Chapter 2: MLP Fundamentals - Concepts, Methods, Ecosystem

We systematize the concepts of learning potential energy surfaces and the key aspects of GNNs that satisfy equivariance. We also cover the key considerations for training data design.

💡 Supplement: Equivariance is a property meaning "the meaning remains the same even if the coordinate system changes." This ensures generalization with data containing diverse structures.

Learning Objectives

By reading this chapter, you will master the following:
- Understand the precise definition of MLP and its relationship with related fields (quantum chemistry, machine learning, molecular dynamics)
- Explain 15 frequently encountered technical terms in MLP research
- Understand the 5-step MLP workflow (data collection, descriptor design, model training, validation, simulation)
- Compare the characteristics and application scenarios of 3 types of descriptors (symmetry functions, SOAP, graphs)

2.1 What is MLP: Precise Definition

In Chapter 1, we learned that MLP is a revolutionary technology that "combines DFT accuracy with the speed of empirical force fields." Here, we define it more precisely.

Definition

Machine Learning Potential (MLP) is:

A method that trains machine learning models using datasets of atomic configurations, energies, and forces obtained from quantum mechanical calculations (mainly DFT) to rapidly and accurately predict the potential energy and forces for arbitrary atomic configurations.

Expressed mathematically:

Training data: D = {(R₁, E₁, F₁), (R₂, E₂, F₂), ..., (Rₙ, Eₙ, Fₙ)}

R: Atomic configuration (3N-dimensional vector, N=number of atoms)
E: Energy (scalar)
F: Forces (3N-dimensional vector)

Objective: Learn function f_MLP
  E_pred = f_MLP(R)
  F_pred = -∇_R f_MLP(R)

Constraints:
  - |E_pred - E_DFT| < several meV (millielectronvolts)
  - |F_pred - F_DFT| < tens of meV/Å
  - Computational time: 10⁴-10⁶ times faster than DFT

Three Essential Elements of MLP

1. Data-Driven
- Requires large amounts of data (thousands to tens of thousands of configurations) from DFT calculations
- Data quality and quantity determine model performance
- No need for "manual parameter tuning" like traditional empirical force fields

2. High-Dimensional Function Approximation
- The Potential Energy Surface (PES) is ultra-high-dimensional
- Example: 100-atom system → 300-dimensional space
- Neural networks efficiently learn this complex function

3. Physical Constraints Incorporation
- Energy conservation
- Translational and rotational invariance (energy remains unchanged when the entire system is moved)
- Atomic permutation symmetry (energy remains unchanged when atoms of the same element are swapped)
- Force-energy consistency (F = -∇E)

Position in Related Fields

Contributions from Quantum Chemistry:
- DFT, quantum chemistry methods (CCSD(T), etc.)
- Concept of potential energy surfaces
- Chemical insights (bonding, reaction mechanisms)

Contributions from Machine Learning:
- Neural Networks (NN), Graph Neural Networks (GNN)
- Optimization algorithms (Adam, SGD, etc.)
- Regularization, overfitting countermeasures

Contributions from Molecular Dynamics:
- MD integration methods (Verlet method, etc.)
- Ensemble theory (NVT, NPT)
- Statistical mechanics analysis methods

2.2 MLP Glossary: 15 Important Concepts

We briefly explain technical terms frequently encountered in MLP research.

Top 5 Most Important Terms

Terms beginners should understand first:

1. Potential Energy Surface (PES): What MLP learns
2. Descriptor: Numerical representation of atomic configurations
3. Invariance and Equivariance: Mathematical properties for satisfying physical laws
4. MAE: Quantitative evaluation of model accuracy
5. Active Learning: Efficient data collection strategy

2.3 Input to MLP: Types of Atomic Configuration Data

MLP training requires diverse atomic configurations. What types of data are used?

