Moment Tensor Potential (MTP) Training for Crystal and Amorphous Structures

Version: W-2024.09-SP1

Downloads

c-am-TiSi-MTP-training.hdf5
MTP_TiSi_basis400_final.mtp
c-TiSi-DFT-EOS.hdf5
c-TiSi-MTP-EOS.hdf5
am-TiSi-MTP-validation.hdf5
c-am-TiSi-MTP-training-validation-results.zip

In this introductory tutorial, you will learn how to use QuantumATK to

• Set-up a workflow for training a machine-learned force field (ML FF), MTP, for crystal and amorphous TiSi and TiSi₂ structures, which can be used for molecular dynamics (MD), geometry optimization, and other force-field-based simulations.

• Analyze the MTP training results.

• Validate the quality of the trained MTP.

Unlike many empirical potentials with physical functional forms, ML FFs such as MTPs are not highly transferable and must be trained with the target application in mind. For example, an MTP trained for bulk structures will not describe accurately free surfaces or nanoparticles of that material without the addition of such structures to the training set. However, the outlined training protocol can be easily adapted or extended to train an accurate MTP for different geometries, stoichiometries, element composition and other materials by explicitly including relevant training data for such structures.

Prerequisites

• Tutorial Geometry Optimization

• Tutorial Molecular Dynamics Simulations for Generating Amorphous Structures

Overview of the MTP Training Workflow

ML FFs are trained to predict ab initio, i.e., DFT, potential energy surfaces (PES) and vastly improve the accuracy of classical dynamics simulations while keeping the computational cost at the level of empirical force fields. To train a ML FF, one first needs to calculate training data (energy, forces, stress) for generated training configurations (typically, of relatively small sizes) with the reference DFT calculator and then fit an MTP to that DFT training data. Simulations with trained ML FFs can reach the high accuracy, long simulation times and large system sizes that capture the complexity of realistic systems, e.g., amorphous and alloy materials, and processes, e.g., diffusion, crystallization or deposition.

QuantumATK implemented Moment Tensor Potentials (MTPs) [1] as they provide high performance for a given accuracy when compared to other ML FFs models in the literature [2]. MTP advantages include:

• Very fast and supports large scale simulations.​

• Allows training to multi-element-materials.​

• Systematically improvable.

Tip

1. Since the MTP is trained to match the DFT reference calculator results, the accuracy of the final MTP is tied to this choice. Thus, it is important to choose the reference calculator that gives a good description of the system and application you want to study.

2. Learn more about ML FFs and their benefits here: Glossary page on What are Machine-Learned Force Fields

MTP training for crystal and amorphous structures in QuantumATK is structured into three stages:

1. Crystal Training Data - generates crystal training configurations with random displacements and calculates training data (energy, forces, stress) with the reference calculator DFT. Crystal training data forms the bulk of the training set, giving a fundamental description of the material in question. It will also be used to fit an initial MTP in the 2^nd MTP training stage.

2. Amorphous Active Learning - fits an initial MTP to the crystal training data, which is then used to run iterative Active learning geometry optimization and MD simulations to identify relevant configurations which are not represented in the initial training set. This workflow actively adds extrapolating amorphous configurations, incl. DFT data, to the training data set. The combined amorphous + crystal training data is then used in the 3^rd stage to fit an ensemble of MTP models.

3. Final MTP Fitting - generates an ensemble of MTP models and selects the best fitted MTP that gives lowest and comparable training and testing errors with respect to the reference DFT method.

The chosen MTP can then be used in production simulations, such as MD, geometry optimization, and other force-field-based simulations.

1. Download the ready-to-use MTP training workflow c-am-TiSi-MTP-training.hdf5 and open it in the Workflow Builder.

2. The workflow consists of three aforementioned stages (each contained in a so-called Blocks of Blocks) and is applicable to any crystal and amorphous material:

1. Crystal Training Data

2. Amorphous Active Learning

3. Final MTP Fitting

3. The LCAOCalculator block placed above the three Blocks of Blocks is used for defining a reference DFT-LCAO calculator (using the PBE xc-functional) that is used to calculate training data (energy, forces and stress) throughout the MTP training simulations.

• Under the Main tab, the k-point density is increased from the default medium to high, i.e., from 4Å x 4Å x 4Å to 7Å x 7Å x 7Å, to reduce the impact of sampling discontinuities when slightly changing the cell size during random displacements of the structure.

• Under the Iteration Control tab, the SCF Tolerance is decreased from the default 1e-04 to 5e-05 to reduce the noise in energy, forces, stress, and the Max steps for SCF is increased from default 100 to 200 so that SCF also converges even if the configuration needs more steps due to the tighter tolerance.

• Under the Algorithm tab ‣ Parameters, the default Optimize diagonalization for memory is switched to Optimize diagonalization for speed.

  

We will now go through the three stages step-by-step, explain the settings, and provide tips and recommendations.

Crystal Training Data

We will begin with the 1^st stage of the MTP training workflow - Crystal Training Data. This includes generation of crystal training configurations, i.e., repeated and randomly displaced crystal structures of different phases of titanium silicide using the crystalTrainingRandomDisplacements protocol, and calculation of training data with DFT for these displaced configurations.

1. The Initial Crystal Structures ConfigurationTable(List) block contains three crystal structures of titanium silicide, with two different stoichiometries, which we will include in the MTP training: TiSi, TiSi₂-C54 and TiSi₂-C49. These structures will be used for generating randomly displaced structures - the training configurations.

