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State Predictive Information Bottleneck

Dedi Wang, Pratyush Tiwary

arXiv:2011.10127v2physics.chem-phcond-mat.stat-mechphysics.bio-phphysics.comp-ph

TL;DR

High-dimensional MD trajectories make it challenging to identify reaction coordinates that distinguish metastable states and capture slow dynamics. SPIB predicts future metastable states from a learned bottleneck representation; the authors report a connection to the committor and useful control of state coarse-graining through the time delay.

  • Problem

    High-dimensional molecular dynamics trajectories create a challenge in finding low-dimensional reaction coordinates that distinguish metastable states and capture slow dynamics.

    Source: §I · p. 1

  • Method

    SPIB learns a reaction coordinate by compressing trajectory information into a bottleneck representation that predicts future discrete states.

    Source: §I · p. 2

  • Results

    The learned bottleneck is related to the committor and can identify transition states, while the time delay controls the degree of metastable-state coarse-graining.

    Source: §IV · p. 10

  • Takeaways & Limitations

    The authors conclude that SPIB can provide an accurate and understandable description of metastable-state interconversion dynamics.

    Source: §I · p. 2

    Interpreting the learned reaction coordinate is not trivial, and the authors propose using representation-learning approaches to address this issue.

    Source: §IV · p. 11

Abstract

from arXiv · show

The ability to make sense of the massive amounts of high-dimensional data generated from molecular dynamics (MD) simulations is heavily dependent on the knowledge of a low dimensional manifold (parameterized by a reaction coordinate or RC) that typically distinguishes between relevant metastable states and which captures the relevant slow dynamics of interest. Methods based on machine learning and artificial intelligence have been proposed over the years to deal with learning such low-dimensional manifolds, but they are often criticized for a disconnect from more traditional and physically interpretable approaches. To deal with such concerns, in this work, we propose a deep learning based State Predictive Information Bottleneck (SPIB) approach to learn the RC from high dimensional molecular simulation trajectories. We demonstrate analytically and numerically how the RC learnt in this approach is deeply connected to the committor in chemical physics, and can be used to accurately identify transition states. A crucial hyperparameter in this approach is the time-delay, or how far into the future the algorithm should make predictions about. Through careful comparisons for benchmark systems, we demonstrate that this hyperparameter choice gives useful control over how coarse-grained we want the metastable state classification of the system to be. We thus believe that this work represents a step forward in systematic application of deep learning based ideas to molecular simulations in a way that bridges the gap between artificial intelligence and traditional chemical physics.

I. INTRODUCTION

Molecular dynamics produces high-dimensional trajectories, and low-dimensional reaction coordinates can make the data more understandable and capture slow dynamics. The introduction motivates SPIB as an information-bottleneck approach that predicts future states while seeking a reaction coordinate connected to the committor.

I. INTRODUCTION

  • Committor probability pB(x) is the probability that a trajectory from conformation X reaches product state B before reactant state A.Committor distributions have been used to study phenomena including ion solvation and biomolecular isomerization,2,4–6Source: §I · p. 1
  • Transition path sampling (TPS) samples pathways between metastable states, and subsequent reaction-coordinate methods can depend heavily on human intuition to generate trial coordinates.3,7,8,9,10,11Source: §I · p. 1
  • Slow-mode approaches including diffusion maps, VAC/TICA, VAMPnets, and SGOOP seek interpretable reaction coordinates, but optimization can be difficult without simplifying assumptions.12–18Examples include representing coordinates as linear combinations of pre-selected order parameters in TICA and SGOOP.16,18Source: §I · p. 1
  • SPIB learns a reaction coordinate from MD trajectories and connects its bottleneck variable to the committor, with the goal of identifying transition states and metastable states.Its discrete-state representation is learned during training and focuses the coordinate on state-to-state transitions.Source: §I · p. 2

A. Information Bottleneck

  • The information bottleneck seeks a representation that uses minimal information from input X to predict target y, trading prediction capacity I(z, y) against model complexity I(X, z).Direct optimization is computationally expensive; a variational lower bound is used, with β controlling the trade-off.21,23Source: A. Information Bottleneck · p. 2

B. State Predictive Information Bottleneck

SPIB adapts the information bottleneck to predict a trajectory’s future metastable state rather than its exact future configuration. The resulting coordinate targets state transitions, while a learned multimodal prior supports separation among states.

