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Sparse learning of stochastic dynamic equations

Lorenzo Boninsegna, Feliks Nüske, Cecilia Clementi

arXiv:1712.02432v1stat.MLcond-mat.stat-mech

TL;DR

SINDy must be adapted for noisy, non-deterministic dynamics in (macro)molecular systems. The paper presents a stochastic extension with sparse-regression strategies and demonstrates solutions that preserve predictive power while recovering key properties of two example systems.

  • Problem

    Noise is a central obstacle to applying SINDy to (macro)molecular systems because their dynamics are usually non-deterministic.

  • Method

    The paper extends SINDy to derive stochastic dynamical equations from data and uses sparsity constraints and cross-validation-based sparse regression.

  • Results

    The optimal sparse solutions preserve model predictive power, approximate the potential gradient, and reproduce thermodynamics and long-timescale kinetics in the projected dynamics.

  • Takeaways & Limitations

    Cross-validation helps identify effective sparse-regression solutions, while the resulting models retain key static and dynamical properties in the demonstrated systems.

Abstract

from arXiv · show

With the rapid increase of available data for complex systems, there is great interest in the extraction of physically relevant information from massive datasets. Recently, a framework called Sparse Identification of Nonlinear Dynamics (SINDy) has been introduced to identify the governing equations of dynamical systems from simulation data. In this study, we extend SINDy to stochastic dynamical systems, which are frequently used to model biophysical processes. We prove the asymptotic correctness of stochastics SINDy in the infinite data limit, both in the original and projected variables. We discuss algorithms to solve the sparse regression problem arising from the practical implementation of SINDy, and show that cross validation is an essential tool to determine the right level of sparsity. We demonstrate the proposed methodology on two test systems, namely, the diffusion in a one-dimensional potential, and the projected dynamics of a two-dimensional diffusion process.

I. INTRODUCTION

Large datasets often yield descriptive patterns without a functional link to observed behavior or a straightforward physical interpretation. The paper extends SINDy to learn stochastic dynamical equations from data, including projected variables, and emphasizes cross-validation for sparse model selection.

  • Large datasets can reveal patterns but often do not provide a functional link between variables and observed behavior.
  • The paper seeks methods that infer explicit analytical dynamical equations from available data for physical interpretation.
  • SINDy combines sparse regression and compressed sensing to identify differential-equation terms from time-dependent data using a function library.
  • Noise and nondeterministic molecular dynamics limit direct application of deterministic SINDy to molecular systems.
  • The proposed extension derives stochastic equations for microscopic or transformed variables and uses projected stochastic dynamics for reduced representations.
  • Extensive cross-validation is required for successful sparsification when noise is significant or data are limited.

A. Sparse Identification of Dynamical Systems

Deterministic SINDy represents an unknown vector field as a sparse linear combination of library functions fitted to sampled trajectories. Least-squares regression alone is generally dense, so sparsity must be explicitly enforced.

  • SINDy learns an unknown vector field from sampled state trajectories by expressing it as a linear combination of preselected basis functions.
  • Evaluating the function dictionary along the data produces a regression system whose coefficients determine the learned dynamical equations.
  • Least-squares fitting generally produces a non-sparse coefficient vector, despite the goal of identifying a compact functional form.
  • An L1-norm penalty with multiplier ρ enforces sparsity during regression.

B. Sparse Identification of Stochastic Dynamics

The paper extends sparse identification from deterministic equations to Ito diffusion processes by regressing drift and diffusion terms from trajectory variations. It also formulates effective stochastic dynamics for projected variables and proves convergence of the resulting coefficients under increasing data and decreasing time step.

  • The stochastic extension models diffusion-process drift and diffusion as linear combinations of basis functions estimated from trajectory data.
  • Kramers–Moyal expansion estimates drift and diffusion from conditional linear and quadratic variations of the process.
  • B. Sparse Identification of Stochastic Dynamics: Projected dynamics learn an effective Ito stochastic differential equation using projected observations and equilibrium distributions restricted to projection level sets.
  • B. Sparse Identification of Stochastic Dynamics: A memoryless effective SDE is meaningful when projected variables capture the slowest processes and the system has separated timescales.
  • Equilibrium sampling automatically supplies the averages needed for regression of projected drift and diffusion terms.
  • C. Convergence Result: As N →∞ and s →0, the regression solutions converge to the coefficient vectors of the best approximation problems.