Major Input Data Types

1. Equilibrium Structures and Vicinity
- Description: Most stable structure (energy minimum) and its neighborhood
- Use: Properties of stable structures, vibrational spectra
- Generation: DFT structure optimization + small displacements
- Data Volume: Hundreds to thousands of configurations
- Examples: Crystal structures, optimized molecular structures

2. Molecular Dynamics Trajectories
- Description: Time-series configurations from ab initio MD (AIMD)
- Use: Dynamic behavior, high-temperature properties
- Generation: Short-time (tens of ps) AIMD at various temperatures
- Data Volume: Thousands to tens of thousands of configurations
- Examples: Liquids, melting, diffusion processes

3. Reaction Pathways
- Description: Paths from reactants to products including transition states
- Generation: Nudged Elastic Band (NEB) method, String method
- Use: Catalytic reactions, chemical reaction mechanisms
- Data Volume: Hundreds to thousands of configurations (including multiple paths)
- Examples: CO₂ reduction reactions, hydrogen generation reactions

4. Random Sampling
- Description: Random exploration of configuration space
- Generation: Adding large displacements to existing structures, Monte Carlo sampling
- Use: Improving generalization performance, covering unknown regions
- Data Volume: Thousands to tens of thousands of configurations
- Caution: May include high-energy regions causing unstable calculations

5. Defects and Interface Structures
- Description: Crystal defects, surfaces, grain boundaries, nanoparticles
- Use: Material fracture, catalytic active sites
- Generation: Systematically introduce defects and perform DFT calculations
- Data Volume: Hundreds to thousands of configurations
- Examples: Vacancies, dislocations, surface adsorption sites

Example Data Type Combinations

Typical dataset composition (Cu catalyst CO₂ reduction reaction):

2.4 MLP Ecosystem: Understanding the Big Picture

MLP does not function in isolation but operates within an ecosystem comprising data generation, training, simulation, and analysis.

Details of Each Ecosystem Phase

Phase 1: Data Generation (DFT Calculations)
- Tools: VASP, Quantum ESPRESSO, CP2K, GPAW
- Computational Time: Several days to weeks on supercomputers
- Output: Atomic configurations, energies, forces, stress tensors

Phase 2: Model Training
- Tools: SchNetPack, NequIP, MACE, DeePMD-kit
- Computational Resources: 1 to several GPUs
- Training Time: Several hours to days
- Output: Trained MLP model (.pth file, etc.)

Phase 3: Simulation
- Tools: LAMMPS, ASE, i-PI
- Computational Resources: 1 to tens of GPUs
- Simulation Time: Nanoseconds to microseconds
- Output: Trajectory files (.xyz, .lammpstrj)

Phase 4: Analysis
- Tools: Python (NumPy, MDAnalysis, MDTraj), OVITO, VMD
- Analysis Content:
- Structure: Radial Distribution Function (RDF), coordination number, cluster analysis
- Dynamics: Diffusion coefficient, reaction rate constants, residence time
- Thermodynamics: Free energy, entropy

2.5 MLP Workflow: 5 Steps

Research projects using MLP proceed through the following 5 steps.

Step 1: Data Collection

Objective: Generate high-quality DFT data needed for MLP training

Specific Tasks:
1. System Definition: Determine target chemical system, size, composition
2. Sampling Strategy: Plan what configurations to calculate
- Equilibrium structures and vicinity
- AIMD (multiple temperatures)
- Reaction pathways
- Random sampling
3. DFT Calculation Settings:
- Functional (PBE, HSE06, etc.)
- Basis functions (plane waves, localized orbitals)
- Cutoff energy, k-point mesh
4. Parallel Calculation Execution: Calculate thousands of configurations on supercomputers

Keys to Success:
- Diversity: Include various configurations (monotonous data degrades generalization)
- Balance: Balance between low- and high-energy regions
- Quality Control: Check SCF convergence, force convergence

Typical Cost: 5,000 configurations → 3-7 days on supercomputer

Step 2: Descriptor Design

Objective: Convert atomic configurations into numerical vectors suitable for machine learning

Major Descriptor Types (detailed in 2.6):
- Symmetry Functions: Manual design, Behler-Parrinello
- SOAP (Smooth Overlap of Atomic Positions): Mathematically refined representation
- Graph Neural Networks (GNN): Automatic learning, SchNet, DimeNet

Descriptor Selection Criteria:
- Data Efficiency: Can high accuracy be achieved with little data?
- Computational Cost: Computational time during inference (prediction)
- Physical Interpretability: Can chemical insights be obtained?

