizer.lr_scheduler.ExponentialDecay(
        **cfg.TRAIN.lr_scheduler
    )()
    optimizer = ppsci.optimizer.Adam(lr_scheduler)(model)

    # set validator
    eval_data_dict = reader.load_csv_file(
        cfg.EVAL_CSV_PATH,
        ("x", "y", "z", "u", "v", "w", "p"),
        {
            "x": "Points:0",
            "y": "Points:1",
            "z": "Points:2",
            "u": "U:0",
            "v": "U:1",
            "w": "U:2",
            "p": "p",
        },
    )
    input_dict = {
        "x": (eval_data_dict["x"] - cfg.CENTER[0]) * cfg.SCALE,
        "y": (eval_data_dict["y"] - cfg.CENTER[1]) * cfg.SCALE,
        "z": (eval_data_dict["z"] - cfg.CENTER[2]) * cfg.SCALE,
    }
    if "area" in input_dict.keys():
        input_dict["area"] *= cfg.SCALE ** (equation["NavierStokes"].dim)

    label_dict = {
        "p": eval_data_dict["p"],
        "u": eval_data_dict["u"],
        "v": eval_data_dict["v"],
        "w": eval_data_dict["w"],
    }
    eval_dataloader_cfg = {
        "dataset": {
            "name": "NamedArrayDataset",
            "input": input_dict,
            "label": label_dict,
        },
        "sampler": {"name": "BatchSampler"},
        "num_workers": 1,
    }
    sup_validator = ppsci.validate.SupervisedValidator(
        {**eval_dataloader_cfg, "batch_size": cfg.EVAL.batch_size.sup_validator},
        ppsci.loss.MSELoss("mean"),
        {
            "p": lambda out: out["p"],
            "u": lambda out: out["u"],
            "v": lambda out: out["v"],
            "w": lambda out: out["w"],
        },
        metric={"MSE": ppsci.metric.MSE()},
        name="ref_u_v_w_p",
    )
    validator = {sup_validator.name: sup_validator}

    # set visualizer(optional)
    visualizer = {
        "visualize_u_v_w_p": ppsci.visualize.VisualizerVtu(
            input_dict,
            {
                "p": lambda out: out["p"],
                "u": lambda out: out["u"],
                "v": lambda out: out["v"],
                "w": lambda out: out["w"],
            },
            batch_size=cfg.EVAL.batch_size.sup_validator,
            prefix="result_u_v_w_p",
        ),
    }

    # initialize solver
    solver = ppsci.solver.Solver(
        model,
        constraint,
        cfg.output_dir,
        optimizer,
        lr_scheduler,
        cfg.TRAIN.epochs,
        cfg.TRAIN.iters_per_epoch,
        save_freq=cfg.TRAIN.save_freq,
        log_freq=cfg.log_freq,
        eval_during_train=True,
        eval_freq=cfg.TRAIN.eval_freq,
        seed=cfg.seed,
        equation=equation,
        geom=geom,
        validator=validator,
        visualizer=visualizer,
        pretrained_model_path=cfg.TRAIN.pretrained_model_path,
        checkpoint_path=cfg.TRAIN.checkpoint_path,
        eval_with_no_grad=cfg.EVAL.eval_with_no_grad,
    )
    # train model
    solver.train()
    # evaluate after finished training
    solver.eval()
    # visualize prediction after finished training
    solver.visualize()


def evaluate(cfg: DictConfig):
    # set model
    model = ppsci.arch.MLP(**cfg.MODEL)

    # set validator
    eval_data_dict = reader.load_csv_file(
        cfg.EVAL_CSV_PATH,
        ("x", "y", "z", "u", "v", "w", "p"),
        {
            "x": "Points:0",
            "y": "Points:1",
            "z": "Points:2",
            "u": "U:0",
            "v": "U:1",
            "w": "U:2",
            "p": "p",
        },
    )
    input_dict = {
        "x": (eval_data_dict["x"] - cfg.CENTER[0]) * cfg.SCALE,
        "y": (eval_data_dict["y"] - cfg.CENTER[1]) * cfg.SCALE,
        "z": (eval_data_dict["z"] - cfg.CENTER[2]) * cfg.SCALE,
    }

