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PhyLSTM

# linux
wget -c https://paddle-org.bj.bcebos.com/paddlescience/datasets/PhyLSTM/data_boucwen.mat
# windows
# curl https://paddle-org.bj.bcebos.com/paddlescience/datasets/PhyLSTM/data_boucwen.mat -o data_boucwen.mat
python phylstm2.py

# linux
wget -c https://paddle-org.bj.bcebos.com/paddlescience/datasets/PhyLSTM/data_boucwen.mat
# windows
# curl https://paddle-org.bj.bcebos.com/paddlescience/datasets/PhyLSTM/data_boucwen.mat -o data_boucwen.mat
python phylstm3.py

# linux
wget -c https://paddle-org.bj.bcebos.com/paddlescience/datasets/PhyLSTM/data_boucwen.mat
# windows
# curl https://paddle-org.bj.bcebos.com/paddlescience/datasets/PhyLSTM/data_boucwen.mat -o data_boucwen.mat
python phylstm2.py mode=eval EVAL.pretrained_model_path=https://paddle-org.bj.bcebos.com/paddlescience/models/phylstm/phylstm2_pretrained.pdparams

# linux
wget -c https://paddle-org.bj.bcebos.com/paddlescience/datasets/PhyLSTM/data_boucwen.mat
# windows
# curl https://paddle-org.bj.bcebos.com/paddlescience/datasets/PhyLSTM/data_boucwen.mat -o data_boucwen.mat
python phylstm3.py mode=eval EVAL.pretrained_model_path=https://paddle-org.bj.bcebos.com/paddlescience/models/phylstm/phylstm3_pretrained.pdparams
Pretrained Model Metrics
phylstm2_pretrained.pdparams loss(sup_valid): 0.00799
phylstm3_pretrained.pdparams loss(sup_valid): 0.03098

1. Background Introduction

We introduce an innovative physics-informed LSTM framework for metamodeling of nonlinear structural systems lacking data. The basic concept is to integrate available but incomplete physical knowledge (such as physical laws, scientific principles) into a deep Long Short-Term Memory (LSTM) network, which restricts and facilitates learning within a feasible solution space. Physical constraints are embedded in the loss function to enforce model training to accurately capture potential system non-linearities even when available training datasets are very limited. Especially for dynamic structures, physical laws of equations of motion, state dependence, and hysteresis constitutive relationships are considered to construct physical losses. Embedded physics can alleviate overfitting problems, reduce the need for large training datasets, and improve the robustness of trained models, making them capable of extrapolation, thereby making more reliable predictions. Therefore, the physics-knowledge-guided deep learning paradigm outperforms traditional non-physics-guided data-driven neural networks.

2. Problem Definition

Metamodeling of structural systems aims to develop low-fidelity (or low-order) models to effectively capture potential nonlinear input-output behaviors. Metamodels can be trained on datasets obtained from high-fidelity simulations or actual system sensing. To illustrate better, we consider a building-type structure and assume that seismic dynamics are governed by low-fidelity nonlinear equations of motion (EOM):

\[ \mathbf{M} \ddot{\mathbf{u}}+\underbrace{\mathbf{C} \dot{\mathbf{u}}+\lambda \mathbf{K u}+(1-\lambda) \mathbf{K r}}_{\mathbf{h}}=-\mathbf{M} \Gamma a_g \]

Where M is the mass matrix; C is the damping matrix; K is the stiffness matrix.

The governing equation can be rewritten in a more general form:

\[ \ddot{\mathbf{u}}+\mathrm{g}=-\Gamma a_g \]

3. Problem Solving

Next, we will explain how to convert the problem into PaddleScience code step by step and solve the problem using deep learning methods. In order to quickly understand PaddleScience, only key steps such as model construction, equation construction, and computational domain construction are described below, while other details please refer to API Documentation.

3.1 Model Construction

In the PhyLSTM problem, establish an LSTM network Deep LSTM network, expressed in PaddleScience code as follows

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model = ppsci.arch.DeepPhyLSTM(
    cfg.MODEL.input_size,
    eta.shape[2],
    cfg.MODEL.hidden_size,
    cfg.MODEL.model_type,
)

DeepPhyLSTM parameters input_size is input size, output_size is output size, hidden_size is hidden layer size, model_type is model type.