Hyperparameters:
- Cutoff radius (5-10Å)
- Number of radial basis functions (10-50)
- Angular resolution

Step 3: Model Training

Objective: Optimize neural network to learn PES

Training Process:
1. Data Split: Training (80%), validation (10%), test (10%)
2. Loss Function Definition:
```
Loss = w_E × MSE(E_pred, E_true) + w_F × MSE(F_pred, F_true) w_E: Energy weight (typically 1)
w_F: Force weight (typically 100-1000, for unit conversion)
```
3. Optimization: Adam, SGD, etc. for thousands to tens of thousands of epochs
4. Regularization: L2 regularization, dropout to prevent overfitting

Hyperparameter Tuning:
- Learning rate: 10⁻³ to 10⁻⁵
- Batch size: 32-256
- Network depth: 3-6 layers
- Hidden layer size: 64-512 nodes

Computational Resources: Several hours to 2 days on 1 GPU

Step 4: Validation

Objective: Evaluate whether model has sufficient accuracy and generalization performance

Quantitative Metrics:
| Metric | Target Value | Description |
|------|--------|------|
| Energy MAE | < 1-5 meV/atom | Mean absolute error (test set) |
| Force MAE | < 50-150 meV/Å | Force error on atoms |
| Stress MAE | < 0.1 GPa | For solid materials |
| R² (coefficient of determination) | > 0.99 | Energy correlation |

Qualitative Validation:
- Extrapolation Test: Check accuracy on configurations outside training data range
- Physical Quantity Reproduction: Do lattice constants, elastic constants, vibrational spectra match DFT?
- Short-Time MD Test: Run 10-100 ps MD, check energy conservation

If Failed:
- Add data (Active Learning)
- Hyperparameter tuning
- Change to more powerful model (SchNet → NequIP)

Step 5: Production Simulation

Objective: Execute scientifically meaningful simulations with trained MLP

Typical MLP-MD Simulation Settings:

System size: 10³-10⁴ atoms
Temperature: 300-1000 K (depending on purpose)
Pressure: 1 atm or constant volume
Time step: 0.5-1.0 fs
Total simulation time: 1-100 ns
Ensemble: NVT (canonical), NPT (isothermal-isobaric)

Execution Time Estimates:
- 1,000 atoms, 1 ns simulation → 1-3 days on 1 GPU
- Further speedup possible with parallelization

Cautions:
- Energy Drift: Monitor for monotonic energy increase/decrease in long simulations
- Unlearned Regions: Accuracy may degrade when encountering configurations not in training data
- Ensemble Uncertainty: Evaluate prediction variance with multiple independent MLP models

2.6 Types of Descriptors: Numericalizing Atomic Configurations

MLP performance heavily depends on descriptor design. Here we compare three major approaches.

1. Symmetry Functions

Proposed by: Behler & Parrinello (2007)

Basic Idea:
- Represent the local environment of each atom i using functions of radial (distance) and angular information
- Designed to satisfy rotational and translational invariance

Radial Symmetry Functions:

G_i^rad = Σ_j exp(-η(r_ij - R_s)²) × f_c(r_ij)

r_ij: distance between atoms i and j
η: parameter determining Gaussian width
R_s: center distance (use multiple values)
f_c: cutoff function (smoothly decays influence of distant atoms)

Angular Symmetry Functions:

G_i^ang = 2^(1-ζ) Σ_(j,k≠i) (1 + λcosθ_ijk)^ζ ×
          exp(-η(r_ij² + r_ik² + r_jk²)) ×
          f_c(r_ij) × f_c(r_ik) × f_c(r_jk)

θ_ijk: angle formed by atoms j-i-k
ζ, λ: parameters controlling angular resolution

Advantages:
- Physically interpretable (corresponds to radial and angular distributions)
- Rotational and translational invariance guaranteed
- Relatively simple implementation

Disadvantages:
- Manual design: Need to manually select parameters like η, R_s, ζ, λ
- High-dimensional: Requires 50-100 dimensional descriptors (increased computational cost)
- Low data efficiency: May need tens of thousands of configurations

Application Examples: Water, silicon, metal surfaces

2. SOAP (Smooth Overlap of Atomic Positions)

Proposed by: Bartók et al. (2013)

Basic Idea:
- Approximate electron density around atoms with "Gaussian density"
- Calculate "overlap" of these density distributions as descriptor
- Mathematically rigorous rotational invariance

Mathematical Definition (Simplified):

ρ_i(r) = Σ_j exp(-α|r - r_j|²)  (Gaussian density around atom i)

SOAP_i = integral[ρ_i(r) × ρ_i(r') × kernel(r, r')]

kernel: radial/angular basis functions

In practice, efficiently calculated using spherical harmonics expansion.

