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Learning unknown ODE models with Gaussian processes

Markus Heinonen, Cagatay Yildiz, Henrik Mannerström, Jukka Intosalmi, Harri Lähdesmäki

arXiv:1803.04303v1stat.ML

TL;DR

Many systems have governing dynamics that are partially or completely unknown, limiting conventional parametric ODE modeling. npODE learns unknown nonlinear dynamics using Gaussian process vector fields in the exact ODE formalism, and reports state-of-the-art performance on human motion datasets.

  • Problem

    Many systems have partially or completely unknown governing dynamics, and no model was identified for learning nonlinear ODE functions against true forward solutions.

    Source: §1 · pp. 1–2

  • Method

    npODE learns unknown nonlinear differential functions with an augmented Gaussian process vector field, optimizing exact forward-simulated responses against data.

    Source: §1 · p. 2

  • Results

    npODE demonstrates state-of-the-art performance on human motion datasets.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The model accurately learns arbitrary nonlinear continuous-time dynamics from observational data without assumptions about the underlying system, and forecasts into the future.

    Source: §6 · p. 9

    The forecasting comparison is not direct because methods optimize latent spaces separately, and low-dimensional representations necessarily introduce reconstruction errors.

    Source: §5.1 · p. 7

Abstract

from arXiv · show

In conventional ODE modelling coefficients of an equation driving the system state forward in time are estimated. However, for many complex systems it is practically impossible to determine the equations or interactions governing the underlying dynamics. In these settings, parametric ODE model cannot be formulated. Here, we overcome this issue by introducing a novel paradigm of nonparametric ODE modelling that can learn the underlying dynamics of arbitrary continuous-time systems without prior knowledge. We propose to learn non-linear, unknown differential functions from state observations using Gaussian process vector fields within the exact ODE formalism. We demonstrate the model's capabilities to infer dynamics from sparse data and to simulate the system forward into future.

1 Introduction

ODE models are widely used to describe and predict system dynamics, but conventional models require a known parametric driving function. The paper introduces npODE to learn unknown nonlinear differential functions from observations by fitting exact forward-simulated responses.

1 Introduction

  • Dynamical systems models are used across experimental sciences to understand system behavior and predict future states.The governing dynamics may be only partially known or completely unknown.Source: §1 · p. 1
  • Conventional ODE modeling assumes a known parametric driving function, whose parameters are estimated from data.Evaluating system responses requires a computationally intensive forward solution.Source: §1 · p. 1
  • The central gap is the absence of a model that learns nonlinear ODE functions over the state while matching true forward solutions to observations.Prior approaches used limiting approximations, including gradient matching.Source: §1 · p. 2
  • npODE learns arbitrary, a priori unknown nonlinear differential functions from data by optimizing exact forward-simulated responses rather than using gradient matching or another approximation.The method uses a Gaussian process vector field with inducing points and sensitivity equations for gradient computation.Source: §1 · p. 2

2 Nonparametric ODE model

The model represents an unknown ODE vector field with a Gaussian process and fits its forward-simulated trajectories to noisy observations. Inducing points make the field tractable, while a non-centered parameterization supports optimization.

2 Nonparametric ODE model

  • The unknown vector field is modeled as a vector-valued Gaussian process, where similar states imply similar differential values according to a matrix-valued kernel.Inducing locations and vectors define the interpolated field, which is compatible with deterministic ODE trajectories and can represent arbitrary vector fields with appropriate choices.Source: §2 · pp. 2–3

2.1 Operator-valued kernels

  • Operator-valued kernels provide matrix-valued covariances between states; the identity decomposable kernel is one simple choice, while other kernels can encode prior system characteristics.Divergence-free and curl-free kernels induce different vector-field properties; input-dependent lengthscales and spectral kernels offer other modeling options.Source: §2.1 · p. 3

2.2 Joint model

  • The joint model uses a Gaussian likelihood for observations around forward-simulated responses, with inducing vectors given a Gaussian process prior.The estimated parameters include the initial state, inducing vectors, observation noise, and kernel hyperparameters.Source: §2.2 · pp. 3–4

2.3 Noncentral parameterisation

  • A non-centered parameterization maps standard-normal latent inducing vectors through a Cholesky factor to inducing values, decoupling their contributions from kernel parameters during optimization.The model finds a MAP estimate while keeping inducing locations fixed on a grid; lengthscales are selected by cross-validation.Source: §2.3 · p. 4

3 Sensitivity equations

Sensitivity equations provide gradients of simulated states with respect to vector-field parameters and the initial state. Solving the sensitivities alongside the state system avoids inefficient finite differences for these derivatives.

3 Sensitivity equations

  • The derivatives of the simulated response with respect to inducing vectors are represented by sensitivities that evolve in an additional ODE system.The state and sensitivity systems are concatenated and solved jointly; the implementation uses CVODES to solve them numerically.Source: §3 · pp. 4–5
  • Sensitivity equations provide computationally efficient analytical gradients of simulated states with respect to inducing vectors and the initial state.Finite differences are possible but highly inefficient and reported to cause instability.Source: §3 · pp. 4–5
  • The sensitivity equation is obtained by differentiating the full nonparametric ODE system with respect to the inducing vectors.Its gradients are exact up to the numerical accuracy of the ODE solver.Source: §3 · pp. 4–5

4 Simple simulated dynamics

On simulated Van der Pol, FitzHugh–Nagumo, and Lotka–Volterra oscillators, npODE learns dynamics from sparse observations and forecasts future cycles. Its vector-field estimates are most accurate near observed points, with a few errors in data-free border regions.