3.2 Data Construction

Before running the code for this problem, please download data_boucwen.mat according to the following command.

wget -c -P ./ https://paddle-org.bj.bcebos.com/paddlescience/datasets/PhyLSTM/data_boucwen.mat

This case involves reading data construction, as shown below

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mat = scipy.io.loadmat(cfg.DATA_FILE_PATH)
ag_data = mat["input_tf"]  # ag, ad, av
u_data = mat["target_X_tf"]
ut_data = mat["target_Xd_tf"]
utt_data = mat["target_Xdd_tf"]
ag_data = ag_data.reshape([ag_data.shape[0], ag_data.shape[1], 1])
u_data = u_data.reshape([u_data.shape[0], u_data.shape[1], 1])
ut_data = ut_data.reshape([ut_data.shape[0], ut_data.shape[1], 1])
utt_data = utt_data.reshape([utt_data.shape[0], utt_data.shape[1], 1])

t = mat["time"]
dt = t[0, 1] - t[0, 0]

ag_all = ag_data
u_all = u_data
u_t_all = ut_data
u_tt_all = utt_data

# finite difference
N = u_data.shape[1]
phi1 = np.concatenate(
    [
        np.array([-3 / 2, 2, -1 / 2]),
        np.zeros([N - 3]),
    ]
)
temp1 = np.concatenate([-1 / 2 * np.identity(N - 2), np.zeros([N - 2, 2])], axis=1)
temp2 = np.concatenate([np.zeros([N - 2, 2]), 1 / 2 * np.identity(N - 2)], axis=1)
phi2 = temp1 + temp2
phi3 = np.concatenate(
    [
        np.zeros([N - 3]),
        np.array([1 / 2, -2, 3 / 2]),
    ]
)
phi_t0 = (
    1
    / dt
    * np.concatenate(
        [
            np.reshape(phi1, [1, phi1.shape[0]]),
            phi2,
            np.reshape(phi3, [1, phi3.shape[0]]),
        ],
        axis=0,
    )
)
phi_t0 = np.reshape(phi_t0, [1, N, N])

ag_star = ag_all[0:10]
eta_star = u_all[0:10]
eta_t_star = u_t_all[0:10]
eta_tt_star = u_tt_all[0:10]
ag_c_star = ag_all[0:50]
lift_star = -ag_c_star

eta = eta_star
ag = ag_star
lift = lift_star
eta_t = eta_t_star
eta_tt = eta_tt_star
ag_c = ag_c_star
g = -eta_tt - ag
phi_t = np.repeat(phi_t0, ag_c_star.shape[0], axis=0)

3.3 Constraint Construction

Set training dataset and loss calculation function, return fields, code is as follows:

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sup_constraint_pde = ppsci.constraint.SupervisedConstraint(
    {
        "dataset": {
            "name": "NamedArrayDataset",
            "input": input_dict_train,
            "label": label_dict_train,
        },
        "sampler": {
            "name": "BatchSampler",
            "drop_last": True,
            "shuffle": True,
        },
        "batch_size": 1,
        "num_workers": 0,
    },
    ppsci.loss.FunctionalLoss(functions.train_loss_func2),
    {
        "eta_pred": lambda out: out["eta_pred"],
        "eta_dot_pred": lambda out: out["eta_dot_pred"],
        "g_pred": lambda out: out["g_pred"],
        "eta_t_pred_c": lambda out: out["eta_t_pred_c"],
        "eta_dot_pred_c": lambda out: out["eta_dot_pred_c"],
        "lift_pred_c": lambda out: out["lift_pred_c"],
    },
    name="sup_train",
)
constraint_pde = {sup_constraint_pde.name: sup_constraint_pde}

3.4 Validator Construction

Set evaluation dataset and loss calculation function, return fields, code is as follows:

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sup_validator_pde = ppsci.validate.SupervisedValidator(
    {
        "dataset": {
            "name": "NamedArrayDataset",
            "input": input_dict_val,
            "label": label_dict_val,
        },
        "batch_size": 1,
        "num_workers": 0,
    },
    ppsci.loss.FunctionalLoss(functions.train_loss_func2),
    {
        "eta_pred": lambda out: out["eta_pred"],
        "eta_dot_pred": lambda out: out["eta_dot_pred"],
        "g_pred": lambda out: out["g_pred"],
        "eta_t_pred_c": lambda out: out["eta_t_pred_c"],
        "eta_dot_pred_c": lambda out: out["eta_dot_pred_c"],
        "lift_pred_c": lambda out: out["lift_pred_c"],
    },
    metric={"metric": ppsci.metric.FunctionalMetric(functions.metric_expr)},
    name="sup_valid",
)
validator_pde = {sup_validator_pde.name: sup_validator_pde}

3.5 Hyperparameter Setting

Next, we need to specify the number of training epochs. Here we use 100 training epochs based on experimental experience.

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# training settings
TRAIN:
  epochs: 100

3.6 Optimizer Construction

The training process will call the optimizer to update model parameters. Here, the Adam optimizer is selected and learning_rate is set to 1e-3.

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# initialize solver
optimizer = ppsci.optimizer.Adam(cfg.TRAIN.learning_rate)(model)

3.7 Model Training and Evaluation

After completing the above settings, you only need to pass the instantiated objects to ppsci.solver.Solver in order.

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solver = ppsci.solver.Solver(
    model,
    constraint_pde,
    optimizer=optimizer,
    validator=validator_pde,
    cfg=cfg,
)

Finally, start training and evaluation:

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# train model
solver.train()
# evaluate after finished training
solver.eval()

4. Complete Code

phylstm2.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.

"""
Reference: https://github.com/zhry10/PhyLSTM.git
"""

from os import path as osp

import functions
import hydra
import numpy as np
import scipy.io
from omegaconf import DictConfig

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, "train.log"), "info")

    mat = scipy.io.loadmat(cfg.DATA_FILE_PATH)
    ag_data = mat["input_tf"]  # ag, ad, av
    u_data = mat["target_X_tf"]
    ut_data = mat["target_Xd_tf"]
    utt_data = mat["target_Xdd_tf"]
    ag_data = ag_data.reshape([ag_data.shape[0], ag_data.shape[1], 1])
    u_data = u_data.reshape([u_data.shape[0], u_data.shape[1], 1])
    ut_data = ut_data.reshape([ut_data.shape[0], ut_data.shape[1], 1])
    utt_data = utt_data.reshape([utt_data.shape[0], utt_data.shape[1], 1])

    t = mat["time"]
    dt = t[0, 1] - t[0, 0]

    ag_all = ag_data
    u_all = u_data
    u_t_all = ut_data
    u_tt_all = utt_data

    # finite difference
    N = u_data.shape[1]
    phi1 = np.concatenate(
        [
            np.array([-3 / 2, 2, -1 / 2]),
            np.zeros([N - 3]),
        ]
    )
    temp1 = np.concatenate([-1 / 2 * np.identity(N - 2), np.zeros([N - 2, 2])], axis=1)
    temp2 = np.concatenate([np.zeros([N - 2, 2]), 1 / 2 * np.identity(N - 2)], axis=1)
    phi2 = temp1 + temp2
    phi3 = np.concatenate(
        [
            np.zeros([N - 3]),
            np.array([1 / 2, -2, 3 / 2]),
        ]
    )
    phi_t0 = (
        1
        / dt
        * np.concatenate(
            [
                np.reshape(phi1, [1, phi1.shape[0]]),
                phi2,
                np.reshape(phi3, [1, phi3.shape[0]]),
            ],
            axis=0,
        )
    )
    phi_t0 = np.reshape(phi_t0, [1, N, N])

    ag_star = ag_all[0:10]
    eta_star = u_all[0:10]
    eta_t_star = u_t_all[0:10]
    eta_tt_star = u_tt_all[0:10]
    ag_c_star = ag_all[0:50]
    lift_star = -ag_c_star