   

Note

The required input for the Initial Crystal Structures depends on the application of the final trained MTP. If you intend to use the trained MTP for simulating other stoichiometries and crystal phases than TiSi, TiSi₂-C54 and TiSi₂-C49, then you should include structures with these “other” stoichiometries and crystal phases as well, to ensure accurate simulations for those materials.

Tip

1. Crystal structures can be imported from the NanoLab’s Internal Database in the Builder or from the Materials Project database in the Databases.

2. It is recommended to optimize the geometry of crystal structures with the reference (DFT) calculator before using them for the MTP training.

2. The CrystalTrainingRandomDisplacements block sets parameters for randomly displacing configurations around the equilibrium of the provided crystal structures of titanium silicide.

• System sizes: Define the target sizes for the supercells in which the atoms will be displaced. It is recommended to use at least 2 different sizes (in this case, 20 and 50), so that the MTP model can better learn the system-size dependence of the total energy.

• Target sample size: Set the minimum for how many displaced configurations should be generated. Note that depending on various input choices, the final number of training configurations can be significantly higher. Here, we choose 200 (default).

• Base / max atomic rattling amplitudes: Define by what magnitude the atoms are displaced. Should be chosen so most displaced configurations have max. atomic forces below 5 eV / Ang, and that the max. atomic forces peaks are between 20 – 40 eV / Ang. The default values (0.15 Å/ 0.4 Å) work well for most materials, but for pure metals larger values can be chosen (e.g. 0.25 Å / 0.55 Å).

• Include strain: Should be enabled (by default), unless not needed for a specific application.

  

3. The MomentTensorPotentialTraining block sets parameters for calculating DFT training data (energy, forces, stress) for the randomly displaced configurations.

• Only the parameters in purple boxes, i.e., related to the calculation of the DFT training data, are used. The remaining parameters are related to the MTP fitting, which we do not carry out in this stage.

• Number of processes per task: Specify how many MPI processes should be used for each DFT calculation (default - all available processes). Since the DFT calculations are mostly independent from each other, multiple DFT calculations can be done at the same time. For example, when submitting a calculation on 40 MPI processes, number of processes per task could be 8, thus resulting in 5 DFT calculations running at the same time. This can speed up the entire calculation, but will use more memory.

• LCAOCalculator and MomentTensorPotentialTraining blocks must be connected, so that the defined LCAOCalculator is used to calculate training data.

  

4. The TrainingSet block exports the generated training data with DFT as a TrainingSet object.

• Sample size: Keep this at All configurations (default) to include all configurations from the CrystalTrainingRandomDisplacements block in the dataset.

• Recalculate Training Data: Changed to False, because the DFT data has already been calculated. Only if the same configurations are used with a different reference calculator, should this parameter be set to True.

  

Note

In case of training an MTP model for the use of crystal applications only, one could take the TrainingSet generated above and directly move to the 3^rd stage - Final MTP Fitting. The resulting MTP would be able to describe the trained crystal phases up to medium temperatures (i.e., well below the melting point where the crystal is still stable). Crystal properties, such as lattice constants, elastic moduli, or phonons, are typically described very well using this training data.

Amorphous Active Learning

In the 1^st stage of the MTP training we generated crystal training data for the initial MTP fitting. In the 2^nd stage, Amorphous Active Learning, we will include training data for amorphous configurations. Here, it is not so straightforward to identify all relevant amorphous configurations beforehand, thus we will employ the ActiveLearningSimulation approach for on-the-fly improvement of training dataset by actively adding extrapolating amorphous training configurations from dynamic simulations, such as geometry optimization and melt-quench MD.

Note

1. Active learning is a strategy for iteratively finding and selecting new atomic configurations for which the MTP has the highest uncertainty, i.e., highest prediction errors (so-called extrapolating configurations). These configurations are most likely to improve the MTP’s accuracy when added to the training data set. Since Active learning targets specific configuration spaces instead of randomly sampling configurations, it reduces the need for expensive DFT training data calculations, which leads to efficient and fast training.

2. It is recommended to include Active learning when training an MTP for amorphous systems​, interfaces​, surface processes, and systems at high temperatures.

3. In addition to geometry optimization and MD, methods such as NEB, AKMC and TFMC can also detect extrapolating configurations and add them to the training data set.

During Amorphous Active learning, first, one fits an initial MTP to the crystal training data and then iteratively repeats the flow below until there are no more extrapolating configurations to add, as also shown in the figure below:

• Perform geometry optimization and MD melt-quench (melting/cooling) simulations with the current MTP to find additional extrapolating amorphous training configurations.

• Calculate DFT training data for these added training configurations.

• Fit a preliminary MTP to the improved training data.

The actual workflow for the Amorphous Active Learning stage in the Workflow Builder looks like this:

1. The Initial Amorphous Structures ConfigurationTable(List) block contains initial amorphous structures of titanium silicide, where each of them will be used as a starting point for Active learningGeometryOptimization and ActiveLearningMolecularDynamics simulations. Since we have eight initial amorphous Ti₃₉Si₃₉ and Ti₂₆Si₅₂ structures, QuantumATK will perform eight Active learning geometry optimization and eight subsequent Active learning MD melt-quench simulations.

   

Tip

Initial amorphous configurations can be generated using Packmol in the Builder as showcased in the tutorial Molecular Dynamics Simulations for Generating Amorphous Structures or can be generated on-the-fly using the AmorphousPrebuilder block in the Workflow Builder.

Now let’s set up the generation of MTP fitting parameters that will be fitted to the training set data by minimizing the error (energy, forces and stresses) with respect to the reference DFT calculator.