    label_dict = {
        "p": eval_data_dict["p"],
        "u": eval_data_dict["u"],
        "v": eval_data_dict["v"],
        "w": eval_data_dict["w"],
    }
    eval_dataloader_cfg = {
        "dataset": {
            "name": "NamedArrayDataset",
            "input": input_dict,
            "label": label_dict,
        },
        "sampler": {"name": "BatchSampler"},
        "num_workers": 1,
    }
    sup_validator = ppsci.validate.SupervisedValidator(
        {**eval_dataloader_cfg, "batch_size": cfg.EVAL.batch_size.sup_validator},
        ppsci.loss.MSELoss("mean"),
        {
            "p": lambda out: out["p"],
            "u": lambda out: out["u"],
            "v": lambda out: out["v"],
            "w": lambda out: out["w"],
        },
        metric={"MSE": ppsci.metric.MSE()},
        name="ref_u_v_w_p",
    )
    validator = {sup_validator.name: sup_validator}

    # set visualizer
    visualizer = {
        "visualize_u_v_w_p": ppsci.visualize.VisualizerVtu(
            input_dict,
            {
                "p": lambda out: out["p"],
                "u": lambda out: out["u"],
                "v": lambda out: out["v"],
                "w": lambda out: out["w"],
            },
            batch_size=cfg.EVAL.batch_size.sup_validator,
            prefix="result_u_v_w_p",
        ),
    }

    # initialize solver
    solver = ppsci.solver.Solver(
        model,
        output_dir=cfg.output_dir,
        log_freq=cfg.log_freq,
        seed=cfg.seed,
        validator=validator,
        visualizer=visualizer,
        pretrained_model_path=cfg.EVAL.pretrained_model_path,
        eval_with_no_grad=cfg.EVAL.eval_with_no_grad,
    )
    # evaluate
    solver.eval()
    # visualize prediction
    solver.visualize()


def export(cfg: DictConfig):
    # set model
    model = ppsci.arch.MLP(**cfg.MODEL)

    # initialize solver
    solver = ppsci.solver.Solver(
        model,
        pretrained_model_path=cfg.INFER.pretrained_model_path,
    )
    # export model
    from paddle.static import InputSpec

    input_spec = [
        {key: InputSpec([None, 1], "float32", name=key) for key in model.input_keys},
    ]
    solver.export(input_spec, cfg.INFER.export_path, with_onnx=False)


def inference(cfg: DictConfig):
    from deploy.python_infer import pinn_predictor

    predictor = pinn_predictor.PINNPredictor(cfg)
    eval_data_dict = reader.load_csv_file(
        cfg.EVAL_CSV_PATH,
        ("x", "y", "z", "u", "v", "w", "p"),
        {
            "x": "Points:0",
            "y": "Points:1",
            "z": "Points:2",
            "u": "U:0",
            "v": "U:1",
            "w": "U:2",
            "p": "p",
        },
    )
    input_dict = {
        "x": (eval_data_dict["x"] - cfg.CENTER[0]) * cfg.SCALE,
        "y": (eval_data_dict["y"] - cfg.CENTER[1]) * cfg.SCALE,
        "z": (eval_data_dict["z"] - cfg.CENTER[2]) * cfg.SCALE,
    }
    output_dict = predictor.predict(input_dict, cfg.INFER.batch_size)

    # mapping data to cfg.INFER.output_keys
    output_dict = {
        store_key: output_dict[infer_key]
        for store_key, infer_key in zip(cfg.MODEL.output_keys, output_dict.keys())
    }

    ppsci.visualize.save_vtu_from_dict(
        "./aneurysm_pred.vtu",
        {**input_dict, **output_dict},
        input_dict.keys(),
        cfg.MODEL.output_keys,
    )


@hydra.main(version_base=None, config_path="./conf", config_name="aneurysm.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()

5. 结果展示

对于血管瘤测试集(共 2,962,708 个三维坐标点),模型预测结果如下所示。

aneurysm_compare.jpg

左侧为PaddleScience预测结果,中间为OpenFOAM求解器预测结果,右侧为两者的差值

可以看到对于管壁压力\(p(x,y,z)\),模型的预测结果和 OpenFOAM 结果基本一致。

6. 参考资料