Machine-Learning Potentials in Atomistic Simulations

• Machine-learning potentials are data-driven models that approximate potential energy surfaces in chemistry and materials science using methods like RKHS and deep neural networks.
• They employ symmetry-invariant descriptors and analytic gradients to enable efficient molecular dynamics and quantum simulations with sub–1 kcal/mol accuracy.
• By balancing computational cost with high-fidelity predictions, these approaches facilitate accurate simulations of reactive events and complex atomistic processes.

Machine-learning potentials (MLPs) are data-driven models that approximate potential energy surfaces (PES) for complex atomistic systems, primarily in chemistry and materials science. By interpolating or regressing energies from large sets of quantum mechanical calculations, MLPs enable atomistic simulations—such as molecular dynamics (MD) and quantum dynamics—with accuracy approaching that of ab initio methods but at a fraction of the computational cost. Core approaches include representations rooted in reproducing kernel Hilbert spaces (RKHS) and deep neural networks (NNs), both of which can achieve sub–1 kcal/mol accuracy and analytic gradients suitable for MD, facilitating high-fidelity simulations of reactive and flexible molecular systems.

1. Mathematical Foundations: RKHS and NN Frameworks

MLPs rest on the formalism of expressing the PES as a continuous mapping from nuclear geometries $x \in \mathbb{R}^D$ to energies,

$f(x): \mathbb{R}^D \to \mathbb{R},$

where $D$ is the dimensionality of the configuration space.

RKHS Approach

The RKHS method constructs $f(x)$ as a linear combination of kernel functions centered at each reference point:

$f(x) = \sum_{i=1}^N c_i K(x, x_i), \tag{1}$

where $c_i$ are coefficients and $K$ is a symmetric, positive-definite kernel. Analytical gradients follow due to the differentiable nature of $K$, facilitating efficient computation of forces for MD. For direct solution, the system

$$f(x_j) = \sum_{i=1}^N c_i K(x_j, x_i), \quad j = 1, ..., N \tag{2}$$

is solved (often via Cholesky decomposition).

Physical insight is embedded by constructing multidimensional kernels as tensor products of tailored univariate kernels:

$$K(x, x') = \prod_{d=1}^D k^{(d)}(x^{(d)}, x'^{(d)}), \tag{3}$$

where $k^{(d)}$ can be chosen with known decay (e.g., reciprocal power or exponential decay) to enforce correct long-range asymptotics. Explicit forms, such as reciprocal power decay or exponential decay kernels, can be tuned to reflect dispersion or other physical forces and are parameterized using hypergeometric and beta functions for ensured smoothness and correct asymptotic limits.

Deep Neural Network (DNN) Approach

NN-based MLPs define a deep mapping. For a fully-connected architecture,

$y = W x + b, \tag{6}$

with non-linear activations ($\sigma$), the model can be recursively described as

$h = \sigma(W_1 x + b_1), \tag{7}$

$y = W_2 h + b_2, \tag{8}$

and, for deep architectures, by stacking such transformations. NNs use either engineered local descriptors (e.g., radial Gaussians times spherical harmonics ensuring symmetry invariance) or learnable descriptors (e.g., as in PhysNet), in which descriptors are constructed dynamically through network layers and attention mechanisms, often with explicit bias toward locality (e.g., using $\exp(-r_{ij})$).

State-of-the-art models sum atomic energy contributions,

$E_{\text{tot}} = \sum_i E_i,$

where $E_i$ is predicted by an NN that processes the atomic environment encoded by descriptors. This partitioning enables scaling to larger systems and preserves size-consistency.

2. Descriptor Design and Incorporation of Physical Symmetries

Both kernels and NN-based MLPs rely on descriptors to encode molecular geometry and composition. Essential requirements are invariance to translation, rotation, and permutation of equivalent atoms.

• RKHS-based kernels: Descriptor design manifests as the construction of kernels that mix distances and angles with correct decay. The multidimensional product ensures that long-range and high-dimensionality effects are captured.
• NN-based models:
  • Hand-crafted local descriptors: Radial and angular functions (e.g., symmetry functions, products of Gaussians and spherical harmonics) encode local atom environments with explicit symmetry.
  • Learnable descriptors: Models such as PhysNet include an embedding of atomic numbers and an attention mechanism for distance-based invariance, with architectures (e.g., interaction blocks) designed such that translation, rotation, and permutation symmetries are automatically respected.