2. The NonLinearCoefficientsParameters block contains settings for randomly generating non-linear coefficients for the radial functions for each element pair (see NonLinearCoefficientsParameters for more info), that will be used in the next step - calculation of MTP fitting parameters (see MomentTensorPotentialFittingParameters for more info).

• Initial coefficients: Set to Random to use random values for initial non-linear coefficients.

• Random seed: Set to an arbitrary number, but use the same number for different Active learning runs. This way consistent results are achieved between different Active learning runs.

• Perform optimization: recommend to keep it unchecked in the Amorphous Active Learning stage as non-linear MTP training (optimizing non-linear coefficients) usually does not give much improvement in accuracy and it will increase the calculation time.

  

3. The MomentTensorPotentialFittingParameters block contains settings for MTP fitting using the non-linear coefficients generated in the previous step.

• Basis size: Use 400 basis function in this example. Generally, it is recommended to use either Predefined Basis Small (default), or between 300 – 800 basis functions. More basis functions usually provide a better accuracy (lower training error), but the simulation can become less stable, especially in out-of-equilibrium situations, e.g., at high temperatures or energies.

• Outer cutoff radii: Set to 4.5 Ang in this example. This value specifies the range of the atomic environments. Typically values between 4-5 Ang are recommended. Similar to basis size, larger values can increase accuracy, whereas smaller values can provide more stability.

• MTP filename: One should always specify a filename with .mtp extension. This is where the trained MTP parameters are stored.

• Forces cap: Specify this value in order to exclude configurations with very large forces, i.e., above the Force cap, from the training. For Active learning it is always recommended to set this to a relatively large value, e.g. 500 eV / Ang, to include closer atom distances with larger forces in order to make sure the repulsive behavior is predicted correctly in such situations. Increasing the Forces cap (within some limit) makes the simulation more stable, but less accurate. Thus, while setting the Forces cap, one needs to balance between simulation stability and accuracy.

  

Note

Within the MTP framework, the total energy of a configuration is computed as the sum of energy contributions from atomic environments. Atomic environments are defined as the representation of the immediate chemical environment of each atom in the system within a cutoff radius. Moment tensors include the radial, angular, and many-body distribution of atoms within a cutoff sphere. Their contractions into scalar basis functions provide an invariant representation of atomic environments (‘structure’) as input for MTP training [1].

Now let’s set up Active learning simulation parameters.

4. The ActiveLearningSimulation block contains settings for running Active learning simulations. MTP fitting parameters from the previous step and the initial training data from the crystal training are used as an input.

• Candidate/ Retrain Threshold: These are numerical measures governing how much configurations can differ from the training set before they are considered to be extrapolating. If the current configuration has an extrapolation grade higher than the candidate threshold (default 1), it will be kept as a possible candidate to be included in the next training. If the value is larger than the retrain threshold (default 3), the current simulation will be stopped, extrapolating configurations will be added to the training data set, and then Active learning will be re-started now using an improved MTP fitted to the extended training data set.

• Limit candidates: Set to 100. Limit the number of recorded candidates for each simulation. This can be beneficial in case the simulation moves between candidate and retrain threshold for a long period and many candidates are registered. In that case the candidates are typically very correlated and not all of them are needed. The limited candidates are then taken from the raw list of candidates at a regular interval, before a structural filter extracts the most different configurations and adds them to the training set.

• Max. Iterations: The maximum number of re-training iterations, i.e., how many times the simulation will start again with a new MTP. Stick to the default value. It can be increased if more iterations are expected.

• Extrapolation grade algorithm: Select which algorithm is used to calculate the extrapolation grade. Unless there is only a single element in the simulation, we always recommend to set this to QueryByCommitteeForces.

• LCAOCalculator and ActiveLearningSimulation blocks must be connected, so that the defined LCAOCalculator is used to calculate training data during Active learning.

  

5. The ActiveLearningOptimizeGeometry block contains settings for performing Active learning geometry optimization. All initial amorphous configurations from the Initial Amorphous Structures ConfigurationTable(List) block will be used as a starting point for the ActiveLearningOptimizeGeometry. The optimizations will be run independently and in parallel (as much as the parallel settings allow that, otherwise sequentially). The DFT calculations and training are done collectively, using the combined extrapolating candidate configurations selected from all optimizations.

• Force tolerance: set a bit higher than the default, i.e., to 0.5 eV / Ang, as this part should primarily do a quick pre-optimization to avoid large forces in the active learning MD part.

• Optimize cell: this is disabled to make the optimization more robust.

  

The combined crystal training + amorphous geometry optimization training data is used to train an improved MTP, which is then used for subsequent Active learning melt-quench MD simulations.

6. The ActiveLearningMolecularDynamics-Melting block contains settings for performing Active learning MD melting simulations. Here, we want to run MD at relatively high temperatures to melt the disordered configurations and sample many different relevant atomic environments.

• Steps: set to 100 000 MD steps and set Log interval to 1000 MD steps.

• Method: set to Langevin.

• Reservoir temperature and Temperature of initial velocities is set to 2500 K to melt the disordered configurations.

  

Tip

One can also use a different thermostat type than Langevin, however, to ensure the simulation is relatively robust one should pick a robust thermostat, like Berendsen or Bussi-Donadio-Parrinello. At such high temperatures, it is recommended to run NVT simulations, whereas NPT simulations should primarily be run at lower temperatures.

Note

• The chosen Active learning MD temperature should reflect a physical process (e.g., chemical reactions, phase transitions, or material properties) that occur within a specific temperature range and that you want to simulate in production simulations with a trained MTP.

