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Learning interpretable continuous-time models of latent stochastic dynamical systems

Lea Duncker, Gergo Bohner, Julien Boussard, Maneesh Sahani

arXiv:1902.04420v1stat.MLcs.LGmath.DS

TL;DR

Learning interpretable continuous-time stochastic dynamics from indirect, noisy, unevenly sampled observations is challenging, and flexible models can leave interpretation to a separate analysis stage. The paper conditions a sparse variational Gaussian-process drift model on learned fixed points and local Jacobians while inferring latent paths. On simulated nonlinear systems, it recovers meaningful fixed points and nearby dynamics, with inferred models approximating the true systems over large state-space regions.

  • Problem

    Identifying nonlinear continuous-time SDEs from intermittent, indirect, noisy observations is challenging, and flexible dynamics models may leave interpretation to a separate analytic stage.

    Source: §1 · p. 1

  • Method

    The method jointly infers latent paths and a continuous-time SDE whose GP drift is conditioned on learned fixed points and local Jacobians.

    Source: §1 · p. 1

  • Results

    On simulated systems, the method recovered meaningful fixed points and nearby dynamics even with sparse data, and approximated true dynamics over large state-space regions.

    Source: §6 · p. 8

  • Takeaways & Limitations

    The conditioned GP representation provides a dynamical portrait based on fixed points and local Jacobians without requiring a specific parametric dynamical form.

    Source: §6 · p. 8

    Demonstrations use unevenly sampled multivariate Gaussian and point-process observations; extension to other processes, including marked point processes, is proposed rather than demonstrated.

    Source: §6 · p. 8

Abstract

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We develop an approach to learn an interpretable semi-parametric model of a latent continuous-time stochastic dynamical system, assuming noisy high-dimensional outputs sampled at uneven times. The dynamics are described by a nonlinear stochastic differential equation (SDE) driven by a Wiener process, with a drift evolution function drawn from a Gaussian process (GP) conditioned on a set of learnt fixed points and corresponding local Jacobian matrices. This form yields a flexible nonparametric model of the dynamics, with a representation corresponding directly to the interpretable portraits routinely employed in the study of nonlinear dynamical systems. The learning algorithm combines inference of continuous latent paths underlying observed data with a sparse variational description of the dynamical process. We demonstrate our approach on simulated data from different nonlinear dynamical systems.

1. Introduction

The paper addresses learning continuous-time stochastic dynamics from indirect, noisy observations while avoiding the interpretation gap of flexible models. It represents dynamics through a Gaussian process conditioned on learned fixed points and local Jacobians, and jointly infers latent paths and model parameters.

1. Introduction

  • Intermittent, indirect, noisy observations make nonlinear continuous-time SDEs difficult to identify, while discrete-time state-space models assume a grid can represent observations and critical dynamics.Flexible nonparametric or recurrent descriptions may predict effectively but leave interpretation to a separate analytic stage.Source: §1 · p. 1
  • The model uses a nonlinear SDE for latent dynamics and noisy, unevenly sampled observations whose expected values follow g(Cx(t_i)+d).The task is to infer latent paths alongside dynamical parameters and an interpretable representation of the drift mapping.Source: §1 · p. 1
  • A Gaussian-process drift conditioned on learned fixed points and local Jacobians makes the dynamical portrait itself the interpretable representation being optimized.This implicitly integrates over details of the drift function rather than requiring fixed points to be found numerically after fitting a general function.Source: §1 · p. 1

2. Background

The background reviews discrete-time GP state-space models, variational inference for continuous-time SDEs, and sparse GP inducing points. These provide context for the paper’s continuous-time latent-path inference and sparse variational treatment of dynamics.

2.1. Gaussian Process State-Space-Model

  • A GP state-space model (GP-SSM) models latent evolution over a fixed time step, whereas this paper retains continuous-time structure for latent-path inference.Approximate inference methods for GP-SSMs include assumed density filtering, expectation propagation, variational inference, and recurrent recognition networks.Source: §2.1 · p. 2

2.2. Gaussian Process Approximation to SDEs

  • Prior variational SDE inference uses a Markov-structured Gaussian approximation to the latent posterior, represented through a linear time-varying SDE or an equivalent GP.Its free-energy optimization yields fixed-point updates that may be closed-form or require forward and backward ODE solutions; the paper extends this framework.Source: §2.2 · p. 2

2.3. Sparse Gaussian Processes using inducing points

  • Sparse GP inducing-point methods condition a GP on pseudo-observations at selected locations, with computation cubic in the number of inducing points and linear in input points.The inducing points can remain auxiliary variational variables; integrating them out recovers the original model.Source: §2.3 · p. 3

3. Interpretable priors on nonlinear dynamics

The paper makes nonlinear drift dynamics interpretable by conditioning a Gaussian-process prior on fixed points and local Jacobians. It uses sparse inducing points for inference and optimizes uncertainty parameters to select the number of fixed points.