Advantages:
- Mathematically refined representation
- Fewer parameters than symmetry functions
- Good compatibility with kernel methods (Gaussian Process Regression)

Disadvantages:
- Somewhat higher computational cost
- More complex combination with neural networks than SchNet

Application Examples: Crystalline materials, nanoclusters, complex alloys

3. Graph Neural Networks

Representative Methods: SchNet (2017), DimeNet (2020), PaiNN (2021)

Basic Idea:
- Represent molecules as graphs
- Nodes (vertices) = atoms
- Edges = interatomic interactions
- Message passing: Propagate information between adjacent atoms
- No manual descriptor design, neural network learns automatically

SchNet Architecture (Conceptual):

Initial state:
  Atom i feature vector h_i^(0) = Embedding(Z_i)  (Z_i: atomic number)

Message passing (repeat L layers):
  for l = 1 to L:
    m_ij = NN_filter(r_ij) × h_j^(l-1)  (distance-dependent filter)
    h_i^(l) = h_i^(l-1) + Σ_j m_ij      (message aggregation)
    h_i^(l) = NN_update(h_i^(l))        (nonlinear transformation)

Energy prediction:
  E_i = NN_output(h_i^(L))              (per-atom energy)
  E_total = Σ_i E_i

Advantages:
- End-to-end learning: No descriptor design needed
- Flexibility: Applicable to various systems
- Data efficiency: High accuracy with less data than symmetry functions

Disadvantages:
- Black box nature (difficult to interpret)
- Requires computational resources for training

Evolution: DimeNet (adding angular information):

DimeNet = SchNet + explicit consideration of bond angles θ_ijk

Embed angular information in messages:
  m_ij = NN(r_ij, {θ_ijk}_k)

Latest: E(3) Equivariant GNN (NequIP, MACE):
- Implement equivariance rather than rotational invariance
- Propagate vector/tensor fields as messages
- Dramatically improved data efficiency (thousands of configurations sufficient)

Application Examples: Organic molecules, catalytic reactions, complex biomolecules

Descriptor Comparison Table

Selection Guidelines:
- Beginners, small systems: Symmetry functions (easy to understand)
- Complex crystals, alloys: SOAP (combine with kernel methods)
- Organic molecules, catalysis: GNN (SchNet, DimeNet)
- Cutting-edge, data scarcity: E(3) equivariant GNN (NequIP, MACE)

2.7 Comparison of Major MLP Architectures

We compare representative MLP methods using the descriptors learned so far.

Evolution Timeline

Accuracy vs Data Efficiency Tradeoff

Typical performance (guidelines for 100-atom molecular systems):

Key Observations:
- Data efficiency improved 10-fold (30,000 → 3,000 configurations)
- Accuracy also improved 10-fold (5 meV → 0.5 meV/atom)
- Training time remains similar (several hours to 1 day)

2.8 Column: Efficient Data Collection with Active Learning

Standard MLP training prepares large amounts of DFT data in advance. However, Active Learning can achieve high accuracy with minimal necessary data.

Active Learning Workflow

Uncertainty Evaluation Methods

Ensemble Method:

# 5 independently trained MLP models
models = [MLP_1, MLP_2, MLP_3, MLP_4, MLP_5]

# Predict energy for a configuration R
energies = [model.predict(R) for model in models]

# Mean and standard deviation
E_mean = mean(energies)
E_std = std(energies)  # Uncertainty metric

# Add DFT calculation if above threshold
if E_std > threshold:
    E_DFT = run_DFT(R)
    add_to_dataset(R, E_DFT)

Advantages:
- Data collection efficiency improves 3-5 times
- Automatically discovers important configurations (transition states, defects, etc.)
- Objective sampling not relying on human intuition

Success Examples:
- Si phase transition: Initial 500 configs → Active Learning +1,500 configs → Total 2,000 configs achieves DFT accuracy (normally requires 10,000)
- Cu catalyst CO₂ reduction: Automatically discovered reaction intermediates, reduced DFT cost by 60%

2.9 Chapter Summary

What We Learned

1. Precise Definition of MLP
   - Data-driven high-dimensional function approximation
   - Importance of physical constraints (invariance, equivariance, force-energy consistency)
   - Fusion technology of quantum chemistry, machine learning, and molecular dynamics

