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[Fix] Adapt preprocess transforms to multiple arguments (#572)
* adapt preprocess transforms to multiple arguments * update code * update docs * fix document * refine code in IDE and FPDE
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‎docs/zh/examples/volterra_ide.md

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<a href="https://aistudio.baidu.com/aistudio/projectdetail/6622866?sUid=438690&shared=1&ts=1691582831601" class="md-button md-button--primary" style>AI Studio快速体验</a>
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=== "模型训练命令"
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``` sh
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python volterra_ide.py
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```
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=== "模型评估命令"
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``` sh
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python volterra_ide.py mode=eval EVAL.pretrained_model_path=https://paddle-org.bj.bcebos.com/paddlescience/models/volterra_ide/volterra_ide_pretrained.pdparams
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```
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| 预训练模型 | 指标 |
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|:--| :--|
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| [volterra_ide_pretrained.pdparams](https://paddle-org.bj.bcebos.com/paddlescience/models/volterra_ide/volterra_ide_pretrained.pdparams) | loss(L2Rel_Validator): 0.00023 <br> L2Rel.u(L2Rel_Validator): 0.00023 |
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## 1. 背景简介
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Volterra integral equation(沃尔泰拉积分方程)是一种积分方程，即方程中含有对待求解函数的积分运算，其有两种形式，如下所示
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在上述问题中，我们确定了输入为 $x$，输出为 $u(x)$，因此我们使用，用 PaddleScience 代码表示如下：
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``` py linenums="37"
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``` py linenums="39"
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--8<--
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examples/ide/volterra_ide.py:37:38
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examples/ide/volterra_ide.py:39:40
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```
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Volterra_IDE 问题的积分域是 $a$ ~ $t$，其中 `a` 为固定常数 0，`t` 的范围为 0 ~ 5，因此可以使用PaddleScience 内置的一维几何 `TimeDomain` 作为计算域。
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``` py linenums="40"
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``` py linenums="42"
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examples/ide/volterra_ide.py:40:42
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examples/ide/volterra_ide.py:42:43
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```
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### 3.3 方程构建
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由于 Volterra_IDE 使用的是积分方程，因此可以直接使用 PaddleScience 内置的 `ppsci.equation.Volterra`，并指定所需的参数：积分下限 `a`、`t` 的离散取值点数 `num_points`、一维高斯积分点的个数 `quad_deg`、$K(t,s)$ 核函数 `kernel_func`、$u(t) - f(t)$ 等式右侧表达式 `func`。
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``` py linenums="44"
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``` py linenums="45"
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examples/ide/volterra_ide.py:44:64
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examples/ide/volterra_ide.py:45:61
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```
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由于等式左侧涉及到积分计算（实际采用高斯积分近似计算），因此在 0 ~ 5 区间内采样出多个 `t_i` 点后，还需要计算其用于高斯积分的点集，即对每一个 `(0,t_i)` 区间，都计算出一一对应的高斯积分点集 `quad_i` 和点权 `weight_i`。PaddleScience 将这一步作为输入数据的预处理，加入到代码中，如下所示
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``` py linenums="66"
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``` py linenums="63"
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examples/ide/volterra_ide.py:66:108
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examples/ide/volterra_ide.py:63:117
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```
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因此可以加入 `t=0` 时的初值条件，代码如下所示
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``` py linenums="110"
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``` py linenums="119"
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examples/ide/volterra_ide.py:110:128
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examples/ide/volterra_ide.py:119:137
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```
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在微分方程约束、初值约束构建完毕之后，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。
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``` py linenums="129"
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``` py linenums="138"
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examples/ide/volterra_ide.py:129:133
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examples/ide/volterra_ide.py:138:142
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```
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### 3.5 超参数设定
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接下来我们需要指定训练轮数和学习率，此处我们按实验经验，让 `L-BFGS` 优化器进行一轮优化即可，但一轮优化内的 `max_iters` 数可以设置为一个较大的一个数 `15000`。
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``` py linenums="135"
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``` yaml linenums="39"
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examples/ide/volterra_ide.py:135:136
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examples/ide/conf/volterra_ide.yaml:39:57
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```
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### 3.6 优化器构建
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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 `LBFGS` 优化器。
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``` py linenums="138"
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``` py linenums="144"
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examples/ide/volterra_ide.py:138:146
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examples/ide/volterra_ide.py:144:145
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```
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### 3.7 评估器构建
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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。
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``` py linenums="148"
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``` py linenums="147"
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examples/ide/volterra_ide.py:148:163
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examples/ide/volterra_ide.py:147:161
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```
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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。
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``` py linenums="165"
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``` py linenums="163"
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examples/ide/volterra_ide.py:165:181
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examples/ide/volterra_ide.py:163:181
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```
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``` py linenums="183"
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examples/ide/volterra_ide.py:183:
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examples/ide/volterra_ide.py:183:194
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```
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‎examples/fpde/fractional_poisson_2d.py