3. Interpretable priors on nonlinear dynamics

  • Fixed points and locally linearized Jacobians are standard objects for analyzing nonlinear dynamics, motivating an interpretable prior over the drift function.The fixed points satisfy f(s_i)=0, and the Jacobians describe local behavior around them.Source: §3 · p. 3

3.1. A Gaussian Process prior for dynamics

  • The drift GP prior is conditioned directly on fixed-point locations and their Jacobians, which become hyperparameters specifying its mean and covariance.The model augments this conditioned prior with inducing points for a sparse variational approximation.Source: §3.1 · p. 3

3.2. Automatic selection of the number of fixed points

  • To handle an unknown number of fixed points, the method introduces extra candidates and optimizes a noise variance for each point to prune superfluous ones.Large optimized uncertainty makes conditioning on a superfluous point have essentially no effect on drift predictions.Source: §3.2 · p. 3

4. Variational inference and learning

The learning procedure alternates variational inference for the latent path with updates to the drift GP, observation mapping, and hyperparameters. Its inference stage uses coupled ODEs and symmetric covariance variations, while model parameters receive closed-form or optimization-based updates as available.

4. Variational inference and learning

  • The variational Bayes algorithm alternates latent-path inference, learning the drift and affine output mapping, and hyperparameter learning.The variational distribution factorizes over the latent path and drift-related variables, with a variational free energy as the optimization objective.Source: §4 · p. 4

4.1. Inference

  • Latent inference extends earlier SDE variational methods to broader observation models and a nonparametric conditioned drift GP, solving coupled ODEs for posterior moments and update terms.Symmetric covariance variations modify the fixed-point equations and are reported to improve numerical stability; the ODEs are solved forward and backward on a time grid.Source: §4.1 · p. 4

4.2.1. DYNAMICS

  • Given latent posterior quantities, the drift stage has closed-form updates for Jacobians and variational parameters associated with the inducing points.The updates include inducing-point and Jacobian parameters jointly, with one-dimensional integrals computable by methods such as Gauss–Legendre quadrature.Source: §4.2.1 · pp. 4–5

4.2.2. OUTPUT MAPPING

  • The output-mapping parameters C and d enter through the expected log-likelihood; Gaussian observations admit optimal updates, while other models may require optimization.For observation models without closed-form solutions, the free energy can be maximized using standard optimization approaches.Source: §4.2.2 · p. 5

4.2.3. HYPERPARAMETERS

  • Kernel hyperparameters and fixed-point locations are learned by direct variational-free-energy optimization, while inducing-point locations are held fixed on a chosen grid in the reported examples.Source: §4.2.3 · p. 5

5. Experiments

Across simulated systems, the method inferred latent trajectories and interpretable dynamics from noisy observations, recovering key qualitative structures in the double-well, limit-cycle, and neural examples.

5.1. Double-well dynamics

  • The double-well example recovered the latent SDE path and dynamics, automatically selecting three fixed points from a prior conditioned on four.The inferred model parameters C and d remained near their good initial values.Source: §5.1 · p. 5, Figure 1 · p. 6

5.2. Van der Pol’s oscillator

  • The learned dynamics successfully captured the generative Van der Pol limit cycle around an unstable fixed point.Data came from 20 repeated trials with 20 output dimensions and Gaussian measurement noise of unknown variance 2.25.Source: §5.2 · p. 5, Figure 2 · p. 7

5.3. Neural population dynamics

  • Across both neural regimes, the method recovered their main qualitative dynamical features; extra hypothesized fixed points became uncertain or moved to sparsely observed regions.The regimes were two stable and one unstable fixed point, or a single stable spiral.Source: §5.3 · p. 6, Figure 3 · p. 7

5.4. Multistable chemical reaction dynamics

  • For the iodine reaction example, Figure 4 presents true and learned concentration dynamics alongside spectroscopy measurements and inferred latent paths.The data comprised 20 trials with different initial conditions and 50 unevenly spaced samples from 13 spectroscopy measurements per trial.Source: §5.4 · pp. 6–7, Figure 4 · p. 8

6. Discussion

The approach recovers fixed points and nearby dynamics from sparse simulated data while approximating the true systems over large state-space regions. It combines continuous-time latent-process inference with GP dynamics conditioned on fixed points and local Jacobians, and the discussion describes its scope and distinctions from related approaches.