• The temperature should not be too low as it would miss configurations relevant at high temperatures (e.g., disordered amorphous structures).

• The temperature should also not be too unphysically high, as in that case the system may visit configurations that are physically unrealistic, or not representative of the system’s true equilibrium state. In addition, due to too high temperature, the system may also fail to properly sample the configuration space by jumping over energy barriers rather than exploring the free energy surface in a smooth, gradual manner.

• To sum it up, too low or too high temperature could result in the reduced accuracy of Active learning and thus the trained MTP.

The combined crystal training + amorphous geometry optimization + amorphous MD melting training data is used to train a further improved MTP, which is then used in subsequent Active learning MD cooling simulations.

7. The ActiveLearningMolecularDynamics-Cooling block contains settings for performing Active learning MD cooling simulations - to simulate cooling the liquid configuration to room temperature.

• Steps: set to 100 000 MD steps and set Log interval to 1000 MD steps.

• Method: set to Langevin.

• Reservoir temperature and Temperature of initial velocities is set to 2500 K, but now we specify a negative heating rate of -2200/100 000 K/fs to decrease the temperature from 2500 K to 300 K during the 100 ps of MD.

  

8. The Final-Training-Set-Table block picks up the Training-Set-Table containing the combined data from crystal displacements and Active learning and exposes it under a more obvious name, so that it can easily be used in the following Final training stage.

Final MTP Fitting

In the 2^nd Amorphous Active Learning stage we did preliminary MTP fittings to generate amorphous training data for titanium silicide on-the-fly. For this, we used only one set of initial non-linear MTP coefficients. In the 3^rd stage, Final MTP Fitting, we will fit an ensemble of MTP models to the crystal+amorphous training data and select the best fitted MTP. We will scan over 30 different initial guesses of non-linear MTP coefficients to generate different fitted MTPs based on these initial guesses. We then select the one which gives the lowest, and comparable, training and testing errors with respect to DFT. This can then be used in production simulations.

The actual workflow for the Final MTP Fitting stage in the Workflow Builder looks like this:

1. The ScanOverNonLinearCoefficients block contains settings for scanning over different initial guesses of non-linear MTP coefficients and obtaining MTP fitting parameters for these different initial guesses. scanOverNonLinearCoefficients is essentially an outer loop over NonLinearCoefficientsParameters + MomentTensorPotentialFittingParameters that loops over different random seeds.

• Number of initial guesses: Defines how many different initial values should be considered. The default value of 30 is usually sufficient. Increasing the number means a better chance of finding the best possible MTP at a cost of an approximately a linear increase in the calculation time.

• Basis size: Use 400 basis functions for this example (same as in active learning). Generally, it is recommended to use either Predefined-Small, or between 300 – 800 basis functions. More basis functions usually provide a better accuracy (lower training error), but the simulation can become less stable, especially in out-of-equilibrium situations, e.g., at high temperatures or energies. For calculations which require a high accuracy, but not stable MD, it can be meaningful to increase the number of basis functions beyond this range. In rare cases, mainly for calculations which require a very high accuracy, such as precise phonon calculations or defect energies, it can be meaningful to increase the number of basis functions to as much as 5000.

• Cutoff radii: Inner and tapering cutoff radii rarely need to be changed from their defaults. The outer cutoff radius should be set to the same as in Active learning (4.5 Å), as the extrapolation behavior will depend on this choice.

• MTP filename suffix: Specify the filename, where the MTP parameters are written to. The different fits will create files such as 0_mtp_filename_suffix.mtp, 1_mtp_filename_suffix.mtp, etc.

• Forces cap: Specify this value in order to exclude configurations with large forces. In this stage it does not have to be as large as for Active learning, we recommend 100 -200 eV / Ang for most cases, unless the application explicitly requires to be able to deal with very large forces (e.g. impact of high-energy ions). We choose 200 here, so that some large forces are included to train the MTP to be able to deal with close atoms, that might occur during the melting MD simulations.

• Perform optimization: Run a separate, local optimization of the non-linear coefficients. This takes much longer, and in most cases it gives very little improvement in accuracy, therefore it is recommended to have it disabled (default).

  

Tip

• ScanningOverNonLinearCoefficients is kind of a non-linear MTP optimization, but a crude one in that one just tests many initial guesses and then picks the best one (without the explicit local optimization of non-linear coefficients). For systems that do not have too many elements it works well.

• If the number of elements becomes larger (5+ different elements), it can be beneficial to use the particle-swarm-optimization (PSO) method ParticleSwarmOptimizationParameters to optimize the non-linear parameters instead of scanning over different initial guesses. This will take longer, but provides a more systematic optimization and often results in a better accuracy for complex systems. Since we only have 2 elements in the system we use here, we will not use PSO in this tutorial.

2. The MomentTensorPotentialTraining block contains settings for taking care of running and analyzing the MTP fitting runs based on 30 different sets of MTP fitting parameters. See MomentTensorPotentialTraining to learn more.

• Train test split: Set it to 0.9 to use 90 % of the calculated reference data for training and 10 % for testing. The split is done randomly. Specify an arbitrary number for the random seed to make it deterministic and reproducible, e.g., if one wants to change other hyperparameters like a cutoff radii or a basis size later and compare the results.

• Committee size: Number of committee models to train and store using the same data and hyperparameters for the MTP MD uncertainty predictions. This can be set to a small number (we recommend 6). The committee model can then be used in MD simulations to obtain an estimate of the energy and forces error during the production simulations with a trained MTP, using the MolecularDynamicsErrorPredictionHook.

