{
"cells": [
{
"attachments": {},
"cell_type": "markdown",
"metadata": {
"pycharm": {
"name": "#%% md\n"
}
},
"source": [
"# 基于FNO求解二维Navier-Stokes\n",
"\n",
"[](https://mindspore-website.obs.cn-north-4.myhuaweicloud.com/notebook/r2.2/mindflow/zh_cn/data_driven/mindspore_navier_stokes_FNO2D.ipynb) [](https://mindspore-website.obs.cn-north-4.myhuaweicloud.com/notebook/r2.2/mindflow/zh_cn/data_driven/mindspore_navier_stokes_FNO2D.py) [](https://gitee.com/mindspore/docs/blob/r2.2/docs/mindflow/docs/source_zh_cn/data_driven/navier_stokes_FNO2D.ipynb)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {
"pycharm": {
"name": "#%% md\n"
}
},
"source": [
"## 概述\n",
"\n",
"计算流体力学是21世纪流体力学领域的重要技术之一，其通过使用数值方法在计算机中对流体力学的控制方程进行求解，从而实现流动的分析、预测和控制。传统的有限元法（finite element method，FEM）和有限差分法（finite difference method，FDM）常用于复杂的仿真流程（物理建模、网格划分、数值离散、迭代求解等）和较高的计算成本，往往效率低下。因此，借助AI提升流体仿真效率是十分必要的。\n",
"\n",
"近年来，随着神经网络的迅猛发展，为科学计算提供了新的范式。经典的神经网络是在有限维度的空间进行映射，只能学习与特定离散化相关的解。与经典神经网络不同，傅里叶神经算子（Fourier Neural Operator，FNO）是一种能够学习无限维函数空间映射的新型深度学习架构。该架构可直接学习从任意函数参数到解的映射，用于解决一类偏微分方程的求解问题，具有更强的泛化能力。更多信息可参考[Fourier Neural Operator for Parametric Partial Differential Equations](https://arxiv.org/abs/2010.08895)。\n",
"\n",
"本案例教程介绍利用傅里叶神经算子的纳维-斯托克斯方程（Navier-Stokes equation）求解方法。"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {
"pycharm": {
"name": "#%% md\n"
}
},
"source": [
"## 纳维-斯托克斯方程（Navier-Stokes equation）\n",
"\n",
"纳维-斯托克斯方程（Navier-Stokes equation）是计算流体力学领域的经典方程，是一组描述流体动量守恒的偏微分方程，简称N-S方程。它在二维不可压缩流动中的涡度形式如下：\n",
"\n",
"$$\n",
"\\partial_t w(x, t)+u(x, t) \\cdot \\nabla w(x, t)=\\nu \\Delta w(x, t)+f(x), \\quad x \\in(0,1)^2, t \\in(0, T]\n",
"$$\n",
"\n",
"$$\n",
"\\nabla \\cdot u(x, t)=0, \\quad x \\in(0,1)^2, t \\in[0, T]\n",
"$$\n",
"\n",
"$$\n",
"w(x, 0)=w_0(x), \\quad x \\in(0,1)^2\n",
"$$\n",
"\n",
"其中$u$表示速度场，$w=\\nabla \\times u$表示涡度，$w_0(x)$表示初始条件，$\\nu$表示粘度系数，$f(x)$为外力合力项。"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"## 问题描述\n",
"\n",
"本案例利用Fourier Neural Operator学习某一个时刻对应涡度到下一时刻涡度的映射，实现二维不可压缩N-S方程的求解：\n",
"\n",
"$$\n",
"w_t \\mapsto w(\\cdot, t+1)\n",
"$$"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"## 技术路径\n",
"\n",
"MindSpore Flow求解该问题的具体流程如下：\n",
"\n",
"1. 创建数据集。\n",
"2. 构建模型。\n",
"3. 优化器与损失函数。\n",
"4. 模型训练。"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"## Fourier Neural Operator\n",
"\n",
"Fourier Neural Operator模型构架如下图所示。图中$w_0(x)$表示初始涡度，通过Lifting Layer实现输入向量的高维映射，然后将映射结果作为Fourier Layer的输入，进行频域信息的非线性变换，最后由Decoding Layer将变换结果映射至最终的预测结果$w_1(x)$。\n",
"\n",
"Lifting Layer、Fourier Layer以及Decoding Layer共同组成了Fourier Neural Operator。\n",
"\n",
"\n",
"\n",
"Fourier Layer网络结构如下图所示。图中V表示输入向量，上框表示向量经过傅里叶变换后，经过线性变换R，过滤高频信息，然后进行傅里叶逆变换；另一分支经过线性变换W，最后通过激活函数，得到Fourier Layer输出向量。\n",
"\n",
""
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [],
"source": [
"import os\n",
"import time\n",
"import numpy as np\n",
"\n",
"import mindspore\n",