6. Discussion

  • Conditioning a nonparametric GP on fixed points and local Jacobians yields a flexible dynamics model with a directly interpretable dynamical portrait.The framework integrates over the dynamics rather than imposing a specific parametric form.Source: §6 · p. 8
  • Across simulated physical, biological, and chemical systems, the method recovered fixed points and nearby dynamics from sparse data and approximated true dynamics over large state-space regions.The evaluation used realistic but known nonlinear dynamical systems.Source: §6 · p. 8
  • Continuous-time inference represents the variational posterior over latent paths with coupled ODEs, whose numerical solution can use adaptive step sizes.Any discretisation error is described as a numerical issue tied to the ODE solver, rather than an assumption of a discretised model.Source: §6 · p. 8
  • Independent GPs for each latent dimension simplify finite-point inference but cannot capture cross-dimensional latent correlations, reducing descriptive power [Damianou et al., 2011].Source: §6 · p. 8
  • Unlike approaches requiring known dynamics or a known parameterization, this method uses nonparametric GP dynamics while treating the entire SDE as latent; its fixed-point/Jacobian representation is novel.Batz et al. (2018) model the drift with a GP but assume direct observations of the SDE path; Archambeau et al. (2007; 2008) use known dynamics or a known parameterization.Source: §6 · p. 8
  • The algorithm was demonstrated with unevenly sampled multivariate Gaussian and point-process observations, while marked point processes are described as a possible extension.The authors anticipate applications in neuroscience, chemistry, and finance.Source: §6 · p. 8

A. Variational Lower Bound

The section derives a variational lower bound and expresses it using a factorized variational distribution; it also describes discretizing SDE paths to evaluate a path-distribution KL divergence.

A. Variational Lower Bound

B.1. Lagrangian

The section presents the variational free-energy Lagrangian and separates expected log-likelihood contributions according to whether they vary continuously in time or occur at observation times.

B.1. Lagrangian

  • The full Lagrangian is obtained after integration by parts is applied to the constraints.Source: §B.1 · p. 10
  • The variational free-energy expression includes a term denoted F* and Kullback–Leibler divergences.Source: §B.1 · p. 10
  • Expected log-likelihood terms are divided into continuous-in-state contributions and contributions depending only on latent-state evaluations at observation times.Source: §B.1 · pp. 10–11

B.1.1. EXAMPLE: GAUSSIAN LIKELIHOOD

The appendix gives likelihood examples and supporting variational-inference derivations, including symmetric covariance variations, expectations of dynamics, and updates for several dynamics and output models.

B.1.1. EXAMPLE: GAUSSIAN LIKELIHOOD

  • The Gaussian-likelihood example has no continuous term in the likelihood.Source: §B.1.1 · p. 11

B.1.2. EXAMPLE: MULTIVARIATE POISSON PROCESS LIKELIHOOD

  • The multivariate Poisson-process example uses an exponential inverse-link function and observed event times.Source: §B.1.2 · p. 11

B.2. Symmetric variations in Sx

  • The derivation varies the symmetric covariance matrix Sx symmetrically and accounts for its symmetry in the free-energy derivatives.The text contrasts this treatment with Archambeau et al. (2007).Source: §B.2 · p. 11

B.3. Expected values of dynamics

  • The inference algorithm requires expectations under qx and qf; covariance-function expectations and derivatives can be computed analytically for choices such as the exponentiated quadratic covariance.Source: §B.3 · p. 11

B.4. Inference algorithm

  • The inference algorithm solves ODEs forward and backward with the forward Euler method on a time grid, where event-time delta-function factors cancel the corresponding step-size terms.Source: §B.4 · p. 11

C.1. Conditioned Sparse Gaussian Process dynamics

  • The conditioned sparse-GP derivation gives updates involving inducing-point and Jacobian blocks, with one-dimensional integrals evaluated by Gauss–Legendre quadrature.Source: §C.1 · p. 12, §C.1.2 · pp. 12–13

C.2. Sparse Gaussian Process dynamics

  • For classic sparse-GP dynamics without fixed-point and Jacobian conditioning, closed-form updates are available.Source: §C.2 · p. 13

C.3. Linear dynamics; C.4. Output mapping

  • The appendix also gives closed-form updates for continuous-time linear dynamics and derives output-mapping updates by differentiation.Source: §C.3 · p. 13, §C.4 · p. 13

D. Chemical reaction dynamics

The section identifies the chemical reaction dynamical system used to generate data and refers to the simulation parameter settings.

D. Chemical reaction dynamics

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