• Ensure that the MomentTensorPotentialTraining block is linked to the Training-Set-Table, ScanOverNonLinearCoefficients and LCAOCalculator blocks.

  

Note

• We use random numbers to initialize the coefficients for the MTP training, randomly displacing crystal configurations and training-test set splitting. Therefore, reproducibility of the results to the numerical accuracy is only ensured if you use the same Random seed while re-running the MTP training simulation.

• Many different sets of MTP parameters could result in similar results as there could be many degenerate minima in the hyper-parameter surface. Thus it is not a cause for concern if you get different sets of MTP fitting parameters with similar accuracy.

3. The Select best MTP Parameters block is for selecting the best fitted MTP from 30 generated MTPs based on training and testing errors.

   • MTP filename: The parameters for the best fit are written to this file. It should have the extension .mtp. This mtp file is what you will use for setting up production simulations with the trained MTP.

   

Submitting the MTP Training Calculation

The MTP training set-up is now finished and you are ready to submit this job.

1. Change the Workflow name and Filename as needed (e.g., c-am-TiSi-MTP-training-results.hdf5).

2. Click on the Send content to other tools button and choose Jobs as a script to send the workflow to the Jobs tool to submit this job on a local machine or a remote cluster. It is recommended to use MPI parallelization for MTP training simulations, as both training data calculations with DFT and MTP fitting benefit from MPI parallelization.

3. This also generates a Python script c-am-TiSi-MTP-training-results.py, based on this workflow. It can be opened and edited with the Editor tool and then sent to the Jobs tool, if you need to make manual changes.

4. Estimate of the MTP training run-time: in our test it took 5 hours on the 40-core machine.

Analyzing MTP Training Results

Once the MTP training calculation has finished, we are ready to analyze the results.

Download the following output files if needed: c-am-TiSi-MTP-training-results.log, c-am-TiSi-MTP-training-results.hdf5, c-am-TiSi-MTP-training-results_4.hdf5, active_learning_molecular_dynamics.4.log and open them in the Data View.

1. c-am-TiSi-MTP-training-results.log: The main .log file contains a lot of information about the progress of the overall simulation, while the DFT training data, Active learning geometry optimization and MD log outputs are written to separate files. The most important parts of the c-am-TiSi-MTP-training-results.log file are:

• Completion of Active learning geometry optimization and melt-quench MD simulations, reporting the number of added extrapolating amorphous training configurations. For titanium silicide, 14, 8 and 0 extrapolating amorphous configurations were added during Active learning geometry optimization, MD melting and cooling phases, respectively.

  +------------------------------------------------------------------------------+
  | Active learning geometry optimization completed successfully.                |
  | Number of iterations for MTP training 2                                      |
  | Number of additional training configurations: 14                             |
  +------------------------------------------------------------------------------+
  +------------------------------------------------------------------------------+
  | Active learning MD completed successfully.                                   |
  | Number of iterations for MTP training: 2                                     |
  | Number of additional training configurations: 8                              |
  +------------------------------------------------------------------------------+
  +------------------------------------------------------------------------------+
  | Active learning MD completed successfully.                                   |
  | Number of iterations for MTP training: 1                                     |
  | Number of additional training configurations: 0                              |
  +------------------------------------------------------------------------------+
  

• MTP training report at the end of the .log file. This displays the size of the training set (216 configurations ∼90% of all training configurations) and testing set (25 configurations ∼10% of all training configurations) for each fit. It also shows a list with all the training and testing RSME errors and r² values for energy, forces, and stress for all 30 fits with respect to the reference DFT calculator. In this case, each fit corresponds to a different initial guess for the non-linear coefficients. One can inspect the errors manually and select the best fitted MTP with lowest RMSE errors.

• Best MTP fit, reported by QuantumATK based on the automatic analysis of all the errors. In this case, QuantumATK reports the MTP fit number 19 to be the best, corresponding to the lowest overall training and testing errors. The best fitted MTP is written to MTP_TiSi_basis400_final.mtp, which can then be used for validation and production simulations. All the 30 fitted MTPs are written to files 0,...,29_MTP_TiSi_final_basis400.mtp.