C. Learning of Effective Potentials

The paper addresses learning physically interpretable potentials rather than fitting drift components independently. For reversible dynamics, the free-energy relation also applies after projection, supporting effective-potential estimation.

  • For overdamped Langevin dynamics, the regression solution converges to the best approximation of the potential gradient by the chosen basis functions.
  • The projected case is discussed because estimating the full dynamics is a special case of the effective-dynamics problem.
  • Independent drift-component learning may fail to ensure that the estimated field is generated by a scalar potential in high-dimensional systems.
  • The method therefore estimates potential energy directly using a dictionary of differentiable multivariate basis functions.
  • For reversible diffusion processes, a scalar free energy links drift and diffusion, and this relation also holds for effective projected dynamics.

III. METHODS

The paper formulates sparse regression for stochastic drift and diffusion identification, using iterative coefficient removal and cross validation to select sparsity. The resulting Stepwise Sparse Regressor is parameter-free after cross-validation selects the solution size.

  • Sparse representations are sought for the drift and diffusion terms of stochastic systems.
  • The L0-constrained sparse-regression problem is non-convex and NP-hard, motivating L1 relaxations and alternative algorithms.
  • Unlike deterministic SINDy thresholding, the threshold-based approach is reported as not robust for stochastic cases.
  • A. Sparsity enforcement: Stepwise Sparse Regressor removes one coefficient per iteration, refits the remaining coefficients, and increases sparsity progressively.
  • A. Sparsity enforcement: SSR repeats coefficient removal until cross validation identifies the optimal iteration and solution size.
  • A. Sparsity enforcement: After cross validation selects the optimally sparse solution, SSR is parameter-free and requires no preliminary training phase.

B. Cross Validation

Cross validation evaluates SSR models across solution sizes by alternating training and test subsets. The selected size balances sparsity and predictive accuracy, while score gaps signal when additional sparsification compromises predictivity.

  • Each model is trained on one subset and evaluated by predicting the held-out subset, with the score averaged across folds.
  • The cross-validation score measures prediction deviation and identifies models with low values as optimal within the model family.
  • The validated model family consists of SSR solutions with different solution sizes n = K −q, or equivalently different sparsities q.
  • K-fold cross validation splits the dataset into k disjoint subsets, each serving alternatively as a test set.
  • Intermediate solutions balance sparsity and accuracy, whereas under- and over-fitting regimes have larger cross-validation scores.
  • A gap between successive score values indicates that increasing sparsity by one unit compromises model predictive power.

IV. HOMOGENEOUS DIFFUSION IN A DOUBLE WELL POTENTIAL

The double-well diffusion experiment applies SSR to recover the drift from trajectory data and compares dictionaries with different basis functions. Cross validation selects a four-term analytic solution for the original dictionary, while both dictionaries capture the double-well potential feature.

  • Five independent trajectories are used to recover the drift of overdamped homogeneous diffusion in a one-dimensional double-well polynomial potential.
  • For dictionary Θ, the cross-validation score remains stable as terms are removed until reducing the solution from n = 4 to n = 3 increases it by several orders of magnitude.
  • The optimal sparsity is n = 4 with CV score ˜δΘ = 1.49 · 10−4, retaining the analytic basis functions [1, x, x2, x3].
  • The n = 5 solution adds a small oscillatory term and approximates the gradient with δ = 7.8·10−5, whereas n = 3 performs poorly with δ = 9 · 10−1.
  • Using dictionary Θ′ removes the clear CV-score gap, placing the candidate optimum between n = 5 and n = 10 and retaining three analytic terms but not f0 = 1.
  • Both dictionaries capture the double-well feature, but Θ′ is less parsimonious and has a larger cross-validation score, so Θ is favored.

A. Greedy search

The greedy search evaluates sparse dictionaries through cross-validation to identify solutions balancing accuracy and parsimony. For the two-well potential, it selects the analytic basis functions as the optimal sparse solution, while deterministic SSR can depend on dictionary choice and become trapped in local minima.