"from mindspore import nn, ops, Tensor, jit, set_seed"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"下述`src`包可以在[applications/data_driven/navier_stokes/fno2d/src](https://gitee.com/mindspore/mindscience/tree/r0.6/MindFlow/applications/data_driven/navier_stokes/fno2d/src)下载。\n",
"配置文件可在[config](https://gitee.com/mindspore/mindscience/blob/r0.6/MindFlow/applications/data_driven/navier_stokes/fno2d/configs/fno2d.yaml)中修改。"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [],
"source": [
"from mindflow.cell import FNO2D\n",
"from mindflow.common import get_warmup_cosine_annealing_lr\n",
"from mindflow.loss import RelativeRMSELoss\n",
"from mindflow.utils import load_yaml_config\n",
"from mindflow.pde import UnsteadyFlowWithLoss\n",
"from src import calculate_l2_error, create_training_dataset\n",
"\n",
"set_seed(0)\n",
"np.random.seed(0)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [],
"source": [
"# set context for training: using graph mode for high performance training with GPU acceleration\n",
"mindspore.set_context(mode=mindspore.GRAPH_MODE, device_target='GPU', device_id=2)\n",
"use_ascend = mindspore.get_context(attr_key='device_target') == \"Ascend\"\n",
"config = load_yaml_config('navier_stokes_2d.yaml')\n",
"data_params = config[\"data\"]\n",
"model_params = config[\"model\"]\n",
"optimizer_params = config[\"optimizer\"]"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {
"pycharm": {
"name": "#%% md\n"
}
},
"source": [
"## 创建数据集\n",
"\n",
"训练与测试数据下载: [data_driven/navier_stokes/dataset](https://download.mindspore.cn/mindscience/mindflow/dataset/applications/data_driven/navier_stokes/dataset/)。\n",
"\n",
"本案例根据Zongyi Li在 [Fourier Neural Operator for Parametric Partial Differential Equations](https://arxiv.org/pdf/2010.08895.pdf) 一文中对数据集的设置生成训练数据集与测试数据集。具体设置如下：\n",
"\n",
"基于周期性边界，生成满足如下分布的初始条件$w_0(x)$：\n",
"\n",
"$$\n",
"w_0 \\sim \\mu, \\mu=\\mathcal{N}\\left(0,7^{3 / 2}(-\\Delta+49 I)^{-2.5}\\right)\n",
"$$\n",
"\n",
"外力项设置为：\n",
"\n",
"$$\n",
"f(x)=0.1\\left(\\sin \\left(2 \\pi\\left(x_1+x_2\\right)\\right)+\\right.\\cos(2 \\pi(x_1+x_2)))\n",
"$$\n",
"\n",
"采用`Crank-Nicolson`方法生成数据，时间步长设置为1e-4，最终数据以每 t = 1 个时间单位记录解。所有数据均在256×256的网格上生成，并被下采样至64×64网格。本案例选取粘度系数$\\nu=1e−5$，训练集样本量为19000个，测试集样本量为3800个。"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Data preparation finished\n"
]
}
],
"source": [
"train_dataset = create_training_dataset(data_params, input_resolution=model_params[\"input_resolution\"], shuffle=True)\n",
"test_input = np.load(os.path.join(data_params[\"path\"], \"test/inputs.npy\"))\n",
"test_label = np.load(os.path.join(data_params[\"path\"], \"test/label.npy\"))"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {
"pycharm": {
"name": "#%% md\n"
}
},
"source": [
"## 构建模型\n",
"\n",
"网络由1层Lifting layer、多层Fourier Layer以及1层Decoding layer叠加组成：\n",
"\n",
"- Lifting layer对应样例代码中`FNO2D.fc0`，将输出数据$x$映射至高维；\n",
"\n",
"- 多层Fourier Layer的叠加对应样例代码中`FNO2D.fno_seq`，本案例采用离散傅里叶变换实现时域与频域的转换；\n",
"\n",
"- Decoding layer对应代码中`FNO2D.fc1`与`FNO2D.fc2`，获得最终的预测值。"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [],
"source": [
"model = FNO2D(in_channels=model_params[\"in_channels\"],\n",