  +------------------------------------------------------------------------------+
  | Moment Tensor Potential Training Report                                      |
  +------------------------------------------------------------------------------+
  | Index                 Training set size(s)             Testing set size(s)   |
  |    19                                  216                              25   |
  +------------------------------------------------------------------------------+
  +------------------------------------------------------------------------------+
  | RMSE (training):                                                             |
  |                                                                              |
  | Index             Filename     Energy/atom           Force          Stress   |
  |                                       (eV)          (eV/Å)        (eV/Å^3)   |
  |     0 0_MTP_TiSi_[...].mtp    0.0025043219    0.1012940348    0.0389318624   |
  |     1 1_MTP_TiSi_[...].mtp    0.0029281002    0.1158755465    0.0361733476   |
  |     2 2_MTP_TiSi_[...].mtp    0.0037681036    0.1250996880    0.0376190870   |
  |     3 3_MTP_TiSi_[...].mtp    0.0028928380    0.1201118210    0.0429722560   |
  |     4 4_MTP_TiSi_[...].mtp    0.0022172316    0.1082486877    0.0278265693   |
  |     5 5_MTP_TiSi_[...].mtp    0.0028463811    0.1041497742    0.0395561149   |
  |     6 6_MTP_TiSi_[...].mtp    0.0022442684    0.1029489858    0.0357410797   |
  |     7 7_MTP_TiSi_[...].mtp    0.0040683096    0.1241173502    0.0543206816   |
  |     8 8_MTP_TiSi_[...].mtp    0.0044881187    0.1169965922    0.0731741130   |
  |     9 9_MTP_TiSi_[...].mtp    0.0019220696    0.0954770279    0.0309075969   |
  |    10 10_MTP_TiSi[...].mtp    0.0022222135    0.1040334579    0.0271600170   |
  |    11 11_MTP_TiSi[...].mtp    0.0020293338    0.0981887142    0.0296488370   |
  |    12 12_MTP_TiSi[...].mtp    0.0037672343    0.1055226664    0.0613020433   |
  |    13 13_MTP_TiSi[...].mtp    0.0027541694    0.0999899064    0.0464718686   |
  |    14 14_MTP_TiSi[...].mtp    0.0028242243    0.1101901201    0.0451052006   |
  |    15 15_MTP_TiSi[...].mtp    0.0033376688    0.1123514802    0.0511654854   |
  |    16 16_MTP_TiSi[...].mtp    0.0035935329    0.1315763184    0.0431630558   |
  |    17 17_MTP_TiSi[...].mtp    0.0028141484    0.1199110297    0.0192196160   |
  |    18 18_MTP_TiSi[...].mtp    0.0027824647    0.1003360462    0.0409380488   |
  |    19 19_MTP_TiSi[...].mtp    0.0015130631    0.0938986276    0.0186246025   |
  |    20 20_MTP_TiSi[...].mtp    0.0037646145    0.1260588897    0.0243364675   |
  |    21 21_MTP_TiSi[...].mtp    0.0035362934    0.1128509267    0.0456808120   |
  |    22 22_MTP_TiSi[...].mtp    0.0024321919    0.1017323905    0.0413082318   |
  |    23 23_MTP_TiSi[...].mtp    0.0025639095    0.1113184768    0.0416981450   |
  |    24 24_MTP_TiSi[...].mtp    0.0017477251    0.1023449077    0.0162820468   |
  |    25 25_MTP_TiSi[...].mtp    0.0035681856    0.1294370831    0.0291967493   |
  |    26 26_MTP_TiSi[...].mtp    0.0032966773    0.1102070614    0.0526647940   |
  |    27 27_MTP_TiSi[...].mtp    0.0021051690    0.0994845053    0.0240463666   |
  |    28 28_MTP_TiSi[...].mtp    0.0032323621    0.1044183263    0.0573661541   |
  |    29 29_MTP_TiSi[...].mtp    0.0021269500    0.1094758067    0.0216593875   |
  +------------------------------------------------------------------------------+
  +------------------------------------------------------------------------------+
  | RMSE (testing):                                                              |
  |                                                                              |
  | Index             Filename     Energy/atom           Force          Stress   |
  |                                       (eV)          (eV/Å)        (eV/Å^3)   |
  |    19 19_MTP_TiSi[...].mtp    0.0010749092    0.0838614510    0.0193958613   |
  +------------------------------------------------------------------------------+
  +------------------------------------------------------------------------------+
  | r^2 (training):                                                              |
  |                                                                              |
  | Index             Filename     Energy/atom           Force          Stress   |
  |    19 19_MTP_TiSi[...].mtp    0.9999999998    0.9937161787    0.0391601952   |
  +------------------------------------------------------------------------------+
  +------------------------------------------------------------------------------+
  | r^2 (testing):                                                               |
  |                                                                              |
  | Index             Filename     Energy/atom           Force          Stress   |
  |    19 19_MTP_TiSi[...].mtp    0.9999999999    0.9924226840   -1.1109615599   |
  +------------------------------------------------------------------------------+
  
  Best fit at index 19.
  Copying the best parameters to file MTP_TiSi_basis400_final.mtp.
  

Note

r² values, coefficient of determination, indicate how good the MTP is at predicting the trained properties in the dataset, with values closer to 1 being better. Looking at the log-file, we can see that all of the trained MTPs have an r² value of over 0.99999999 indicating that they are all accurate within the configurational space sampled in the training data set, thus not helping much in choosing the most accurate fitted MTP. Therefore, to find the best MTP fit, one has to look for an MTP fit with lowest training and testing RMSE errors.

2. c-am-TiSi-MTP-training-results.hdf5: The main .hdf5 file can be used to inspect crystal and amorphous training data, as well as training and testing errors for all 30 fits.

• Amorphous and crystal training configurations, along with DFT calculated energy, forces and stresses. Right-click on the mtp_1 object to open it with the Movie tool. The total number of training configurations for titanium silicide is 241 from which 22 are amorphous configurations from the Amorphous Active Learning stage (in the beginning of the set) and 219 are crystal configurations from the Crystal Training Data stage.

  

• MTP fitting analysis to graphically inspect MAE, RMSE and r² values for all 30 MTP fits. Right-click on the mtp_1 object to open it with the MTP Analyzer. Click on Fit index to choose a fit, e.g., 19 (19 is the best fit in this case), and then click Plot. The MTP Analyzer produces a scatterplot of MTP predicted values vs. the reference DFT values for energy, forces, and stress, for each of the training and testing datasets.

  

The above scatter plot compares the MTP predicted data (y-axis) and reference DFT data (x-axis). The energy, forces and stress values are compared for both training and test sets and indicates high accuracy of the trained MTP: the data points for energy and forces lie along the 45 degree line and for stress there are minor deviations, but overall good correlation.