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import math
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from typing import Dict
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from typing import Tuple
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from typing import Union
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import numpy as np
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return paddle.abs(1 - (out["x"] ** 2 + out["y"] ** 2)) ** (1 + ALPHA / 2)
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return np.abs(1 - (out["x"] ** 2 + out["y"] ** 2)) ** (1 + ALPHA / 2)
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# set input transform
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def fpde_transform(in_: Dict[str, np.ndarray]) -> Dict[str, np.ndarray]:
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# set transform for input data
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def input_data_fpde_transform(
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input: Dict[str, np.ndarray],
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weight: Dict[str, np.ndarray],
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label: Dict[str, np.ndarray],
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) -> Tuple[
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Dict[str, paddle.Tensor], Dict[str, paddle.Tensor], Dict[str, paddle.Tensor]
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]:
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"""Get sampling points for integral.
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Args:
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in_ (Dict[str, np.ndarray]): Raw input dict.
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input (Dict[str, paddle.Tensor]): Raw input dict.
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weight (Dict[str, paddle.Tensor]): Raw weight dict.
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label (Dict[str, paddle.Tensor]): Raw label dict.
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Returns:
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Dict[str, np.ndarray]: Input dict contained sampling points.
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Tuple[ Dict[str, paddle.Tensor], Dict[str, paddle.Tensor], Dict[str, paddle.Tensor] ]:
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Input dict contained sampling points, weight dict and label dict.
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"""
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points = np.concatenate((in_["x"].numpy(), in_["y"].numpy()), axis=1)
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points = np.concatenate((input["x"].numpy(), input["y"].numpy()), axis=1)
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x = equation["fpde"].get_x(points)
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return {
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**in_,
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**{k: paddle.to_tensor(v) for k, v in x.items()},
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}
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return (
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{
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**input,
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**{k: paddle.to_tensor(v) for k, v in x.items()},
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},
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weight,
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label,
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)
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fpde_constraint = ppsci.constraint.InteriorConstraint(
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equation["fpde"].equations,
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"transforms": (
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{
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"FunctionalTransform": {
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"transform_func": fpde_transform,
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"transform_func": input_data_fpde_transform,
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},
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},
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),

‎examples/ide/conf/volterra_ide.yaml

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hydra:
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run:
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# dynamic output directory according to running time and override name
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dir: outputs_volterra_IDE/${now:%Y-%m-%d}/${now:%H-%M-%S}/${hydra.job.override_dirname}
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job:
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name: ${mode} # name of logfile
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chdir: false # keep current working direcotry unchaned
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config:
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override_dirname:
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exclude_keys:
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- TRAIN.checkpoint_path
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- TRAIN.pretrained_model_path
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- EVAL.pretrained_model_path
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- mode
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- output_dir
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- log_freq
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sweep:
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# output directory for multirun
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dir: ${hydra.run.dir}
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subdir: ./
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# general settings
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mode: train # running mode: train/eval
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seed: 42
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output_dir: ${hydra:run.dir}
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log_freq: 20
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# set geometry
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BOUNDS: [0, 5]
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# model settings
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MODEL:
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input_keys: ["x"]
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output_keys: ["u"]
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num_layers: 3
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hidden_size: 20
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activation: "tanh"
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# training settings
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TRAIN:
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epochs: 1
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iters_per_epoch: 1
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save_freq: 1
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eval_during_train: true
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eval_freq: 1
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optimizer:
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learning_rate: 1
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max_iter: 15000
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max_eval: 1250
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tolerance_grad: 1.0e-8
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tolerance_change: 0
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history_size: 100
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quad_deg: 20
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npoint_interior: 12
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npoint_ic: 1
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pretrained_model_path: null
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checkpoint_path: null
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# evaluation settings
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EVAL:
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pretrained_model_path: null
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eval_with_no_grad: true
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npoint_eval: 100

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