" out_channels=model_params[\"out_channels\"],\n",
" resolution=model_params[\"input_resolution\"],\n",
" modes=model_params[\"modes\"],\n",
" channels=model_params[\"width\"],\n",
" depths=model_params[\"depth\"]\n",
" )\n",
"\n",
"model_params_list = []\n",
"for k, v in model_params.items():\n",
" model_params_list.append(f\"{k}-{v}\")\n",
"model_name = \"_\".join(model_params_list)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {
"pycharm": {
"name": "#%% md\n"
}
},
"source": [
"## 优化器与损失函数\n",
"\n",
"使用相对均方根误差作为网络训练损失函数："
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [],
"source": [
"steps_per_epoch = train_dataset.get_dataset_size()\n",
"lr = get_warmup_cosine_annealing_lr(lr_init=optimizer_params[\"initial_lr\"],\n",
" last_epoch=optimizer_params[\"train_epochs\"],\n",
" steps_per_epoch=steps_per_epoch,\n",
" warmup_epochs=optimizer_params[\"warmup_epochs\"])\n",
"\n",
"optimizer = nn.Adam(model.trainable_params(), learning_rate=Tensor(lr))\n",
"\n",
"problem = UnsteadyFlowWithLoss(model, loss_fn=RelativeRMSELoss(), data_format=\"NHWC\")"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {
"pycharm": {
"name": "#%% md\n"
}
},
"source": [
"## 模型训练\n",
"\n",
"使用**MindSpore >= 2.0.0**的版本，可以使用函数式编程范式训练神经网络。"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"pycharm": {
"name": "#%%\n"
}
},
"outputs": [],
"source": [
"def train():\n",
" if use_ascend:\n",
" from mindspore.amp import DynamicLossScaler, auto_mixed_precision, all_finite\n",
" loss_scaler = DynamicLossScaler(1024, 2, 100)\n",
" auto_mixed_precision(model, 'O3')\n",
"\n",
" def forward_fn(train_inputs, train_label):\n",
" loss = problem.get_loss(train_inputs, train_label)\n",
" if use_ascend:\n",
" loss = loss_scaler.scale(loss)\n",
" return loss\n",
"\n",
" grad_fn = mindspore.value_and_grad(forward_fn, None, optimizer.parameters, has_aux=False)\n",
"\n",
" @jit\n",
" def train_step(train_inputs, train_label):\n",
" loss, grads = grad_fn(train_inputs, train_label)\n",
" if use_ascend:\n",
" loss = loss_scaler.unscale(loss)\n",
" if all_finite(grads):\n",
" grads = loss_scaler.unscale(grads)\n",
" loss = ops.depend(loss, optimizer(grads))\n",
" else:\n",
" loss = ops.depend(loss, optimizer(grads))\n",
" return loss\n",
"\n",
" sink_process = mindspore.data_sink(train_step, train_dataset, sink_size=1)\n",
" summary_dir = os.path.join(config[\"summary_dir\"], model_name)\n",
"\n",
" for cur_epoch in range(optimizer_params[\"train_epochs\"]):\n",
" local_time_beg = time.time()\n",
" model.set_train()\n",
" cur_loss = 0.0\n",
" for _ in range(steps_per_epoch):\n",
" cur_loss = sink_process()\n",
"\n",
" print(\"epoch: %s, loss is %s\" % (cur_epoch + 1, cur_loss), flush=True)\n",
" local_time_end = time.time()\n",
" epoch_seconds = (local_time_end - local_time_beg) * 1000\n",
" step_seconds = epoch_seconds / steps_per_epoch\n",
" print(\"Train epoch time: {:5.3f} ms, per step time: {:5.3f} ms\".format\n",
" (epoch_seconds, step_seconds), flush=True)\n",
"\n",