    eta = eta_star
    ag = ag_star
    lift = lift_star
    eta_t = eta_t_star
    eta_tt = eta_tt_star
    ag_c = ag_c_star
    g = -eta_tt - ag
    phi_t = np.repeat(phi_t0, ag_c_star.shape[0], axis=0)

    model = ppsci.arch.DeepPhyLSTM(
        cfg.MODEL.input_size,
        eta.shape[2],
        cfg.MODEL.hidden_size,
        cfg.MODEL.model_type,
    )
    model.register_input_transform(functions.transform_in)
    model.register_output_transform(functions.transform_out)

    dataset_obj = functions.Dataset(eta, eta_t, g, ag, ag_c, lift, phi_t)

    (
        input_dict_train,
        label_dict_train,
        input_dict_val,
        label_dict_val,
    ) = dataset_obj.get(cfg.TRAIN.epochs)

    sup_constraint_pde = ppsci.constraint.SupervisedConstraint(
        {
            "dataset": {
                "name": "NamedArrayDataset",
                "input": input_dict_train,
                "label": label_dict_train,
            },
            "sampler": {
                "name": "BatchSampler",
                "drop_last": True,
                "shuffle": True,
            },
            "batch_size": 1,
            "num_workers": 0,
        },
        ppsci.loss.FunctionalLoss(functions.train_loss_func2),
        {
            "eta_pred": lambda out: out["eta_pred"],
            "eta_dot_pred": lambda out: out["eta_dot_pred"],
            "g_pred": lambda out: out["g_pred"],
            "eta_t_pred_c": lambda out: out["eta_t_pred_c"],
            "eta_dot_pred_c": lambda out: out["eta_dot_pred_c"],
            "lift_pred_c": lambda out: out["lift_pred_c"],
        },
        name="sup_train",
    )
    constraint_pde = {sup_constraint_pde.name: sup_constraint_pde}

    sup_validator_pde = ppsci.validate.SupervisedValidator(
        {
            "dataset": {
                "name": "NamedArrayDataset",
                "input": input_dict_val,
                "label": label_dict_val,
            },
            "batch_size": 1,
            "num_workers": 0,
        },
        ppsci.loss.FunctionalLoss(functions.train_loss_func2),
        {
            "eta_pred": lambda out: out["eta_pred"],
            "eta_dot_pred": lambda out: out["eta_dot_pred"],
            "g_pred": lambda out: out["g_pred"],
            "eta_t_pred_c": lambda out: out["eta_t_pred_c"],
            "eta_dot_pred_c": lambda out: out["eta_dot_pred_c"],
            "lift_pred_c": lambda out: out["lift_pred_c"],
        },
        metric={"metric": ppsci.metric.FunctionalMetric(functions.metric_expr)},
        name="sup_valid",
    )
    validator_pde = {sup_validator_pde.name: sup_validator_pde}

    # initialize solver
    optimizer = ppsci.optimizer.Adam(cfg.TRAIN.learning_rate)(model)
    solver = ppsci.solver.Solver(
        model,
        constraint_pde,
        optimizer=optimizer,
        validator=validator_pde,
        cfg=cfg,
    )

    # train model
    solver.train()
    # evaluate after finished training
    solver.eval()


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, "eval.log"), "info")

    mat = scipy.io.loadmat(cfg.DATA_FILE_PATH)
    ag_data = mat["input_tf"]  # ag, ad, av
    u_data = mat["target_X_tf"]
    ut_data = mat["target_Xd_tf"]
    utt_data = mat["target_Xdd_tf"]
    ag_data = ag_data.reshape([ag_data.shape[0], ag_data.shape[1], 1])
    u_data = u_data.reshape([u_data.shape[0], u_data.shape[1], 1])
    ut_data = ut_data.reshape([ut_data.shape[0], ut_data.shape[1], 1])
    utt_data = utt_data.reshape([utt_data.shape[0], utt_data.shape[1], 1])

    t = mat["time"]
    dt = t[0, 1] - t[0, 0]

    ag_all = ag_data
    u_all = u_data
    u_t_all = ut_data
    u_tt_all = utt_data

