# 3.10. Fit tensor like Dipole and Polarizability

Unlike energy, which is a scalar, one may want to fit some high dimensional physical quantity, like dipole (vector) and polarizability (matrix, shorted as polar). Deep Potential has provided different APIs to do this. In this example, we will show you how to train a model to fit a water system. A complete training input script of the examples can be found in

$deepmd_source_dir/examples/water_tensor/dipole/dipole_input.json$deepmd_source_dir/examples/water_tensor/polar/polar_input.json


The training and validation data are also provided our examples. But note that the data provided along with the examples are of limited amount, and should not be used to train a production model.

Similar to the input.json used in ener mode, training JSON is also divided into model, learning_rate, loss and training. Most keywords remain the same as ener mode, and their meaning can be found here. To fit a tensor, one needs to modify model/fitting_net and loss.

## 3.10.1. Theory

To represent the first-order tensorial properties (i.e. vector properties), we let the fitting network, denoted by $$\mathcal F_{1}$$, output an $$M$$-dimensional vector; then we have the representation,

$(T_i^{(1)})_\alpha = \frac{1}{N_c} \sum_{j=1}^{N_c}\sum_{m=1}^M (\mathcal G^i)_{jm} (\mathcal R^i)_{j,\alpha+1} (\mathcal F_{1}(\mathcal D^i))_m, \ \alpha=1,2,3.$

We let the fitting network $$\mathcal F_{2}$$ output an $$M$$-dimensional vector, and the second-order tensorial properties (matrix properties) are formulated as

$(T_i^{(2)})_{\alpha\beta} = \frac{1}{N_c^2} \sum_{j=1}^{N_c}\sum_{k=1}^{N_c}\sum_{m=1}^M (\mathcal G^i)_{jm} (\mathcal R^i)_{j,\alpha+1} (\mathcal R^i)_{k,\beta+1} (\mathcal G^i)_{km} (\mathcal F_{2}(\mathcal D^i))_m, \ \alpha,\beta=1,2,3,$

where $$\mathcal{G}^i$$ and $$\mathcal{R}^i$$ can be found in se_e2_a. Thus, the tensor fitting network requires the descriptor to have the same or similar form as the DeepPot-SE descriptor. $$\mathcal{F}_1$$ and $$\mathcal F_2$$ are the neural network functions. The total tensor $$\boldsymbol{T}$$ (total dipole $$\boldsymbol{T}^{(1)}$$ or total polarizability $$\boldsymbol{T}^{(2)}$$) is the sum of the atomic tensor:

$\boldsymbol{T} = \sum_i \boldsymbol{T}_i.$

The tensorial models can be used to calculate IR spectrum and Raman spectrum.1

## 3.10.2. The fitting Network

The fitting_net section tells DP which fitting net to use.

The JSON of dipole type should be provided like

	"fitting_net" : {
"type": "dipole",
"sel_type": [0],
"neuron": [100,100,100],
"resnet_dt": true,
"seed": 1,
},


The JSON of polar type should be provided like

	"fitting_net" : {
"type": "polar",
"sel_type": [0],
"neuron": [100,100,100],
"resnet_dt": true,
"seed": 1,
},

• type specifies which type of fitting net should be used. It should be either dipole or polar. Note that global_polar mode in version 1.x is already deprecated and is merged into polar. To specify whether a system is global or atomic, please see here.

• sel_type is a list specifying which type of atoms have the quantity you want to fit. For example, in the water system, sel_type is [0] since 0 represents atom O. If left unset, all types of atoms will be fitted.

• The rest arguments have the same meaning as they do in ener mode.

## 3.10.3. Loss

DP supports a combinational training of the global system (only a global tensor label, i.e. dipole or polar, is provided in a frame) and atomic system (labels for each atom included in sel_type are provided). In a global system, each frame has just one tensor label. For example, when fitting polar, each frame will just provide a 1 x 9 vector which gives the elements of the polarizability tensor of that frame in order XX, XY, XZ, YX, YY, YZ, XZ, ZY, ZZ. By contrast, in an atomic system, each atom in sel_type has a tensor label. For example, when fitting a dipole, each frame will provide a #sel_atom x 3 matrices, where #sel_atom is the number of atoms whose type are in sel_type.

The loss section tells DP the weight of these two kinds of loss, i.e.

loss = pref * global_loss + pref_atomic * atomic_loss


The loss section should be provided like

	"loss" : {
"type":		"tensor",
"pref":		1.0,
"pref_atomic":	1.0
},

• type should be written as tensor as a distinction from ener mode.

• pref and pref_atomic respectively specify the weight of global loss and atomic loss. It can not be left unset. If set to 0, the corresponding label will NOT be included in the training process.

