4.12. Fit energy Hessian 
#
Note
Supported backends: PyTorch
To train a model that takes Hessian matrices, i.e., the second order derivatives of energies w.r.t coordinates as input, you only need to prepare full Hessian matrices and modify the loss section to define the Hessian-specific settings, keeping other sections the same as the normal energy model’s input script.
4.12.1. Energy Hessian Loss#
If you want to train with Hessians, you are expected to add the start and limit prefactors of Hessians, i.e., start_pref_h and limit_pref_h to the loss section in the input.json:
"loss": {
"type": "ener",
"start_pref_e": 0.02,
"limit_pref_e": 1,
"start_pref_f": 1000,
"limit_pref_f": 1,
"start_pref_v": 0,
"limit_pref_v": 0,
"start_pref_h": 10,
"limit_pref_h": 1
},
The options start_pref_e, limit_pref_e, start_pref_f, limit_pref_f, start_pref_v and limit_pref_v determine the start and limit prefactors of energy, force, and virial, respectively. The calculation and definition of Hessian loss are the same as for the other terms.
If one does not want to train with virial, then he/she may set the virial prefactors start_pref_v and limit_pref_v to 0.
4.12.2. Hessian Format in PyTorch#
In the PyTorch backend, Hessian matrices are listed in hessian.npy files, and the data format may contain the following files:
type.raw
set.*/box.npy
set.*/coord.npy
set.*/energy.npy
set.*/force.npy
set.*/hessian.npy
This system contains Nframes frames with the same atom number Natoms, the total number of elements contained in all frames is Ntypes. Most files are the same as those in standard formats, here we only list the distinct ones:
ID | Property | Raw file | Unit | Shape | Description |
|---|---|---|---|---|---|
hessian | Hessian matrices | hessian.npy | eV/Å^2 | Nframes * (Natoms * 3 * Natoms * 3) | Second-order derivatives of energies w.r.t coordinates. |
Note that the hessian.npy should contain the full Hessian matrices with shape of (3Natoms * 3Natoms) for each frame, rather than the upper or lower triangular matrices with shape of (3Natoms * (3Natoms + 1) / 2) for each frame.
4.12.3. Train the Model#
There are two approaches to training a Hessian model. The first method involves training the model from scratch using the same command as in the ener mode within the PyTorch backend:
dp --pt train input.json
The second approach is to train a Hessian model from a pretrained energy model, following the same command as the finetune strategy within the PyTorch backend:
dp --pt train input.json --finetune pretrained_energy.pt
The detailed loss can be found in lcurve.out:
# step rmse_val rmse_trn rmse_e_val rmse_e_trn rmse_f_val rmse_f_trn rmse_h_val rmse_h_trn lr
0 1.05e+02 2.28e+01 2.11e-01 1.59e+00 3.25e+00 3.37e-01 6.00e+00 6.37e+00 1.0e-03
200 1.86e+01 3.23e+01 9.24e-03 1.54e-01 2.51e-01 4.70e-01 5.31e+00 9.05e+00 1.0e-03
400 2.69e+01 2.98e+01 1.03e-01 1.07e-01 5.67e-01 4.17e-01 6.35e+00 8.47e+00 1.0e-03
600 2.00e+01 1.90e+01 7.23e-02 6.90e-03 3.35e-01 2.58e-01 5.37e+00 5.41e+00 1.0e-03
800 1.68e+01 1.48e+01 4.06e-02 2.27e-01 2.35e-01 1.98e-01 4.76e+00 4.24e+00 1.0e-03
1000 1.70e+01 1.81e+01 3.90e-01 1.66e-01 2.02e-01 1.99e-01 4.98e+00 5.37e+00 1.0e-03
4.12.4. Test the Model#
Warning
A model trained with Hessian cannot be frozen. If freezing is enforced, the model will be treated as a standard energy model, and the frozen one will no longer be able to output Hessian predictions.
If one do freeze and test a Hessian model using the commands:
dp --pt freeze -o frozen_model.pth
dp --pt test -m frozen_model.pth -s test_system -d ${output_prefix} -a -n 1
If dp --pt test -d ${output_prefix} -a is specified, the output files will be the same as those in the ener mode, i.e.,
${output_prefix}.e.out ${output_prefix}.e_peratom.out ${output_prefix}.f.out
${output_prefix}.v.out ${output_prefix}.v_peratom.out
If one intends to use the trained model for Hessian predictions, then he/she is supposed to test the model directly without performing a freezing operation:
dp --pt test -m model.pt -s test_system -d ${output_prefix} -a -n 1
If dp --pt test -d ${output_prefix} -a is specified, the predicted Hessian for each frame are output in an additional file in the working directory:
${output_prefix}.h.out
For *.h.out.*, it contains matrix with shape of (2, n_hess):
# frame - 0: data_h pred_h (3Na*3Na matrix in row-major order)
5.897392891323943331e+01 2.909700516268236825e+01
-7.682282297964052376e+00 2.535680817045881774e+00
-1.266442953072092514e+01 -2.127310638041492652e+01
5.442541716174009031e-02 7.202825779190234756e-02
5.198263170894957939e-05 -8.110080221576332349e-02
7.443552765043950914e-02 -2.248597801730128215e-02
1.029910175689553675e+00 1.938646932394622047e-03
1.213862217511276764e+00 5.344132558814301825e-02
-1.221943904909605250e+00 1.602557574981743893e-01
The full Hessian matrices are stored in a flattened form in the row-major order. Here, n_hess is the total number of Hessian matrix elements across all frames, calculated as:
where \(N_{\text{atom}, i}\) represents the number of atoms in the \(i^{\text{th}}\) frame.