    # finite difference
    N = u_data.shape[1]
    phi1 = np.concatenate(
        [
            np.array([-3 / 2, 2, -1 / 2]),
            np.zeros([N - 3]),
        ]
    )
    temp1 = np.concatenate([-1 / 2 * np.identity(N - 2), np.zeros([N - 2, 2])], axis=1)
    temp2 = np.concatenate([np.zeros([N - 2, 2]), 1 / 2 * np.identity(N - 2)], axis=1)
    phi2 = temp1 + temp2
    phi3 = np.concatenate(
        [
            np.zeros([N - 3]),
            np.array([1 / 2, -2, 3 / 2]),
        ]
    )
    phi_t0 = (
        1
        / dt
        * np.concatenate(
            [
                np.reshape(phi1, [1, phi1.shape[0]]),
                phi2,
                np.reshape(phi3, [1, phi3.shape[0]]),
            ],
            axis=0,
        )
    )
    phi_t0 = np.reshape(phi_t0, [1, N, N])

    ag_star = ag_all[0:10]
    eta_star = u_all[0:10]
    eta_t_star = u_t_all[0:10]
    eta_tt_star = u_tt_all[0:10]
    ag_c_star = ag_all[0:50]
    lift_star = -ag_c_star

    eta = eta_star
    ag = ag_star
    lift = lift_star
    eta_t = eta_t_star
    eta_tt = eta_tt_star
    ag_c = ag_c_star
    g = -eta_tt - ag
    phi_t = np.repeat(phi_t0, ag_c_star.shape[0], axis=0)

    model = ppsci.arch.DeepPhyLSTM(
        cfg.MODEL.input_size,
        eta.shape[2],
        cfg.MODEL.hidden_size,
        cfg.MODEL.model_type,
    )
    model.register_input_transform(functions.transform_in)
    model.register_output_transform(functions.transform_out)

    dataset_obj = functions.Dataset(eta, eta_t, g, ag, ag_c, lift, phi_t)

    (
        _,
        _,
        input_dict_val,
        label_dict_val,
    ) = dataset_obj.get(1)

    sup_validator_pde = ppsci.validate.SupervisedValidator(
        {
            "dataset": {
                "name": "NamedArrayDataset",
                "input": input_dict_val,
                "label": label_dict_val,
            },
            "batch_size": 1,
            "num_workers": 0,
        },
        ppsci.loss.FunctionalLoss(functions.train_loss_func2),
        {
            "eta_pred": lambda out: out["eta_pred"],
            "eta_dot_pred": lambda out: out["eta_dot_pred"],
            "g_pred": lambda out: out["g_pred"],
            "eta_t_pred_c": lambda out: out["eta_t_pred_c"],
            "eta_dot_pred_c": lambda out: out["eta_dot_pred_c"],
            "lift_pred_c": lambda out: out["lift_pred_c"],
        },
        metric={"metric": ppsci.metric.FunctionalMetric(functions.metric_expr)},
        name="sup_valid",
    )
    validator_pde = {sup_validator_pde.name: sup_validator_pde}

    # initialize solver
    solver = ppsci.solver.Solver(
        model,
        validator=validator_pde,
        cfg=cfg,
    )
    # evaluate
    solver.eval()


@hydra.main(version_base=None, config_path="./conf", config_name="phylstm2.yaml")
def main(cfg: DictConfig):
    if cfg.mode == "train":
        train(cfg)
    elif cfg.mode == "eval":
        evaluate(cfg)
    else:
        raise ValueError(f"cfg.mode should in ['train', 'eval'], but got '{cfg.mode}'")


if __name__ == "__main__":
    main()
phylstm3.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.