2. 15 Important Terms
   - PES, descriptor, symmetry functions, message passing, equivariance, invariance
   - Cutoff radius, MAE, Active Learning, data efficiency, generalization
   - Ensemble, transfer learning, many-body interactions, E(3) equivariance

3. MLP Input Data Types
   - Equilibrium structures, MD trajectories, reaction pathways, random sampling, defect structures
   - Dataset composition balance (low/high energy regions, diversity)

4. MLP Ecosystem
   - 4 phases: Data generation → Model training → Simulation → Analysis
   - Representative tools for each phase (VASP, SchNetPack, LAMMPS, MDAnalysis)

5. 5-Step Workflow
   - Step 1: Data collection (DFT calculations, sampling strategy)
   - Step 2: Descriptor design (symmetry functions, SOAP, GNN)
   - Step 3: Model training (loss function, optimization, hyperparameters)
   - Step 4: Accuracy validation (MAE, extrapolation tests, physical quantity reproduction)
   - Step 5: Production simulation (MLP-MD, long timescales)

6. 3 Types of Descriptors
   - Symmetry functions: Manual design, high physical interpretability, low data efficiency
   - SOAP: Mathematical rigor, good compatibility with kernel methods, medium data efficiency
   - GNN: Automatic learning, end-to-end, high data efficiency (especially E(3) equivariant types)

7. Major MLP Architectures
   - 2007 Behler-Parrinello → 2022 MACE: 10x data efficiency improvement, 10x accuracy improvement
   - Latest methods (NequIP, MACE) achieve DFT accuracy with thousands of configurations

Key Points

• MLP performance heavily depends on descriptor selection and data quality/quantity
• Latest methods with E(3) equivariance (NequIP, MACE) are best in data efficiency and accuracy
• Active Learning can reduce DFT calculation costs by 50-70%
• MLP is not a standalone technology but part of an ecosystem combining DFT, MD, and machine learning

To the Next Chapter

In Chapter 3, we will experience actual MLP training using SchNet:
- Implementation with Python code
- Training on small dataset (MD17)
- Accuracy evaluation and hyperparameter tuning
- Troubleshooting

Furthermore, in Chapter 4, we will learn advanced techniques of NequIP/MACE and real research applications.

Exercises

Problem 1 (Difficulty: easy)

Create a comparison table for the three descriptors (symmetry functions, SOAP, graph neural networks) from the perspectives of "need for manual design," "data efficiency," and "physical interpretability."

Problem 2 (Difficulty: medium)

You are starting research on methanol oxidation reaction (CH₃OH → HCHO + H₂) on copper catalyst surfaces. Create a concrete work plan following the 5-step MLP workflow (data collection, descriptor design, training, validation, simulation). In particular, explain what types of atomic configurations should be prepared in Step 1 (data collection) and approximately how many, with reasoning.

Problem 3 (Difficulty: hard)

Quantitatively compare how costs and time for the entire Cu catalyst CO₂ reduction reaction project (from data collection to simulation) change between standard MLP training without Active Learning and with Active Learning. Consider both supercomputer time and GPU time.

2.10 Data Licenses and Reproducibility

To ensure research reproducibility, we specify dataset licenses and code environment details used.

2.10.1 Major Dataset Licenses

Notes:
- Commercial Use: CC BY 4.0 license allows commercial use with proper attribution
- Paper Citation: Always cite original papers when using datasets
- Redistribution: Redistribution allowed if license conditions met (MD17, QM9 are public domain)

2.10.2 Environment Information for Code Reproducibility

Version Management for Major MLP Tools:

DFT Calculation Parameter Recording Example:

# DFT settings recording for reproducibility (VASP example)
system:
  name: "Cu(111) + CO2 adsorption"
  composition: "Cu64C1O2"

dft_parameters:
  code: "VASP 6.3.2"
  functional: "PBE"
  dispersion: "D3(BJ)"
  encut: 500  # eV
  kpoints: "4x4x1 (Gamma-centered)"
  smearing: "Methfessel-Paxton, sigma=0.1 eV"
  convergence:
    electronic: 1.0e-6  # eV
    ionic: 1.0e-3  # eV/Angstrom

structure:
  slab: "Cu(111) 4-layer, 4x4 supercell"
  vacuum: 15  # Angstrom
  fixed_layers: 2  # bottom 2 layers

2.10.3 Energy Unit Conversion Table

Conversion table for major energy units used in MLP:

Force Unit Conversion:

Temperature and Energy Relationship:

k_B T (300 K) = 0.02585 eV = 0.596 kcal/mol

Recommended accuracy targets:
- Energy MAE < k_B T @ 300K  (< 0.026 eV = 0.6 kcal/mol)
- Force MAE < 3 × k_B T / 1Å  (< 0.078 eV/Å = 1.8 kcal/mol/Å)

2.11 Practical Cautions: Common Failure Patterns

We summarize frequently encountered problems in MLP research and their solutions.