" if (cur_epoch + 1) % config[\"save_checkpoint_epoches\"] == 0:\n",
" ckpt_dir = os.path.join(summary_dir, \"ckpt\")\n",
" if not os.path.exists(ckpt_dir):\n",
" os.makedirs(ckpt_dir)\n",
" mindspore.save_checkpoint(model, os.path.join(ckpt_dir, model_params[\"name\"]))\n",
"\n",
" if (cur_epoch + 1) % config['eval_interval'] == 0:\n",
" calculate_l2_error(model, test_input, test_label, config[\"test_batch_size\"])\n"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"epoch: 1, loss is 1.7631323\n",
"Train epoch time: 50405.954 ms, per step time: 50.406 ms\n",
"epoch: 2, loss is 1.9283392\n",
"Train epoch time: 36591.429 ms, per step time: 36.591 ms\n",
"epoch: 3, loss is 1.4265916\n",
"Train epoch time: 35085.079 ms, per step time: 35.085 ms\n",
"epoch: 4, loss is 1.8609437\n",
"Train epoch time: 34407.280 ms, per step time: 34.407 ms\n",
"epoch: 5, loss is 1.5222052\n",
"Train epoch time: 34596.965 ms, per step time: 34.597 ms\n",
"epoch: 6, loss is 1.3424721\n",
"Train epoch time: 33847.209 ms, per step time: 33.847 ms\n",
"epoch: 7, loss is 1.607729\n",
"Train epoch time: 33106.981 ms, per step time: 33.107 ms\n",
"epoch: 8, loss is 1.3308442\n",
"Train epoch time: 33051.339 ms, per step time: 33.051 ms\n",
"epoch: 9, loss is 1.3169765\n",
"Train epoch time: 33901.816 ms, per step time: 33.902 ms\n",
"epoch: 10, loss is 1.4149593\n",
"Train epoch time: 33908.748 ms, per step time: 33.909 ms\n",
"================================Start Evaluation================================\n",
"mean rel_rmse_error: 0.15500953359901906\n",
"=================================End Evaluation=================================\n",
"...\n",
"epoch: 141, loss is 0.777328\n",
"Train epoch time: 32549.911 ms, per step time: 32.550 ms\n",
"epoch: 142, loss is 0.7008966\n",
"Train epoch time: 32522.572 ms, per step time: 32.523 ms\n",
"epoch: 143, loss is 0.72377646\n",
"Train epoch time: 32566.685 ms, per step time: 32.567 ms\n",
"epoch: 144, loss is 0.72175145\n",
"Train epoch time: 32435.932 ms, per step time: 32.436 ms\n",
"epoch: 145, loss is 0.6235678\n",
"Train epoch time: 32463.707 ms, per step time: 32.464 ms\n",
"epoch: 146, loss is 0.9351083\n",
"Train epoch time: 32448.413 ms, per step time: 32.448 ms\n",
"epoch: 147, loss is 0.9283789\n",
"Train epoch time: 32472.401 ms, per step time: 32.472 ms\n",
"epoch: 148, loss is 0.7655642\n",
"Train epoch time: 32604.642 ms, per step time: 32.605 ms\n",
"epoch: 149, loss is 0.7233772\n",
"Train epoch time: 32649.832 ms, per step time: 32.650 ms\n",
"epoch: 150, loss is 0.86825275\n",
"Train epoch time: 32589.243 ms, per step time: 32.589 ms\n",
"================================Start Evaluation================================\n",
"mean rel_rmse_error: 0.07437102290522307\n",
"=================================End Evaluation=================================\n",
"predict total time: 15.212349653244019 s\n"
]
}
],
"source": [
"train()"
]
}
],
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"file_extension": ".py",
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