4 Simple simulated dynamics

  • The simulations use Van der Pol, FitzHugh–Nagumo, and Lotka–Volterra oscillators to illustrate behavior against known true dynamics, not coefficient recovery.Source: §4 · p. 5
  • Training uses 25 points from one cycle for VDP and FHN, and 25 points from 1.7 cycles for LV, with noise variance 0.12.The model uses 52 inducing locations and forecasts 4–8 cycles.Source: §4 · p. 5
  • From less than two cycles of data, the model learns the dynamics and reproduces them reliably into the future.The experiments forecast 4–8 future cycles from the true initial state.Source: §4 · p. 5
  • Estimated vector fields are accurate near observed points, with few errors in border regions without data.The most difficult regions are the middle of the FHN loop and the outermost regions of the LV model.Source: §4 · p. 5

5 Unknown system estimation

The section evaluates unknown dynamics without assumptions about system form, using human-motion trajectories for forecasting and missing-frame imputation. npODE performs best on average in forecasting, while npODE and VGPLVM have approximately equal top average imputation performance.

5 Unknown system estimation

  • The evaluation uses CMU human-walking motion data and tests forecasting and missing-frame imputation without assumptions about the underlying system’s structure or form.The 50-dimensional pose measurements are projected into a three-dimensional latent space.Source: §5 · pp. 5–6

5.1 Forecasting

  • Forecasting comparisons use original-space RMSE after latent trajectories are projected back, and the low-dimensional representation necessarily adds reconstruction error.The models optimize their latent spaces separately, so they are not directly comparable.Source: §5.1 · p. 7
  • Across 43 datasets, npODE is the best-performing method on average for forecasting.Table 1 reports means and standard deviations of RMSEs for forecasting and filling experiments.Source: Table 1 · p. 5, §5.1 · p. 8

5.2 Imputation

  • For middle-gap imputation, npODE and VGPLVM have approximately equal top average performance, although npODE’s reconstructions are slightly warped.The task removes roughly 20% of frames from the trajectory.Source: §5.2 · pp. 8–9

6 Discussion

The discussion presents the nonparametric ODE model as learning arbitrary nonlinear continuous-time dynamics from observational data without assumptions about underlying dynamics. It notes the model's scope boundaries and several directions for future work.

6 Discussion

  • The model is described as learning arbitrary nonlinear continuous-time dynamics from purely observational data without assumptions about the underlying dynamics.The authors characterize this as the beginning of a line of studies on nonparametric ODE systems.Source: §6 · p. 9
  • The current model does not handle time-dependent vector fields; future work includes stochastic extensions and MCMC sampling of the inducing-point posterior for trajectory distributions.The discussion also identifies divergence-free, curl-free, or spectral kernels and inputs or controls for interactive settings such as robotics as future avenues.Source: §6 · p. 9
  • Unlike hidden Markov and state-space models restricted to discrete steps, the model operates along continuous-time trajectories and can represent states between observations.The passage specifically contrasts access to arbitrary times with the inability of discrete-time models to represent states between successive timepoints.Source: §6 · p. 9

Sensitivity Equations

The sensitivity equations are formulated in matrix notation, with time-dependent matrices obtained by differentiating vector-valued functions with respect to vectors.

Sensitivity Equations

  • The sensitivity equation is formulated using matrix notation in the main text.The supplied passage introduces this presentation but does not state a further computational result.Source: Sensitivity Equations · p. 12
  • The time-dependent matrices are obtained by differentiating vector-valued functions with respect to vectors.The displayed derivative fragments include state and vector-field components indexed by dimensions and inducing variables.Source: Sensitivity Equations · p. 12
  • The derivative entries include state components x_D(t,U) and vector-field components f(x(t),U)_D with respect to inducing-variable components.These entries appear as parts of the matrix-derivative presentation.Source: Sensitivity Equations · p. 12

Optimization

The optimization section describes optimizing a log posterior over model variables, using sensitivities for some gradients and finite differences for the signal variance. It also specifies log-parameter optimization for positive quantities.

Optimization

  • For the explicit form of the log posterior, the authors introduce u = vec(U) and Ω = diag(ω_1^2, ..., ω_D^2) for notational simplicity.Source: Optimization · p. 12
  • Gradients are sought for the initial state, inducing vectors, kernel parameters, and noise variables, with inducing-vector gradients projected to the white domain.The passage says the inducing-vector gradients are computed and then projected using noncentral parameterisation.Source: Optimization · pp. 12–13
  • The lengthscale is selected from a grid by cross-validation, while finite differences compute the signal-variance gradient using δ = 10^-4.The passage notes that the signal-variance derivative is nontrivial because both the kernel matrix and simulated state depend on the kernel.Source: Optimization · p. 13
  • Noise standard deviations and signal variance are optimized through their logarithms to ensure their positivity.The section identifies positivity as a property not guaranteed by gradient-based optimization itself.Source: Optimization · p. 13

Implementation details

Implementation initializes inducing vectors from empirical gradients, then uses repeated perturbed starts and LBFGS optimization. Inducing locations are placed on a fixed grid, and lengthscales are selected by cross-validation.

Implementation details

  • Inducing vectors are initialized from empirical gradients between successive observations and conditioned with an optimized scale.The whitened initial inducing vector is obtained by applying L^-1 to the conditioned initialization.Source: Implementation details · p. 13
  • Optimization uses 100 restarts from random perturbations of the whitened initialization.The initialized vector fields partially match the trajectory before these restarts.Source: Implementation details · p. 13
  • The implementation uses LBFGS in Matlab, places inducing locations on an equidistant fixed grid, and selects lengthscales by cross-validation from five specified values.The candidate lengthscales are {0.5, 0.75, 1, 1.25, 1.5}; larger values induce smoother models, while lower values cause overfitting.Source: Implementation details · p. 13

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