## 3.10.4. Training Data Preparation

In tensor mode, the identification of the label’s type (global or atomic) is derived from the file name. The global label should be named dipole.npy/raw or polarizability.npy/raw, while the atomic label should be named atomic_dipole.npy/raw or atomic_polarizability.npy/raw. If wrongly named, DP will report an error

ValueError: cannot reshape array of size xxx into shape (xx,xx). This error may occur when your label mismatch it's name, i.e. you might store global tensor in atomic_tensor.npy or atomic tensor in tensor.npy.


In this case, please check the file name of the label.

## 3.10.5. Train the Model

The training command is the same as ener mode, i.e.

dp train input.json


The detailed loss can be found in lcurve.out:

#  step    rmse_val   rmse_trn  rmse_lc_val rmse_lc_trn rmse_gl_val rmse_gl_trn  lr
0     8.34e+00   8.26e+00   8.34e+00   8.26e+00    0.00e+00    0.00e+00   1.0e-02
100     3.51e-02   8.55e-02   0.00e+00   8.55e-02    4.38e-03    0.00e+00   5.0e-03
200     4.77e-02   5.61e-02   0.00e+00   5.61e-02    5.96e-03    0.00e+00   2.5e-03
300     5.68e-02   1.47e-02   0.00e+00   0.00e+00    7.10e-03    1.84e-03   1.3e-03
400     3.73e-02   3.48e-02   1.99e-02   0.00e+00    2.18e-03    4.35e-03   6.3e-04
500     2.77e-02   5.82e-02   1.08e-02   5.82e-02    2.11e-03    0.00e+00   3.2e-04
600     2.81e-02   5.43e-02   2.01e-02   0.00e+00    1.01e-03    6.79e-03   1.6e-04
700     2.97e-02   3.28e-02   2.03e-02   0.00e+00    1.17e-03    4.10e-03   7.9e-05
800     2.25e-02   6.19e-02   9.05e-03   0.00e+00    1.68e-03    7.74e-03   4.0e-05
900     3.18e-02   5.54e-02   9.93e-03   5.54e-02    2.74e-03    0.00e+00   2.0e-05
1000     2.63e-02   5.02e-02   1.02e-02   5.02e-02    2.01e-03    0.00e+00   1.0e-05
1100     3.27e-02   5.89e-02   2.13e-02   5.89e-02    1.43e-03    0.00e+00   5.0e-06
1200     2.85e-02   2.42e-02   2.85e-02   0.00e+00    0.00e+00    3.02e-03   2.5e-06
1300     3.47e-02   5.71e-02   1.07e-02   5.71e-02    3.00e-03    0.00e+00   1.3e-06
1400     3.13e-02   5.76e-02   3.13e-02   5.76e-02    0.00e+00    0.00e+00   6.3e-07
1500     3.34e-02   1.11e-02   2.09e-02   0.00e+00    1.57e-03    1.39e-03   3.2e-07
1600     3.11e-02   5.64e-02   3.11e-02   5.64e-02    0.00e+00    0.00e+00   1.6e-07
1700     2.97e-02   5.05e-02   2.97e-02   5.05e-02    0.00e+00    0.00e+00   7.9e-08
1800     2.64e-02   7.70e-02   1.09e-02   0.00e+00    1.94e-03    9.62e-03   4.0e-08
1900     3.28e-02   2.56e-02   3.28e-02   0.00e+00    0.00e+00    3.20e-03   2.0e-08
2000     2.59e-02   5.71e-02   1.03e-02   5.71e-02    1.94e-03    0.00e+00   1.0e-08


One may notice that in each step, some of the local loss and global loss will be 0.0. This is because our training data and validation data consist of the global system and atomic system, i.e.

	--training_data
>atomic_system
>global_system
--validation_data
>atomic_system
>global_system


During training, at each step when the lcurve.out is printed, the system used for evaluating the training (validation) error may be either with only global or only atomic labels, thus the corresponding atomic or global errors are missing and are printed as zeros.

1

This section is built upon Jinzhe Zeng, Duo Zhang, Denghui Lu, Pinghui Mo, Zeyu Li, Yixiao Chen, Marián Rynik, Li’ang Huang, Ziyao Li, Shaochen Shi, Yingze Wang, Haotian Ye, Ping Tuo, Jiabin Yang, Ye Ding, Yifan Li, Davide Tisi, Qiyu Zeng, Han Bao, Yu Xia, Jiameng Huang, Koki Muraoka, Yibo Wang, Junhan Chang, Fengbo Yuan, Sigbjørn Løland Bore, Chun Cai, Yinnian Lin, Bo Wang, Jiayan Xu, Jia-Xin Zhu, Chenxing Luo, Yuzhi Zhang, Rhys E. A. Goodall, Wenshuo Liang, Anurag Kumar Singh, Sikai Yao, Jingchao Zhang, Renata Wentzcovitch, Jiequn Han, Jie Liu, Weile Jia, Darrin M. York, Weinan E, Roberto Car, Linfeng Zhang, Han Wang, J. Chem. Phys. 159, 054801 (2023) licensed under a Creative Commons Attribution (CC BY) license.