2.11.1 Dataset Design Failures

Failure Example 1: Excessive Bias Toward Low-Energy Regions

# Problematic dataset composition
Equilibrium structures: 8,000 configs (80%)
AIMD trajectories: 1,500 configs (15%)
Reaction pathways: 500 configs (5%)

Problem: Insufficient data in high-energy regions (transition states, dissociation states)
Result: Accuracy collapse in reaction pathway simulations

Solution:
1. Balance adjustment (equilibrium:AIMD:reaction = 40%:40%:20%)
2. Add high-energy configurations with Active Learning
3. Collect rare events with Enhanced Sampling (Metadynamics, etc.)

Failure Example 2: Insufficient Temperature Range

Training data: 300K AIMD only

Problem: Extrapolation errors when using at 500K MD
Solution:
- Cover wide temperature range during training (200K-600K)
- Sample 1,000+ configs at each temperature
- Always perform temperature extrapolation tests

2.11.2 Model Training Failures

Failure Example 3: Cutoff Radius Selection Mistakes

Diagnostic Method:

# Estimate cutoff from Radial Distribution Function (RDF)
import numpy as np
from ase.geometry import get_distances

def estimate_cutoff(atoms, element='Cu'):
    """Estimate cutoff as third peak position in RDF"""
    positions = atoms.get_positions()
    distances = get_distances(positions, cell=atoms.cell, pbc=True)[1]

    # Calculate histogram
    rdf, bins = np.histogram(distances.flatten(), bins=200, range=(0, 12))

    # Find third peak position
    peaks = find_peaks(rdf, height=rdf.max()*0.3)[0]
    if len(peaks) >= 3:
        cutoff_estimate = bins[peaks[2]]
        print(f"Recommended cutoff: {cutoff_estimate:.2f} Å")

    return cutoff_estimate

Failure Example 4: Loss Function Weight Setting Mistakes

# Bad setting example
loss = 1.0 * MSE(E_pred, E_true) + 1.0 * MSE(F_pred, F_true)

Problem: Unit mismatch
- Energy: eV (absolute value 10¹-10³)
- Force: eV/Å (absolute value 10⁻¹-10¹)
→ Energy learning dominates, force accuracy sacrificed

Correct setting:
loss = 1.0 * MSE(E_pred, E_true) + 100.0 * MSE(F_pred, F_true)
# Or normalize both
loss = MSE(E_pred/E_std, E_true/E_std) + MSE(F_pred/F_std, F_true/F_std)

2.11.3 Simulation Execution Failures

Failure Example 5: Ignoring Energy Drift

# Bad example: Run long MD without checking drift
dyn = VelocityVerlet(atoms, timestep=1.0*units.fs)
dyn.run(10000000)  # 10 ns

Problem: System heats/cools due to energy drift
Diagnosis:
total_energy = [a.get_total_energy() for a in traj]
drift_per_step = (total_energy[-1] - total_energy[0]) / len(total_energy)

if abs(drift_per_step) > 0.001:  # eV/step
    print("Warning: Significant energy drift detected")
    print("Solution 1: Halve timestep (1.0 fs → 0.5 fs)")
    print("Solution 2: Improve model accuracy (achieve force MAE < 0.05 eV/Å)")
    print("Solution 3: Use Langevin dynamics with thermostat coupling")

Failure Example 6: Extrapolation to Unlearned Regions

Training data: 300K AIMD (energy range: E₀ ± 0.5 eV)
Simulation: 1000K MD

Problem: Reaching new configuration space at high temperature → Prediction accuracy collapse

Preventive Measures:
1. Ensemble uncertainty evaluation
   - Warn if standard deviation among multiple models exceeds threshold
2. Gradual heating
   - Gradually increase temperature: 300K → 500K → 700K → 1000K
   - Perform Active Learning at each temperature
3. Always perform extrapolation tests
   - Check accuracy on higher-energy configurations than training data

