Neural Network Approaches for Potential Energy Surface Modeling

Potential energy surfaces (PESs) provide the multidimensional energy landscape that governs molecular structure, dynamics and reactions, yet their accurate construction for systems of realistic size has long been a formidable challenge. Recent years have witnessed the rise of neural network (NN) methodologies that learn PESs directly from quantum-chemical data, overcoming the curse of dimensionality through flexible, non-parametric function approximators. Key advances include symmetry-adapted and permutationally invariant architectures that ensure exact treatment of identical atoms, message-passing schemes that capture many-body interactions via graph representations, and hybrid Gaussian process–NN frameworks that provide uncertainty estimates and facilitate robust extrapolation. Δ-learning approaches have allowed correction of lower-level PESs to high-accuracy benchmarks with minimal additional data, while active-learning and automated sampling pipelines streamline the generation of training sets and enable on-the-fly refinement of the model. Together with efficient descriptors—such as atom-centred symmetry functions and fundamental invariant features—these developments have made it possible to build global full-dimensional PESs for molecules exceeding ten atoms, accurately predict reaction mechanisms, and integrate PES models into molecular dynamics and kinetics simulations across chemistry, materials, and biophysics.

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Neural Network Approaches for Potential Energy Surface Modeling publication trend

The graph below shows the total number of articles in neural network approaches for potential energy surface modeling across all publications each year (not limited to Nature Index journals).

Technical terms

Potential energy surface (PES): A multidimensional function mapping atomic coordinates to potential energy, defining the energetic landscape that governs molecular structure and reaction dynamics.

Neural network (NN): A layered, non-linear model that learns to approximate complex functions—here used to infer PES values from training examples of atomic configurations and their energies.

Permutational invariance: A property of a model whereby its output remains unchanged under the exchange of identical atoms, ensuring exact symmetry in PES predictions.

Gaussian process regression (GPR): A non-parametric Bayesian method that models functions via covariance kernels, providing both predictions and associated uncertainties for values of the PES.

References

1. Accurate fundamental invariant-neural network representation of ab initio potential energy surfaces. National Science Review (2023).
2. Asparagus: A toolkit for autonomous, user-guided construction of machine-learned potential energy surfaces. Computer Physics Communications (2025).
3. Neural network Gaussian processes as efficient models of potential energy surfaces for polyatomic molecules. Machine Learning: Science and Technology (2023).