Learning Collective Variables and Coarse Grained Models

Principal Components Analysis/Karhunen-Loewe expansion (PCA) is a time honored method to visualize, understand or simply reduce data size. Why, then the need for other, non-linear dimension reduction methods? And why are there so many of them? This tutorial will explain how to navigate the landscape of modern dimension reduction algorithms like Isomap, Diffusion Maps, t-SNE and UMAP. The focus wil be on how to safely interpret these algorithms’ results in practice. This will be illustrated with data from MD simulations.

Thanks to Stefan Chmiela and Chris Fu, and to the Tkatchenko and Pfaendtner labs

Molecular dynamics is a powerful tool for studying the thermodynamics and kinetics of complex molecular events. However, these simulations can rarely sample the required time scales in practice. Transition path sampling overcomes this limitation by collecting unbiased trajectories and capturing the relevant events. Moreover, the integration of machine learning can boost the sampling while simultaneously learning the committor, a quantitative representation of the mechanism. Still, the resulting trajectories are by construction non-Boltzmann-distributed, preventing the calculation of free energies and rates. We developed an algorithm to approximate the equilibrium path ensemble from machine-learning-guided path sampling data. At the same time, our algorithm provides efficient sampling, mechanism, free energy, and rates of rare molecular events at a very moderate computational cost. We tested the method on the folding of the mini-protein chignolin. Our algorithm is straightforward and data-efficient, opening the door to applications in many challenging molecular systems.

Discovering optimal reaction coordinates and accurately predicting kinetic rates for activated processes are two of the major challenges in molecular dynamics simulations. In this talk, we first address the challenge of transition rate prediction constructing accurate coarse-grained models of the dynamics of complex systems projected on a collective variable. We propose an algorithm for parameter estimation of Langevin equations from transition path sampling (or other unbiased) trajectories: by maximizing the model likelihood based on any explicit expression of the short-time propagator, this methodology efficiently reconstructs free energy landscapes, diffusion coefficients, and kinetic rates. Second, we use this algorithm and a variational principle that states that the optimal reaction coordinate is the one that minimizes the projected dynamics, to automatically optimize collective variables: we obtain Langevin models from linear combinations of trial collective variables and iteratively adjust and evaluate trial reaction coordinates until the model’s rate is minimized. This framework provides quantitative predictions of kinetic rates and optimal reaction coordinates for both benchmarks and complex systems at a competitive computational cost.

The effectiveness of enhanced sampling techniques that apply accelerating biases to selected collective variables (CVs) to drive free energy barrier crossing and rare events depends sensitively on the quality of the chosen CVs. This talk will review the mathematical and algorithmic underpinnings of a variety of popular data-driven techniques to learn CVs from simulation trajectories based on principles of maximal variance or autocorrelation and their deployment within iterative CV discovery and enhanced sampling protocols. This will be followed by a presentation of the molecular latent space simulators (LSS) approach to learn highly efficient and accurate surrogate models for the dynamics of molecular systems by stacking three specialized deep learning networks to (i) encode a molecular system into a slow latent space, (ii) propagate dynamics in this latent space, and (iii) generatively decode a synthetic molecular trajectory. The talk will be followed by a short hands-on session demonstrating some of these CV discovery and enhanced techniques on simple toy and molecular systems within user-friendly Python codes packaged into Jupyter notebooks.

In this lecture, I will present results on how to measure quantitatively the quality of a coarse-graining procedure in computational statistical physics. We typically have in mind two types of coarse-graining techniques: starting from dynamics in a high dimensional space (molecular dynamics with a large number of atoms), one would like either to project the dynamics onto a few collective variables, or to approximate the dynamics by a jump Markov models between the metastable conformations. The former approach is related to the construction of effective dynamics using the Mori-Zwanzig formalism, and to the computation of free energy using free energy adaptive biasing procedures (metadynamics, adaptive biasing force). The latter approach is useful to understand the foundation of Markov State models (or kinetic Monte Carlo), and of the accelerated molecular dynamics à la D. Perez and A.F Voter. From a mathematical viewpoint, the former requires functional inequalities (Poincaré or logarithmic Sobolev inequalities) while the latter relies on the notion of quasi-stationary distribution.

Coarse-grained (CG) molecular dynamics enables the study of biological processes at temporal and spatial scales that would be intractable at an atomistic resolution. However, accurately learning a CG force field remains a challenge. In this work, we leverage connections between score-based generative models, force fields and molecular dynamics to learn a CG force field without requiring any force inputs during training. Specifically, we train a diffusion generative model on protein structures from molecular dynamics simulations, and we show that its score function approximates a force field that can directly be used to simulate CG molecular dynamics. While having a vastly simplified training setup compared to previous work, we demonstrate that our approach leads to improved performance across several protein simulations for systems up to 56 amino acids, reproducing the CG equilibrium distribution, and preserving dynamics of all-atom simulations such as protein folding events.

Despite the success of equivariant neural networks in scientific applications, they require knowing the symmetry group a priori. However, it may be difficult to know which symmetry to use as an inductive bias in practice. Enforcing the wrong symmetry could even hurt the performance. In this talk, I will discuss our effort in developing a deep learning framework that can automatically discover symmetry from data.  Our framework, LieGAN, represents symmetry as interpretable Lie algebra basis and uses a paradigm akin to generative adversarial training. We further generalized it LaLieGAN to discover non-linear symmetries from high-dimensional data.  Empirically, the learned symmetry can also be readily used in  existing equivariant neural networks to improve accuracy and generalization in prediction. It can also improve equation discovery and long-term forecasting for various dynamical systems.

Recent develops in computer simulations have shown an explosion of new methods for coarse graining interactions via machine learning (ML) strategies.  Many of the strategies are based on expanding the effective many-body interactions of a reduced set of coordinates into 2- and 3-body contributions.  However, this strategy has significant weaknesses in soft matter and biosystems  as many of the important interactions in these systems are inherently 4+ body. These include, for instance, depletion interactions in colloid-polymer mixtures when the polymers are relatively large, or steric interactions between colloids covered by a thick deformable polymer layer. Intriguingly, in many of these systems much of the many-body character is captured by the local environment of the particles – which is well captured by the particle’s Voronoi cell.  Here we present an ML-strategy to fit depletion interactions based on Voronoi-cell-based local descriptors.  We test this strategy using a 2D colloid-polymer mixture, with ideal interactions between the polymers.  In this simplified case, we find essentially perfect agreement between our ML-model and brute force simulations.  We also present a simple and highly accurate new method to predict phase transitions between fluids and crystals based on direct coexistence simulations. We apply this method to both the brute-force and coarse-grained colloid-polymer mixture and show excellent agreement.

Graph data is ubiquitous in many application domains, including in material science and molecular biology. The rapid advancements in machine learning also lead to different graph learning frameworks, such as message passing (graph) neural networks (MPNNs), graph transformers and higher order variants. In this talk, I will describe some of our recent journeys in attempting to provide better (theoretical) understanding of these graph learning models (e.g, their representation power and limitations in capturing long range interactions in graphs), the pros and cons of different models, and ways to further enhance them in practice. This talk is based on multiple joint work with various collaborators, whom I will mention in the talk.