2.11.4 Unit Conversion Failures

Failure Example 7: Hartree vs eV Confusion

# Bad example: Not checking DFT software default units
energy_dft = read_dft_output()  # Hartree units

# Direct input to SchNetPack
dataset.add_sample(energy=energy_dft)  # Expects eV but actually Hartree

Error example:
- Training data energy overestimated by 27x
- MAE appears small but actually a unit problem

Correct method:
HARTREE_TO_EV = 27.211386245988  # CODATA 2018
energy_ev = energy_dft * HARTREE_TO_EV
dataset.add_sample(energy=energy_ev)

# Or handle units explicitly
from ase import units
energy_ev = energy_dft * units.Hartree

Checklist: Unit Consistency Verification

1. [ ] Check DFT software output units (VASP: eV, Gaussian: Hartree, ORCA: Hartree)
2. [ ] Check MLP library expected units (SchNetPack: eV, TorchANI: Hartree)
3. [ ] Define conversion factors as constants (avoid magic numbers)
4. [ ] Sanity check: Is energy in physically reasonable range (-10³ ~ 10³ eV)?
5. [ ] Also verify force units simultaneously (eV/Å vs Hartree/Bohr)

2.12 End-of-Chapter Checklist: MLP Fundamentals Quality Assurance

Check whether you understand and can practice the content of this chapter.

2.12.1 Conceptual Understanding

MLP Definition and Positioning

• [ ] Can explain the three essential elements of MLP (data-driven, high-dimensional approximation, physical constraints)
• [ ] Can list three or more differences between MLP and conventional methods (empirical force fields, DFT)
• [ ] Can explain contributions from quantum chemistry, machine learning, and molecular dynamics to MLP
• [ ] Understand the concept of Potential Energy Surface (PES) and can illustrate it

Understanding MLP Terminology

• [ ] Can accurately explain 10 or more of the 15 important terms
• [ ] Can explain the difference between Invariance and Equivariance with concrete examples
• [ ] Understand why E(3) equivariance is important
• [ ] Know MAE target values (energy: <1-5 meV/atom, force: <50-150 meV/Å)

Data and Workflow

• [ ] Can distinguish the uses of 5 input data types (equilibrium structures, AIMD, reaction pathways, random, defects)
• [ ] Can explain the MLP workflow 5 steps in order
• [ ] Can estimate typical computational time and required resources for each step
• [ ] Understand the advantages and workflow of Active Learning

Descriptors and Architectures

• [ ] Can compare characteristics of three types of descriptors: symmetry functions, SOAP, GNN
• [ ] Understand differences in data efficiency (required configurations) for each descriptor
• [ ] Can explain evolution and features of SchNet, DimeNet, NequIP, MACE
• [ ] Can select appropriate descriptor/architecture for own research subject

2.12.2 Practical Skills

Data Preparation

• [ ] Can select DFT calculation settings (functional, cutoff, k-points) needed for own system
• [ ] Can plan balanced dataset composition (low/high energy regions)
• [ ] Can design 5,000-configuration dataset composition for typical Cu catalyst system (see Exercise 2)
• [ ] Can estimate DFT calculation time (configurations × time/configuration)

Model Selection and Hyperparameters

• [ ] Can select cutoff radius based on system characteristics (metal: 5-6Å, hydrogen bonding systems: 6-8Å)
• [ ] Can appropriately set loss function weights (w_E, w_F)
• [ ] Can perform train/validation/test split (typically 80%/10%/10%)
• [ ] Can adjust learning rate, batch size, epoch count

Accuracy Evaluation

• [ ] Can calculate MAE on test set and compare with target values
• [ ] Can create prediction vs true value correlation plots and calculate R²
• [ ] Can design and conduct extrapolation tests (outside training data range)
• [ ] Can verify reproducibility of physical quantities (lattice constants, vibrational spectra)

Troubleshooting

• [ ] Can diagnose causes of insufficient accuracy (data shortage, hyperparameters, model selection)
• [ ] Can implement Active Learning workflow (ensemble uncertainty evaluation)
• [ ] Can accurately perform energy unit conversions (eV ↔ kcal/mol ↔ Hartree)
• [ ] Can estimate appropriate cutoff radius from RDF