B. State Predictive Information Bottleneck

  • SPIB predicts a future state drawn from a learned dictionary instead of the exact future molecular configuration, allowing state locations and counts to be estimated on the fly.The future-state target is lower-dimensional than the full configuration.Source: B. State Predictive Information… · p. 3
  • By simplifying prediction to future state labels, SPIB captures motion related to transitions between states while ignoring fluctuations within metastable states.Source: B. State Predictive Information… · p. 3
  • The SPIB objective predicts the state label at time n+s from a representation of Xn, where the interval between observations is the time delay ∆t.Source: B. State Predictive Information… · p. 3
  • The trajectory is represented by order parameters or features, and mutually exclusive state labels are encoded as one-hot vectors for a softmax decoder.The encoder is a neural network with multivariate Gaussian output, with mean and variance determined from input Xn.Source: B. State Predictive Information… · p. 3
  • SPIB uses a variational mixture of posteriors prior (VampPrior), a weighted mixture of posteriors intended to provide a multimodal latent prior.When the number of metastable states is unknown, the method recommends choosing a large enough number K of pseudo-inputs to make the prior more flexible.Source: B. State Predictive Information… · p. 3

C. Dependence of SPIB on ∆t

SPIB’s time delay sets the temporal resolution of the learned reaction coordinate by suppressing dynamics faster than the delay. Selecting a delay between within-state relaxation and state-to-state transitions can isolate the slow processes of interest.

C. Dependence of SPIB on ∆t

  • SPIB interprets ∆t as the minimum time resolution to retain, because dynamical components with timescales comparable to or shorter than ∆t decay exponentially.A zero delay instead reduces the task to clustering configurations, while positive delays can filter out fast modes.Source: C. Dependence of SPIB on ∆t · pp. 3–4
  • The simplifying assumption is that fluctuations within each metastable state are much faster than transitions between states, enabling future-state prediction to focus on state transitions.Source: C. Dependence of SPIB on ∆t · p. 4
  • An appropriate delay satisfies tm+1 < ∆t ≪ tm, screening out fast within-state processes while retaining the first m slow state-to-state transitions.Robustness across different ∆t values can be checked empirically.Source: C. Dependence of SPIB on ∆t · p. 4

D. Discrete-State Representation and Iterative Retraining Algorithm

SPIB iteratively refines an initial discrete-state labeling using predictions of the state most likely after a chosen time delay. The procedure retrains until the RC, state-transition density, and labels converge, yielding a dynamics-based coarse-grained description.

D. Discrete-State Representation and Iterative Retraining Algorithm

  • Because ∆t is shorter than a typical metastable-state escape time, configurations are expected to retain their state’s highest predicted probability.This assumption motivates the iterative label-refinement rule.Source: D. Discrete-State Representation and… · p. 4
  • SPIB updates each configuration’s label to the state predicted as most probable after delay ∆t, starting from an arbitrary initial set of labels.The predictor approximates the optimal state-transition density K*(X; ∆t).Source: D. Discrete-State Representation and… · p. 4
  • Refinement can leave some initially assigned labels empty, so the initial state count need not persist.The procedure is described as learning the number and locations of states with minimal human intervention.Source: D. Discrete-State Representation and… · p. 4
  • The converged representation is expected to depend on system dynamics and ∆t, screening out transient intermediate states to retain long-lived metastable states.This yields a dynamics-based coarse-grained description.Source: D. Discrete-State Representation and… · p. 4
  • The iterative scheme alternates predictor training with label refinement until the RC, state-transition density, and state labels converge.The algorithm samples minibatches, updates network parameters and pseudo-inputs, then refines labels.Source: D. Discrete-State Representation and… · pp. 4–5

E. State-Transition Density and Committor

SPIB’s state-transition density provides a route to approximate the committor and identify transition-state regions without requiring a large, finely sampled Markov state model. Under timescale separation, the learned representation is connected to the committor and serves as a reaction coordinate.