"""
Reference: https://github.com/zhry10/PhyLSTM.git
"""

from os import path as osp

import functions
import hydra
import numpy as np
import scipy.io
from omegaconf import DictConfig

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, "train.log"), "info")

    mat = scipy.io.loadmat(cfg.DATA_FILE_PATH)
    t = mat["time"]
    dt = 0.02
    n1 = int(dt / 0.005)
    t = t[::n1]

    ag_data = mat["input_tf"][:, ::n1]  # ag, ad, av
    u_data = mat["target_X_tf"][:, ::n1]
    ut_data = mat["target_Xd_tf"][:, ::n1]
    utt_data = mat["target_Xdd_tf"][:, ::n1]
    ag_data = ag_data.reshape([ag_data.shape[0], ag_data.shape[1], 1])
    u_data = u_data.reshape([u_data.shape[0], u_data.shape[1], 1])
    ut_data = ut_data.reshape([ut_data.shape[0], ut_data.shape[1], 1])
    utt_data = utt_data.reshape([utt_data.shape[0], utt_data.shape[1], 1])

    ag_pred = mat["input_pred_tf"][:, ::n1]  # ag, ad, av
    u_pred = mat["target_pred_X_tf"][:, ::n1]
    ut_pred = mat["target_pred_Xd_tf"][:, ::n1]
    utt_pred = mat["target_pred_Xdd_tf"][:, ::n1]
    ag_pred = ag_pred.reshape([ag_pred.shape[0], ag_pred.shape[1], 1])
    u_pred = u_pred.reshape([u_pred.shape[0], u_pred.shape[1], 1])
    ut_pred = ut_pred.reshape([ut_pred.shape[0], ut_pred.shape[1], 1])
    utt_pred = utt_pred.reshape([utt_pred.shape[0], utt_pred.shape[1], 1])

    N = u_data.shape[1]
    phi1 = np.concatenate(
        [
            np.array([-3 / 2, 2, -1 / 2]),
            np.zeros([N - 3]),
        ]
    )
    temp1 = np.concatenate([-1 / 2 * np.identity(N - 2), np.zeros([N - 2, 2])], axis=1)
    temp2 = np.concatenate([np.zeros([N - 2, 2]), 1 / 2 * np.identity(N - 2)], axis=1)
    phi2 = temp1 + temp2
    phi3 = np.concatenate(
        [
            np.zeros([N - 3]),
            np.array([1 / 2, -2, 3 / 2]),
        ]
    )
    phi_t0 = (
        1
        / dt
        * np.concatenate(
            [
                np.reshape(phi1, [1, phi1.shape[0]]),
                phi2,
                np.reshape(phi3, [1, phi3.shape[0]]),
            ],
            axis=0,
        )
    )
    phi_t0 = np.reshape(phi_t0, [1, N, N])

    ag_star = ag_data
    eta_star = u_data
    eta_t_star = ut_data
    eta_tt_star = utt_data
    ag_c_star = np.concatenate([ag_data, ag_pred[0:53]])
    lift_star = -ag_c_star

    eta = eta_star
    ag = ag_star
    lift = lift_star
    eta_t = eta_t_star
    eta_tt = eta_tt_star
    g = -eta_tt - ag
    ag_c = ag_c_star

    phi_t = np.repeat(phi_t0, ag_c_star.shape[0], axis=0)

    model = ppsci.arch.DeepPhyLSTM(
        cfg.MODEL.input_size,
        eta.shape[2],
        cfg.MODEL.hidden_size,
        cfg.MODEL.model_type,
    )
    model.register_input_transform(functions.transform_in)
    model.register_output_transform(functions.transform_out)

    dataset_obj = functions.Dataset(eta, eta_t, g, ag, ag_c, lift, phi_t)
    (
        input_dict_train,
        label_dict_train,
        input_dict_val,
        label_dict_val,
    ) = dataset_obj.get(cfg.TRAIN.epochs)

    sup_constraint_pde = ppsci.constraint.SupervisedConstraint(
        {
            "dataset": {
                "name": "NamedArrayDataset",
                "input": input_dict_train,
                "label": label_dict_train,
            },
            "sampler": {
                "name": "BatchSampler",
                "drop_last": False,
                "shuffle": False,
            },
            "batch_size": 1,
            "num_workers": 0,
        },
        ppsci.loss.FunctionalLoss(functions.train_loss_func3),
        {
            "eta_pred": lambda out: out["eta_pred"],
            "eta_dot_pred": lambda out: out["eta_dot_pred"],
            "g_pred": lambda out: out["g_pred"],
            "eta_t_pred_c": lambda out: out["eta_t_pred_c"],
            "eta_dot_pred_c": lambda out: out["eta_dot_pred_c"],
            "lift_pred_c": lambda out: out["lift_pred_c"],
            "g_t_pred_c": lambda out: out["g_t_pred_c"],
            "g_dot_pred_c": lambda out: out["g_dot_pred_c"],
        },
        name="sup_train",
    )
    constraint_pde = {sup_constraint_pde.name: sup_constraint_pde}