2.12.3 Application Ability

Project Design

• [ ] Can make plans to apply MLP to new chemical system (own research subject)
• [ ] Can estimate total project computational cost (DFT time, GPU time, personnel costs)
• [ ] Can quantitatively compare costs between Active Learning vs standard training (see Exercise 3)
• [ ] Can optimize dataset composition balance (equilibrium:AIMD:reaction pathways)

Literature Understanding and Critical Evaluation

• [ ] Can understand method sections of MLP papers (descriptor, training data, accuracy)
• [ ] Can judge whether reported MAE values are reasonable (compare with system size, complexity)
• [ ] Can point out dataset biases (temperature range, energy distribution)
• [ ] Can fairly compare performance of different architectures (SchNet vs NequIP)

Research Strategy

• [ ] Can formulate data collection strategy with limited computational resources
• [ ] Understand MLP limitations (extrapolation performance, transfer learning difficulties) and can take countermeasures
• [ ] Can explain MLP's utility and limitations to experimental researchers
• [ ] Aware of trends in next-generation methods (Foundation Models, Universal MLP)

Self-Assessment of Achievement

References

1. Behler, J., & Parrinello, M. (2007). "Generalized neural-network representation of high-dimensional potential-energy surfaces." Physical Review Letters, 98(14), 146401.
   DOI: 10.1103/PhysRevLett.98.146401
   

2. Bartók, A. P., et al. (2013). "On representing chemical environments." Physical Review B, 87(18), 184115.
   DOI: 10.1103/PhysRevB.87.184115
   

3. Schütt, K. T., et al. (2017). "SchNet: A continuous-filter convolutional neural network for modeling quantum interactions." Advances in Neural Information Processing Systems, 30.
   arXiv: 1706.08566
   

4. Klicpera, J., et al. (2020). "Directional message passing for molecular graphs." International Conference on Learning Representations (ICLR).
   arXiv: 2003.03123
   

5. Batzner, S., et al. (2022). "E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials." Nature Communications, 13(1), 2453.
   DOI: 10.1038/s41467-022-29939-5
   

6. Batatia, I., et al. (2022). "MACE: Higher order equivariant message passing neural networks for fast and accurate force fields." Advances in Neural Information Processing Systems, 35.
   arXiv: 2206.07697
   

7. Smith, J. S., et al. (2017). "ANI-1: an extensible neural network potential with DFT accuracy at force field computational cost." Chemical Science, 8(4), 3192-3203.
   DOI: 10.1039/C6SC05720A
   

8. Zhang, L., et al. (2018). "End-to-end symmetry preserving inter-atomic potential energy model for finite and extended systems." Advances in Neural Information Processing Systems, 31.
   arXiv: 1805.09003
   

9. Schütt, K. T., et al. (2019). "Unifying machine learning and quantum chemistry with a deep neural network for molecular wavefunctions." Nature Communications, 10(1), 5024.
   DOI: 10.1038/s41467-019-12875-2
   

10. Musaelian, A., et al. (2023). "Learning local equivariant representations for large-scale atomistic dynamics." Nature Communications, 14(1), 579.
    DOI: 10.1038/s41467-023-36329-y
    

Author Information

Author: MI Knowledge Hub Content Team
Created: 2025-10-17
Version: 1.0 (Chapter 2 initial version)
Series: MLP Introduction Series

Update History:
- 2025-10-19: v1.1 Quality improvement revision
- Added data license and reproducibility sections (MD17, OC20, QM9, ANI-1x, Materials Project)
- Added code reproducibility information (specified SchNetPack 2.0.3, NequIP 0.5.6, MACE 0.3.4 versions)
- Added energy/force unit conversion tables (eV, kcal/mol, Hartree, kJ/mol)
- Added practical cautions section (7 failure examples and solutions)
- Added end-of-chapter checklist (16 conceptual understanding items, 16 practical skill items, 12 application ability items)
- Added DFT calculation parameter recording example (VASP settings YAML format)
- 2025-10-17: v1.0 Initial Chapter 2 version
- Precise definition of MLP and positioning among related fields
- 15 important term glossary (concise table format)
- 5 major input data types
- MLP ecosystem diagram (Mermaid)
- 5-step workflow (detailed for each step)
- Descriptor types (symmetry functions, SOAP, GNN) and comparison table
- Comparison of major MLP architectures (7 methods)
- Active Learning column
- 4 learning objectives, 3 exercises (easy, medium, hard)
- 10 references

License: Creative Commons BY-NC-SA 4.0