Note

We recommend to look at the training configurations to aid potential debugging. In case your production simulations with the trained MTP give unexpected results, check if the training data set includes all the important configurations that needs to be included.

3. Look at c-am-TiSi-MTP-training-results_x.hdf5, active_learning_molecular_dynamics.x.log and active_learning_geometry_optimization.x.log to inspect Active learning geometry optimization and melt-quench MD trajectories. Since we started from 8 different initial amorphous configurations, we have 8 different .hdf5/.log files, where x=0-7.

• Each .hdf5 file contains an alopt, md and md_2 objects, which can opened with the Movie tool

• The alopt object is for inspecting the Active learning geometry optimization trajectory along with the calculated energy and forces.

• The md object is for the Active learning MD melting trajectory along with the calculated temperature, energy, pressure, extrapolation grade, etc. every 10 MD steps.

• The md_2 object is for the Active learning MD cooling trajectory along with the calculated temperature, energy, pressure, extrapolation grade, etc. every 10 MD steps.

• Open, e.g., the md object from the c-am-TiSi-MTP-training-results_4.hdf5 with the Movie Tool and active_learning_molecular_dynamics.4.log with Text Representation to inspect MD trajectory during the Active learning MD melting phase.

  | MD      23000       23000.0    -4.79677e+04          2418.3        2500.0    |
  | MD      24000       24000.0    -4.79693e+04          2148.1        2500.0    |
  | MD      25000       25000.0    -4.79616e+04          2224.4        2500.0    |
  | MD      26000       26000.0    -4.79685e+04          2301.4        2500.0    |
  | MD      27000       27000.0    -4.79666e+04          2644.5        2500.0    |
  | MD      28000       28000.0    -4.79659e+04          2918.8        2500.0    |
  | MD      29000       29000.0    -4.79632e+04          2733.6        2500.0    |
  | MD      30000       30000.0    -4.79684e+04          2812.7        2500.0    |
  | Step 30850: Extrapolation grade 1.7276932788031036. Adding to candidates     |  
  | Step 30860: Extrapolation grade 2.797831465597506. Adding to candidates      |   
  | Step 30870: Extrapolation grade 18.301103354747035.                          | 
  | Stopping simulation and starting new training.                               |   
  +------------------------------------------------------------------------------+
  | Selecting the most different candidate configurations to add to the training | 
  | data. The selection is started from 3 candidate configurations.              |
  | 3 configurations have been selected to be added to the training data.        |
  +------------------------------------------------------------------------------+
  +------------------------------------------------------------------------------+
  | TremoloX Domain Decomposition Report                                         |
  +------------------------------------------------------------------------------+
  | Enabled: False                                                               |
  | Number of Domains: 1                                                         |
  | Decomposition Pattern: 1x1x1                                                 |
  | CPU Information:                                                             |
  |  Process ID 0 at de02tcadhpc16 (1 thread)                                    |
  +------------------------------------------------------------------------------+
  +------------------------------------------------------------------------------+
  | MD       step          time       potential   instantaneous     reservoir    |  
  | MD                                   energy     temperature   temperature    |  
  | MD                     (fs)            (eV)             (K)           (K)    |
  +------------------------------------------------------------------------------+
  | MD          0           0.0    -4.79820e+04          2500.0        2500.0    | 
  | MD       1000        1000.0    -4.79636e+04          2243.0        2500.0    | 
  | MD       2000        2000.0    -4.79629e+04          2377.8        2500.0    |
  | MD       3000        3000.0    -4.79615e+04          2435.5        2500.0    |
  | MD       4000        4000.0    -4.79622e+04          2207.2        2500.0    |
  

  

• The extrapolation grade becomes 1.73 at step 30850 and 2.80 at step 30860. As these are larger than the candidate threshold of 1, but smaller than the retrain threshold of 3, these extrapolating configurations are kept as candidates to be included in the next training and the simulation proceeds.

• When the extrapolation grade becomes 18.3 at step 30870, thus larger than the retrain threshold, the current simulation will be stopped, extrapolating configurations will be added to the training data set, and then Active learning MD will be re-started now using an improved MTP fitted to the extended training data set.

Note

As shown in this tutorial, Active learning targets PES areas where the model is uncertain and selects the extrapolating amorphous configurations (22 in this case), for which to calculate DFT training data, thus minimizing the number of expensive DFT calculations needed. Randomly sampling PES using MD alone would require much longer simulation time scales, which would then become computationally unfeasible if DFT is used for each MD step.

MTP Validation for Crystals

To validate the trained MTP for crystals, one can use the trained MTP to calculate lattice constants and equation-of-state (EOS) for the relevant crystal phases of titanium silicide, TiSi, TiSi₂-C54 and TiSi₂-C49 and compare with the corresponding DFT results.

1. Download c-TiSi-DFT-EOS.hdf5 and c-TiSi-MTP-EOS.hdf5 workflow files for DFT and MTP, respectively, and open them in the Workflow Builder.

2. In the Set ForceFieldCalculator block, specify the location and name of the final trained MTP, MTP_TiSi_basis400_final.mtp, which will be used for the validation.

   

Run the workflows, and once the calculations of optimizing lattice constants and obtaining EOS are finished, we are ready to analyze the results.

3. Download c-TiSi-DFT-EOS-results.hdf5 / c-TiSi-MTP-EOS-results.hdf5 and then in the Data View window select the optgeom, optgeom_1 and optgeom_2, which contain optimized geometries of the three different crystal phases.