E. State-Transition Density and Committor

  • SPIB can estimate an approximate committor and identify correct transition-state regions using relatively few discrete states compared with MSM approaches.MSM committor estimation usually needs many states and well-sampled transitions to estimate its transition matrix accurately.Source: E. State-Transition Density and… · p. 5
  • When state transitions are separated in timescale from within-state fluctuations, the transition density can be factored into state equilibrium densities and state-transition densities.The state-transition density is the function K*(X; ∆t) defined earlier in the paper.Source: E. State-Transition Density and… · p. 5
  • Configurations with approximately equal probabilities of reaching two different states after ∆t can be identified as transition-state ensemble members.This state-transition-density criterion is presented as valid like the committor-based definition; the ensemble forms boundaries of the converged discrete states.Source: E. State-Transition Density and… · p. 5
  • The encoder’s informative representation of the learned state-transition density is proposed as a reaction coordinate approximating the committor.For comparison with deterministic prescriptions, the paper uses the representation mean µ as its RC.Source: E. State-Transition Density and… · p. 5

A. Model Systems

The benchmark systems comprise two analytical two-dimensional potentials and alanine dipeptide in vacuum. Their equilibrium trajectories provide the data used to train and test SPIB.

A. Model Systems

  • Benchmarks cover a two-dimensional double-well potential, a two-dimensional four-well potential, and alanine dipeptide in vacuum.The double-well and four-well potentials are shown in Figs. 2 and 5, respectively.Source: A. Model Systems · p. 6
  • Each analytical potential uses a 60,000-time-unit equilibrium trajectory generated by Langevin dynamics at inverse temperature (kBT)^−1 = 3.0 and friction coefficient γ = 4.0.The integration timestep is 0.001 units, with trajectory resolution 0.01 units.Source: A. Model Systems · p. 6
  • Alanine dipeptide simulations use an 800 ns equilibrium trajectory at 450 K, generated with GROMACS 5.0 patched with PLUMED 2.4.The integration timestep is 2 fs, and trajectory resolution is 0.01 ps.Source: A. Model Systems · p. 6

B. Neural network architecture and training

SPIB uses fully connected nonlinear neural networks for both its encoder and decoder. Training uses Adam, while state labels are periodically refined during training.

B. Neural network architecture and training

C. Double-Well Analytical Potential

On the double-well potential, SPIB recovers the state boundary while suppressing noise and within-state fluctuations. The learned state definition remains robust across a broad range of initial boundaries and time delays, subject to stated limits.

C. Double-Well Analytical Potential

D. Four-Well Analytical Potential

On a four-well potential without prior knowledge of the number of metastable states, SPIB learns a discrete-state representation whose coarseness depends on prediction time delay. The delay filters fast dynamics, while a broad range of delays can preserve the same representation.

D. Four-Well Analytical Potential

  • With no prior state-count information, SPIB recovers the correct labels at ∆t = 0.5, then merges C/D at ∆t = 2 and A/B at ∆t = 10.The initial labels discretize the input space finely, with K = 10 and D = 10 despite four actual metastable states.Source: Figure 6 · p. 7, D. Four-Well Analytical Potential · pp. 7–8
  • A broad range of time delays yields the same discrete-state representation, indicating robustness to ∆t selection.The fractional populations of the learned states are plotted to show this robustness.Source: Figure 7 · p. 8, D. Four-Well Analytical Potential · p. 8
  • The implied timescales are t2 = 151.7, t3 = 5.3, and t4 = 1.5; their eigenvectors represent distinct transition modes and the stationary distribution.The second eigenvector mainly represents A-to-D transitions, the third A-to-B, and the fourth C-to-D.Source: Figure 8 · p. 8
  • Increasing ∆t coarse-grains the system from four states to three and then two by filtering out fast dynamical modes.The state mergers reflect the different transition timescales between states.Source: D. Four-Well Analytical Potential · p. 8