    sup_validator_pde = ppsci.validate.SupervisedValidator(
        {
            "dataset": {
                "name": "NamedArrayDataset",
                "input": input_dict_val,
                "label": label_dict_val,
            },
            "sampler": {
                "name": "BatchSampler",
                "drop_last": False,
                "shuffle": False,
            },
            "batch_size": 1,
            "num_workers": 0,
        },
        ppsci.loss.FunctionalLoss(functions.train_loss_func3),
        {
            "eta_pred": lambda out: out["eta_pred"],
            "eta_dot_pred": lambda out: out["eta_dot_pred"],
            "g_pred": lambda out: out["g_pred"],
            "eta_t_pred_c": lambda out: out["eta_t_pred_c"],
            "eta_dot_pred_c": lambda out: out["eta_dot_pred_c"],
            "lift_pred_c": lambda out: out["lift_pred_c"],
            "g_t_pred_c": lambda out: out["g_t_pred_c"],
            "g_dot_pred_c": lambda out: out["g_dot_pred_c"],
        },
        metric={"metric": ppsci.metric.FunctionalMetric(functions.metric_expr)},
        name="sup_valid",
    )
    validator_pde = {sup_validator_pde.name: sup_validator_pde}

    # initialize solver
    optimizer = ppsci.optimizer.Adam(cfg.TRAIN.learning_rate)(model)
    solver = ppsci.solver.Solver(
        model,
        constraint_pde,
        cfg.output_dir,
        optimizer,
        None,
        cfg.TRAIN.epochs,
        cfg.TRAIN.iters_per_epoch,
        save_freq=cfg.TRAIN.save_freq,
        log_freq=cfg.log_freq,
        seed=cfg.seed,
        validator=validator_pde,
        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()


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, "train.log"), "info")

    mat = scipy.io.loadmat(cfg.DATA_FILE_PATH)
    t = mat["time"]
    dt = 0.02
    n1 = int(dt / 0.005)
    t = t[::n1]

    ag_data = mat["input_tf"][:, ::n1]  # ag, ad, av
    u_data = mat["target_X_tf"][:, ::n1]
    ut_data = mat["target_Xd_tf"][:, ::n1]
    utt_data = mat["target_Xdd_tf"][:, ::n1]
    ag_data = ag_data.reshape([ag_data.shape[0], ag_data.shape[1], 1])
    u_data = u_data.reshape([u_data.shape[0], u_data.shape[1], 1])
    ut_data = ut_data.reshape([ut_data.shape[0], ut_data.shape[1], 1])
    utt_data = utt_data.reshape([utt_data.shape[0], utt_data.shape[1], 1])

    ag_pred = mat["input_pred_tf"][:, ::n1]  # ag, ad, av
    u_pred = mat["target_pred_X_tf"][:, ::n1]
    ut_pred = mat["target_pred_Xd_tf"][:, ::n1]
    utt_pred = mat["target_pred_Xdd_tf"][:, ::n1]
    ag_pred = ag_pred.reshape([ag_pred.shape[0], ag_pred.shape[1], 1])
    u_pred = u_pred.reshape([u_pred.shape[0], u_pred.shape[1], 1])
    ut_pred = ut_pred.reshape([ut_pred.shape[0], ut_pred.shape[1], 1])
    utt_pred = utt_pred.reshape([utt_pred.shape[0], utt_pred.shape[1], 1])