By judicious selection or learning of descriptors, these models generalize across composition, size, and conformation.

3. Construction, Training, and Evaluation Procedures

RKHS Construction

• Reference energies at discrete $x_i$ are computed ab initio.
• Kernel coefficients $c_i$ are determined by solving the linear system (Eq. 2).
• Analytical gradients (forces) are immediately available by differentiating Eq. 1.
• The kernel form is chosen and parameterized (e.g., with decay exponent $m$, number of terms $n$, decay rate $\beta$), with physical constraints encoded explicitly.

Neural Network Model Development

• Descriptor computation: For a given structure, local descriptors for each atom are generated.
• Data organization: Tens of thousands of reference points (ab initio geometries) are typically required for high-dimensional systems.
• Training: Network weights are fit by minimizing a loss over the reference dataset, often a sum of energy and force errors (RMSE $\sim 1$ kcal/mol is achieved in practice).
  • $$L = \sum_{k} (\hat{E}_k - E_k^\text{ref})^2 + \lambda_\text{force}\sum_{k, i} \lVert \hat{F}_{k,i} - F_{k,i}^\text{ref}\rVert^2$$.
• Enforcement of symmetry invariance is built into the descriptor and/or architecture.

4. Performance, Scope, and Application Domains

• Accuracy: Both approaches routinely achieve sub–1 kcal/mol errors relative to high-level ab initio data for small molecular systems or reactive event simulations.
• Analytical gradients (for RKHS) and efficient force computations (for NNs) enable direct application in classical and quantum molecular dynamics.
• Illuminative examples:
  • RKHS-based PESs have enabled quantum and semiclassical reaction dynamics for small triatomic and tetraatomic systems.
  • NN-based MLPs (using local or learnable descriptors) have enabled high-fidelity simulations of proton transfer in malonaldehyde and S_N2 reactions.
• Long-range effects and scalability: RKHS excels at embedding physical asymptotics, but the number of training points grows rapidly with the system size. NNs, via local decomposition and flexible representations, scale more efficiently but require larger and more diverse datasets and careful architectural design to handle highly nonlinear mappings.

5. Advantages, Limitations, and Trade-offs

Aspect	RKHS Methods	Deep Neural Networks (NN)
Analytical gradients	Direct, straightforward	Requires automatic differentiation, but also efficient
Physical asymptotics	Explicit control via kernel design	Requires explicit architectural or loss-based constraints
Scalability (high-D)	Challenged by exponential increase in points	Naturally scalable (size-consistent if atomic decomposition)
Data requirements	Effective with modest datasets for small systems	Large, diverse datasets essential for robustness
Flexibility	Limited by kernel construction	Highly flexible and non-linear
Generalization	Challenging for very high-dimensional or highly irregular systems	Adaptable to larger/heterogeneous systems
Incorporation of invariances	Intrinsic via kernel construction	Intrinsic via descriptors and architecture (if constructed properly)

Speech marks and bolding omitted as per fact-only requirement.

6. Outlook and Synthesis

Both RKHS and NN-based machine-learning potentials have established themselves as essential tools for constructing accurate PESs that support detailed simulations of chemical reactivity and molecular dynamics. The RKHS formalism anchors the construction of globally smooth, physically constrained potentials when enough reference data and reasonable dimensionality permits; NNs offer flexible and extensible mappings capable of scaling to high-dimensional, complex molecular configurations, especially when equipped with symmetrized, local, or learned descriptors. The choice between the two depends on system size, available reference data, need for explicit asymptotics, and computational constraints.

Continued progress is expected in hybridizing physical insight (long-range kernels, analytic invariants) with deep learning advances to improve accuracy, efficiency, and transferability across increasingly complex chemical spaces. Both frameworks underpin the modern development of PES representations that are fundamentally shifting the scale and fidelity of tractable atomistic simulation.