4. Right-click on optgeom, optgeom_1 and optgeom_2 objects to Open All with Text Representation. Inspect and compare lattice constants of optimized geometries obtained with the trained MTP and DFT. MTP gives good agreement with DFT results (within 1-2%) for all crystal phases, as shown in the table below.

# Item:     1
# Location: c-TiSi-MTP-EOS.hdf5
# Name:     optgeom
# Type:     BulkConfiguration
+----------------------------------------------------------+
| Bulk Bravais lattice                                     |
+----------------------------------------------------------+
Type:
FaceCenteredOrthorhombic

Lattice constants:
a =     4.779122 Ang
b =     8.162575 Ang
c =     8.694788 Ang

Lattice constants with MTP vs. DFT
	TiSi		TiSi2-C54		TiSi2-C49
	MTP	DFT	MTP	DFT	MTP	DFT
a	4.779 Å	4.829 Å	3.524 Å	3.558 Å	3.657 Å	3.647 Å
b	8.163 Å	8.306 Å	13.298 Å	13.644 Å	4.955 Å	5.025 Å
c	8.695 Å	8.595 Å	3.604 Å	3.591 Å	6.419 Å	6.562 Å

5. Calculated EOS (energy as a function of volume) are saved in these files EOS_tisi2c49_c-TiSi-MTP-validation.hdf5, EOS_tisi2c54_c-TiSi-MTP-validation.hdf5, EOS_tisi_c-TiSi-MTP-validation.hdf5 for MTP and in these files EOS_tisi2c49_c-TiSi-DFT-validation.hdf5, EOS_tisi2c54_c-TiSi-DFT-validation.hdf5, EOS_tisi_c-TiSi-DFT-validation.hdf5 for DFT.

6. In Data View, find the Data Plot tool under Supplementary tools, and then drag-and-drop volumes and dft_energies from the Data View panel to the Data Plot tool x and y drop areas, respectively.

   

7. Click Create line to plot EOS for each crystal phase with DFT and MTP.

8. Repeat the steps 6 and 7 for each crystal phase with DFT and MTP for a total of 6 plots.

9. Drag-and-drop the handle from one plot to another to combine the plots into one.

10. Combined plots for TiSi₂-C54 and TiSi₂-C49 show that the trained MTP reproduces the DFT trend, so that the TiSi₂-C49 crystal phase is lower in energy than the TiSi₂-C54 crystal phase at the minimum.

    

MTP Validation for Amorphous

To validate the trained MTP for amorphous structures, one can use the trained MTP to run some short melt-quench MD simulations (50 ps melting and 50 ps cooling) on, e.g., Ti₂₆Si₅₂, take some ~10 snapshots from each and recalculate these with DFT and calculate the energy, forces and stress RMSE between MTP and DFT.

1. Download am-TiSi-MTP-validation.hdf5 workflow and open it in the Workflow Builder.

   

2. In the Set ForceFieldCalculator block, specify the location and name of the final trained MTP, MTP_TiSi_basis400_final.mtp, which will be used for the validation.

Run the workflow and once the calculation is finished, we are ready to analyze the results.

3. Open the am-TiSi-MTP-validation-results.log and at the end of the file inspect the calculated energy, forces and stress RMSE between MTP and DFT. Small RMSE values shown in the figure below indicate a good accuracy of the trained MTP.

+------------------------------------------------+
| Energy RMSE: 0.0172973844972 eV                |
| Forces RMSE: 0.231592088227 eV/Ang             |
| Stress RMSE: 0.0238206661876 eV/Ang**3         |
+------------------------------------------------+

4. Open the md and md_2 objects from am-TiSi-MTP-validation-results.hdf5 using the Movie Tool to inspect MD trajectories from the 50 ps MD melting and from 50 ps cooling simulations, respectively.

   

Note

It is recommended to check the MD trajectories with the trained MTP to ensure that the trajectories look as expected and appear physical. In this case, check that the amorphous configuration does not crystallize during the cooling part of the simulation. In case of unexpected crystallization, one can consider increasing the size (number of atoms) of the amorphous training configurations and/or add additional amorphous configurations from Active learning to the training data set.

Summary and Outlook

In this introductory tutorial, you have learnt how to set up and analyze MTP training and MTP quality validation calculations for crystal and amorphous materials. In order to adapt the introduced MTP training protocol for different geometries (surfaces, interfaces, nanoparticles), stoichiometries, element composition and other materials, you need to explicitly include relevant training data for such structures.

Depending on the intended MTP application, consider also validating these properties against the reference DFT values:

• Elastic constants, bulk modulus, phonon bandstructure.

• RDF and ADF for amorphous structures (it requires some long DFT-MD simulations, which is expensive).

• Experimental properties that are relevant for the application, if available, but note that since the MTP is trained to reproduce DFT results it can only be as accurate as the underlying DFT model.

References

[1] (1,2)

Alexander V Shapeev. Moment tensor potentials: A class of systematically improvable interatomic potentials. Multiscale Modeling & Simulation, 14(3):1153–1173, 2016.

[2]

Yunxing Zuo, Chi Chen, Xiangguo Li, Zhi Deng, Yiming Chen, Jörg Behler, Gábor Csányi, Alexander V. Shapeev, Aidan P. Thompson, Mitchell A. Wood, and Shyue Ping Ong. Performance and Cost Assessment of Machine Learning Interatomic Potentials. The Journal of Physical Chemistry A, 124(4):731–745, 2020. URL: https://doi.org/10.1021/acs.jpca.9b08723, arXiv:https://doi.org/10.1021/acs.jpca.9b08723, doi:10.1021/acs.jpca.9b08723.