  • Search strategy: Cross-validation compares candidate dictionaries across solution sizes to locate an accuracy–sparsity transition and select the optimal sparse model.The greedy procedure can replace iterative sparse-regression searches by evaluating sampled sub-dictionaries and their cross-validation scores.
  • Search results: Removing one term from the n = 4 solution increases the cross-validation score by several orders of magnitude, indicating that a key component is missing.Sparser n = 3 dictionaries are therefore less accurate than the selected four-term solution.
  • Search results: Solutions with 5 < n < 20 can reproduce the potential gradient comparably, but they are less parsimonious than the four-term solution.The analysis distinguishes equally accurate models by their larger dictionary size.
  • Algorithmic limitations: Deterministic SSR depends on the dictionary, may search the non-convex solution space inefficiently, and can become trapped in sub-optimal local minima.Its performance cannot be reliably inferred from dictionary conditioning alone.
  • Algorithmic limitations: CV-based SSR relaxes to a sparse, though not always sparsest, dictionary-dependent solution that captures the main features of the gradient.This provides a more effective search outcome at the level examined in the paper.

B. Effects of sampling noise on algorithmic performance

The section examines how sampling noise affects sparse-regression convergence. Reduced sampling noise improves recovery of the optimal solution, but performance remains dictionary-specific and high-dimensional applications may require a more robust search algorithm.

  • Assumptions: Finite trajectories only approximately satisfy the Boltzmann-distribution assumption underlying the proof of Theorem 1.This creates a scope boundary between the asymptotic argument and finite-sample data.
  • Sampling error: The median sampling error in the trajectory data is approximately 10−2, with the largest deviation near the energy barrier.The error could be reduced using longer trajectories or lower temperatures.
  • Convergence: As sampling noise decreases, the percentage of dictionaries converging to [1, x, x2, x3] increases, although some remain sub-optimal even without noise.The experiment evaluates 100 randomly selected dictionaries containing the analytic basis functions.
  • Convergence: Sampling quality influences whether SSR becomes trapped in a local minimum, and the effect of noise reduction depends on the dictionary.Some dictionaries reach the global minimum only at extremely low noise levels.
  • Practical boundary: Relevant high-dimensional systems are expected to have large sampling errors, motivating a more robust algorithm for searching the solution space.The section therefore identifies sampling quality as a practical limitation for SSR.

V. “LEARNING” A PROJECTED DYNAMICS

The paper learns sparse drift and diffusion representations for a two-dimensional diffusion projected onto an effective angular coordinate. Cross-validation isolates one-term analytic solutions, and the learned projected dynamics recover the original system’s thermodynamics and long-timescale kinetics.

  • Projected coordinate: The polar angle φ is used as an effective coordinate because it correlates with the first seven eigenvectors describing basin-hopping motions.The benchmark system is the two-dimensional lemon slice potential with κ = 7 minima.
  • Model comparison: Figure 7 compares actual and sparse-represented drift and diffusion, alongside estimates obtained from sampling.Drift appears in the upper panels and diffusion in the lower panels.
  • Sparse learning: Cross-validation identifies solution size n = 1 as optimal for both drift and diffusion.The minimum cross-validation score occurs at n = 1 for each quantity.
  • Sparse learning: ˜b(φ) = 6.39 sin(7φ) achieves δ = 4.0 · 10−2, while ˜a(φ) = 0.95 achieves δ = 2.0 · 10−4.These one-term solutions isolate the analytic basis functions for the drift and diffusion.
  • Validation: The learned projected dynamics accurately recover the first few timescales and agree with the projected equilibrium distributions of the original dynamics.Together, these results recover the full thermodynamics and long-timescale kinetics of the original dynamics.

VI. DISCUSSION

The discussion finds that cross-validation-based sparse regression can recover key stochastic dynamics, while dictionary choice, collinearity, pruning, and sampling errors constrain performance and interpretation.

  • Method and overall performance: The SSR method appears effective for learning stochastic dynamical equations, with cross-validation identifying the sparsity level that preserves predictive power.The optimal solution maximizes sparsity while preserving model predictive power.
  • Test systems: The reconstructed one-dimensional dynamics closely approximate the potential gradient and recover key drift and diffusion features.These features are reported for the double-well potential example.
  • Test systems: The sparse solution for the lemon-slice projected dynamics reproduces the original model’s thermodynamics and long-timescale kinetics.The comparison is shown in Figs. 7c, 7f, and 8.
  • Limitations: The method’s performance and selected sparse solution depend on the basis-function dictionary, with collinearity allowing multiple nearly equivalent solutions driven by numerical noise.Ill-conditioned dictionaries exacerbate convergence problems, while physically meaningful dictionary entries can be lost after orthogonalization.
  • Limitations: Deterministic pruning can remove functions permanently and funnel the algorithm into local minima; stochastic pruning with reintroduction is proposed as a possible improvement.The proposed strategy is described as resembling Monte Carlo techniques and may partially compensate for collinearity.
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