E. Alanine Dipeptide in Vacuum

SPIB learns alanine-dipeptide metastable states at different time resolutions and a one-dimensional reaction coordinate whose transition-state region has a committor centered near 0.5. The learned state-transition density also indicates which dihedral coordinates matter for the transition.

E. Alanine Dipeptide in Vacuum

  • The alanine-dipeptide trajectory was represented by φ, ψ, θ, and ω, with initial state labels defined by dividing φ into 10 grids.The analysis used a 2D RC, K = 10 pseudo-inputs, D = 10 state-label dimensions, and β = 0.01.Source: E. Alanine Dipeptide in Vacuum · p. 8, Figure 9 · p. 8
  • At ∆t = 0.5 ps, SPIB yields three states (f0 = 0.61, f2 = 0.37, f6 = 0.02), while ∆t = 2 ps yields two (f0 = 0.98, f6 = 0.02).The two top-left minima become dynamically indistinguishable when their interconversion time approaches the 2-ps delay.Source: E. Alanine Dipeptide in Vacuum · pp. 8–9, Figure 10 · p. 9
  • SPIB’s 2D RC free-energy surface shows a higher barrier between Ceq and Cax than between the two local minima within Ceq, consistent with distinct interconversion timescales.The reported converged implied timescales are t2 = 79 ps and t3 = 4.2 ps.Source: Figure 10 · p. 9, E. Alanine Dipeptide in Vacuum · p. 9, Figure 12 · p. 9
  • The same metastable-state labels persist across a broad range of time delays, making the coarser SPIB classification robust.The apparent label flip between states 0 and 2 for ∆t > 2 ps does not change the metastable states obtained.Source: Figure 11 · p. 9, E. Alanine Dipeptide in Vacuum · p. 9
  • pCax ∼0.5 is the single committor peak near the SPIB transition state RC = −1.92, supporting identification of the correct transition state.The committor was estimated from 50 1-ps trajectories per configuration under −2.22 < RC < −1.62.Source: Figure 13 · p. 10, E. Alanine Dipeptide in Vacuum · p. 10
  • In the learned transition density to Cax, φ and θ are both needed to identify the transition state, while ψ is nearly irrelevant.The φ–ψ projection shows transition states in the TS region aligned almost parallel to the ψ axis.Source: Figure 13 · p. 10, E. Alanine Dipeptide in Vacuum · p. 10

IV. DISCUSSION

The discussion presents SPIB as a way to learn reaction coordinates and state-transition behavior, with a conditional theoretical connection to the committor. It identifies RC interpretability as a remaining challenge and enhanced sampling as future work.

IV. DISCUSSION

  • The prediction delay sets the dynamical time resolution and controls the degree of metastable-state coarse-graining SPIB learns.SPIB automatically indexes the high-dimensional state space through iterative retraining and characterizes state dynamics with state-transition density.Source: §IV · p. 10
  • SPIB’s bottleneck variable can in principle be equivalent to the committor when state-to-state transitions are timescale-separated from fluctuations within metastable states.Numerical benchmark tests support state-transition density as a reasonable committor substitute and show that the RC can identify transition states.Source: §IV · p. 10
  • Because SPIB uses a nonlinear deep-neural-network encoder, interpreting its learned RC is not trivial; the authors identify representation learning as future work.Source: §IV · pp. 10–11
  • Enhanced sampling is proposed as future work and would involve reweighting biased input trajectories, for example trajectories biased along a trial RC.Source: §IV · p. 10
  • The authors frame SPIB as a step toward understanding complex systems and a tool that may be useful across scientific communities.Source: §IV · p. 11

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