Machine Learning for Molecular and Materials Simulations

Local representations encode an atom’s neighborhood while preserving invariance to translations, rotations, and permutations of atoms of the same species. We will compare atom-centered symmetry functions (ACSFs), the smooth overlap of atomic positions (SOAP), and the atomic cluster expansion (ACE), discussing their differences, advantages, and limitations.

• J. Behler. Atom-centered symmetry functions for constructing high-dimensional neural network potentials. Journal of Chemical Physics 134, 074106 (2011).
• A. P. Bartók, R. Kondor & G. Csányi. On representing chemical environments. Physical Review B 87, 184115 (2013).
• R. Drautz. Atomic cluster expansion for accurate and transferable interatomic potentials. Physical Review B 99, 014104 (2019).

Invariant message-passing models learn atomic representations by exchanging information between neighboring atoms while keeping their features unchanged under rotations. We will compare SchNet, which uses interatomic distances, with DimeNet, which also includes angles, discussing their differences, advantages, and limitations.

• K. T. Schütt, H. E. Sauceda, P.-J. Kindermans, A. Tkatchenko & K.-R. Müller. SchNet – A deep learning architecture for molecules and materials. The Journal of Chemical Physics 148, 241722 (2018).
• J. Gasteiger, J. Groß & S. Günnemann. Directional Message Passing for Molecular Graphs. International Conference on Learning Representations (2020).

Equivariant message passing uses features that transform with rotations of the atomic structure. We will compare NequIP and MACE, focusing on their message construction and the advantages and limitations of each approach.

• S. Batzner, A. Musaelian, L. Sun, M. Geiger, J. P. Mailoa, M. Kornbluth, N. Molinari, T. E. Smidt & B. Kozinsky. E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials. Nature Communications 13, 2453 (2022).
• I. Batatia, D. P. Kovács, G. N. C. Simm, C. Ortner & G. Csányi. MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields. Advances in Neural Information Processing Systems 35, 11423–11436 (2022).

The Point Edge Transformer (PET) learns atomic interactions without imposing rotational equivariance in its architecture. Direct force models predict forces without computing them as derivatives of a potential energy. We will discuss the advantages and limitations of both approaches for molecular dynamics.

• S. N. Pozdnyakov & M. Ceriotti. Smooth, exact rotational symmetrization for deep learning on point clouds. Advances in Neural Information Processing Systems 36, 79469–79501 (2023).
• F. Bigi, M. F. Langer & M. Ceriotti. The dark side of the forces: assessing non-conservative force models for atomistic machine learning. Proceedings of Machine Learning Research 267, 4384–4414 (2025).

Foundation interatomic potentials are trained on diverse datasets and can be applied or fine-tuned to new systems. We will discuss MACE-MP-0, focusing on its transferability, advantages, and limitations.

• I. Batatia et al. A foundation model for atomistic materials chemistry. The Journal of Chemical Physics 163, 184110 (2025).

Electrostatic interactions can couple atoms beyond the range of local models. We will examine how interatomic potentials describe long-range interactions, charge transfer, and polarization, discussing the advantages and limitations of different approaches.

• W. J. Baldwin, I. Batatia, M. Vondrák, J. T. Margraf & G. Csányi. Design Space of Self-Consistent Electrostatic Machine Learning Interatomic Potentials. arXiv preprint arXiv:2603.14700 (2026).

Free-energy barriers can prevent molecular dynamics simulations from exploring relevant configurations within accessible timescales. Metadynamics adds a history-dependent bias along selected collective variables to promote transitions between metastable states. We will compare the original and well-tempered formulations, discussing the choice of collective variables and the recovery of free-energy differences.

• A. Laio & M. Parrinello. Escaping free-energy minima. Proceedings of the National Academy of Sciences 99, 12562–12566 (2002).
• A. Barducci, G. Bussi & M. Parrinello. Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy Method. Physical Review Letters 100, 020603 (2008).

Collective variables provide a low-dimensional description of molecular configurations. We will compare Deep-LDA, which learns to separate predefined metastable states, with Deep-TICA, which uses time-lagged correlations to learn slow collective modes. The discussion will address their training objectives, data requirements, and use in enhanced sampling.

• L. Bonati, V. Rizzi & M. Parrinello. Data-Driven Collective Variables for Enhanced Sampling. The Journal of Physical Chemistry Letters 11, 2998–3004 (2020).
• L. Bonati, G. Piccini & M. Parrinello. Deep learning the slow modes for rare events sampling. Proceedings of the National Academy of Sciences 118, e2113533118 (2021).

Boltzmann generators use invertible models to transform samples from a simple distribution into molecular configurations. We will examine energy-based and data-based training, and how reweighting is used to estimate equilibrium properties. The discussion will address sampling efficiency and the challenge of sampling multiple metastable states.

• F. Noé, S. Olsson, J. Köhler & H. Wu. Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning. Science 365, eaaw1147 (2019).

For molecular sampling, flow matching trains continuous normalizing flows that transform samples from a simple distribution into molecular configurations. We will examine how rotations and permutations of atoms of the same species enter the model and training objective, discussing their effects on training and sampling efficiency. These flows can serve as Boltzmann generators, but their paths are not molecular dynamics trajectories.

• L. Klein, A. Krämer & F. Noé. Equivariant flow matching. Advances in Neural Information Processing Systems 36, 59886–59910 (2023).
• Background: Y. Lipman, R. T. Q. Chen, H. Ben-Hamu, M. Nickel & M. Le. Flow Matching for Generative Modeling. International Conference on Learning Representations (2023).

FlashMD learns to predict atomic positions and momenta over time intervals longer than conventional molecular dynamics steps. We will compare this approach with Timewarp, which uses learned proposals and a Metropolis–Hastings correction to sample equilibrium configurations. The discussion will address the distinction between reproducing dynamical properties and equilibrium distributions, and the advantages and limitations of each approach.

• F. Bigi, S. Chong, A. Kristiadi & M. Ceriotti. FlashMD: long-stride, universal prediction of molecular dynamics. Advances in Neural Information Processing Systems 38, 98379–98416 (2025).
• L. Klein, A. Y. K. Foong, T. E. Fjelde, B. Mlodozeniec, M. Brockschmidt, S. Nowozin, F. Noé & R. Tomioka. Timewarp: Transferable Acceleration of Molecular Dynamics by Learning Time-Coarsened Dynamics. Advances in Neural Information Processing Systems 36, 52863–52883 (2023).

Coarse-grained models describe molecular systems with fewer degrees of freedom. We will discuss how force matching learns effective interactions from atomistic simulations, using CGnets and a transferable protein model as examples. The discussion will address the choice of representation, transfer to new proteins, and why reproducing equilibrium distributions does not guarantee correct dynamics.

• J. Wang, S. Olsson, C. Wehmeyer, A. Pérez, N. E. Charron, G. de Fabritiis, F. Noé & C. Clementi. Machine Learning of Coarse-Grained Molecular Dynamics Force Fields. ACS Central Science 5, 755–767 (2019).
• Further reading: N. E. Charron et al. Navigating protein landscapes with a machine-learned transferable coarse-grained model. Nature Chemistry 17, 1284–1292 (2025).