    N = u_data.shape[1]
    phi1 = np.concatenate(
        [
            np.array([-3 / 2, 2, -1 / 2]),
            np.zeros([N - 3]),
        ]
    )
    temp1 = np.concatenate([-1 / 2 * np.identity(N - 2), np.zeros([N - 2, 2])], axis=1)
    temp2 = np.concatenate([np.zeros([N - 2, 2]), 1 / 2 * np.identity(N - 2)], axis=1)
    phi2 = temp1 + temp2
    phi3 = np.concatenate(
        [
            np.zeros([N - 3]),
            np.array([1 / 2, -2, 3 / 2]),
        ]
    )
    phi_t0 = (
        1
        / dt
        * np.concatenate(
            [
                np.reshape(phi1, [1, phi1.shape[0]]),
                phi2,
                np.reshape(phi3, [1, phi3.shape[0]]),
            ],
            axis=0,
        )
    )
    phi_t0 = np.reshape(phi_t0, [1, N, N])

    ag_star = ag_data
    eta_star = u_data
    eta_t_star = ut_data
    eta_tt_star = utt_data
    ag_c_star = np.concatenate([ag_data, ag_pred[0:53]])
    lift_star = -ag_c_star

    eta = eta_star
    ag = ag_star
    lift = lift_star
    eta_t = eta_t_star
    eta_tt = eta_tt_star
    g = -eta_tt - ag
    ag_c = ag_c_star

    phi_t = np.repeat(phi_t0, ag_c_star.shape[0], axis=0)

    model = ppsci.arch.DeepPhyLSTM(
        cfg.MODEL.input_size,
        eta.shape[2],
        cfg.MODEL.hidden_size,
        cfg.MODEL.model_type,
    )
    model.register_input_transform(functions.transform_in)
    model.register_output_transform(functions.transform_out)

    dataset_obj = functions.Dataset(eta, eta_t, g, ag, ag_c, lift, phi_t)
    (
        _,
        _,
        input_dict_val,
        label_dict_val,
    ) = dataset_obj.get(1)

    sup_validator_pde = ppsci.validate.SupervisedValidator(
        {
            "dataset": {
                "name": "NamedArrayDataset",
                "input": input_dict_val,
                "label": label_dict_val,
            },
            "sampler": {
                "name": "BatchSampler",
                "drop_last": False,
                "shuffle": False,
            },
            "batch_size": 1,
            "num_workers": 0,
        },
        ppsci.loss.FunctionalLoss(functions.train_loss_func3),
        {
            "eta_pred": lambda out: out["eta_pred"],
            "eta_dot_pred": lambda out: out["eta_dot_pred"],
            "g_pred": lambda out: out["g_pred"],
            "eta_t_pred_c": lambda out: out["eta_t_pred_c"],
            "eta_dot_pred_c": lambda out: out["eta_dot_pred_c"],
            "lift_pred_c": lambda out: out["lift_pred_c"],
            "g_t_pred_c": lambda out: out["g_t_pred_c"],
            "g_dot_pred_c": lambda out: out["g_dot_pred_c"],
        },
        metric={"metric": ppsci.metric.FunctionalMetric(functions.metric_expr)},
        name="sup_valid",
    )
    validator_pde = {sup_validator_pde.name: sup_validator_pde}

    # initialize solver
    solver = ppsci.solver.Solver(
        model,
        output_dir=cfg.output_dir,
        seed=cfg.seed,
        validator=validator_pde,
        pretrained_model_path=cfg.EVAL.pretrained_model_path,
        eval_with_no_grad=cfg.EVAL.eval_with_no_grad,
    )

    # evaluate
    solver.eval()


@hydra.main(version_base=None, config_path="./conf", config_name="phylstm3.yaml")
def main(cfg: DictConfig):
    if cfg.mode == "train":
        train(cfg)
    elif cfg.mode == "eval":
        evaluate(cfg)
    else:
        raise ValueError(f"cfg.mode should in ['train', 'eval'], but got '{cfg.mode}'")


if __name__ == "__main__":
    main()

5. Result Display

The PhyLSTM2 case was experimented with the parameter configuration of epoch=100 and learning_rate=1e-3, and the result returned Loss was 0.00799.

The PhyLSTM3 case was experimented with the parameter configuration of epoch=200 and learning_rate=1e-3, and the result returned Loss was 0.03098.

6. References