A Probabilistic State Space Model for Joint Inference from Differential Equations and Data

Schmidt, Jonathan; Krämer, Nicholas; Hennig, Philipp

doi:10.48550/arxiv.2103.10153

Cited by 4 publications

(3 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example, [14] and [15] applied Ensemble Adjustment Kalman Filter (EAKF) algorithm to estimate parameters in a metapopulation SEIR model. [16] proposed an extended Kalman Filter approach based on Gauss-Markov process that can infer time-varying parameter but cannot accommodate time-constant parameters any more. To the best of our knowledge, there is no existing Bayesian inference method that eliminates numerical integration for a general ODE system with both time-constant and time-varying parameters.…”

Section: Review Of Related Literaturementioning

confidence: 99%

Manifold-constrained Gaussian process inference for time-varying parameters in dynamic systems

Sun,

Yang

2023

Stat Comput

View full text Add to dashboard Cite

Identification of parameters in ordinary differential equations (ODEs) is an important and challenging task when modeling dynamic systems in biomedical research and other scientific areas, especially with the presence of time-varying parameters. This article proposes a fast and accurate method, TVMAGI (Time-Varying MAnifold-constrained Gaussian process Inference), to estimate both time-constant and time-varying parameters in the ODE using noisy and sparse observation data. TVMAGI imposes a Gaussian process model over the time series of system components as well as time-varying parameters, and restricts the derivative process to satisfy ODE conditions. Consequently, TVMAGI completely bypasses numerical integration and achieves substantial savings in computation time. By incorporating the ODE structures through manifold constraints, TVMAGI enjoys a principled statistical construct under the Bayesian paradigm, which further enables it to handle systems with missing data or unobserved components. The Gaussian process prior also alleviates the identifiability issue often associated with the time-varying parameters in ODE. Unlike existing approaches, TVMAGI can be applied to general nonlinear systems without specific structural assumptions. Three simulation examples, including an infectious disease compartmental model, Time-varying Manifold Gaussian process inference are provided to illustrate the robustness and efficiency of our method compared with numerical integration and Bayesian filtering methods.

show abstract

Section: Review Of Related Literaturementioning

confidence: 99%

Manifold-constrained Gaussian process inference for time-varying parameters in dynamic systems

Sun,

Yang

2023

Stat Comput

View full text Add to dashboard Cite

show abstract

“…Probabilistic numerical algorithms respond to these challenges by solving problems of numerical simulation with probabilistic inference. For initial value problems, probabilistic solvers share lineartime complexity, adaptive step-size selection, and high polynomial convergence rates with their non-probabilistic counterparts [4][5][6][7], and further provide functionality to quantify uncertainty within probabilistic programs [8,9].…”

Section: Boundary Value Problems In Computational Pipelinesmentioning

confidence: 99%

Linear-Time Probabilistic Solutions of Boundary Value Problems

Krämer,

Hennig

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We propose a fast algorithm for the probabilistic solution of boundary value problems (BVPs), which are ordinary differential equations subject to boundary conditions. In contrast to previous work, we introduce a Gauss-Markov prior and tailor it specifically to BVPs, which allows computing a posterior distribution over the solution in linear time, at a quality and cost comparable to that of wellestablished, non-probabilistic methods. Our model further delivers uncertainty quantification, mesh refinement, and hyperparameter adaptation. We demonstrate how these practical considerations positively impact the efficiency of the scheme. Altogether, this results in a practically usable probabilistic BVP solver that is (in contrast to non-probabilistic algorithms) natively compatible with other parts of the statistical modelling tool-chain.Preprint. Under review.

show abstract

“…Like other filtering-based ODE solvers ("ODE filters"), the algorithm used herein translates numerical approximation of ODE solutions to a problem of probabilistic inference. The resulting (approximate) posterior distribution quantifies the uncertainty associated with the unavoidable discretisation error (Bosch et al, 2021) and provides a language that integrates well with other data inference schemes (Kersting et al, 2020a;Schmidt et al, 2021). The main difference to prior work is that we focus on the setting where the dimension d of the ODE is high, that is, say, d 100.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic ODE Solutions in Millions of Dimensions

Krämer¹,

Bosch²,

Schmidt³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed implementation schemes behind solving high-dimensional ODEs with a probabilistic numerical algorithm. This has not been possible before due to matrix-matrix operations in each solver step, but is crucial for scientifically relevant problems-most importantly, the solution of discretised partial differential equations. In a nutshell, efficient high-dimensional probabilistic ODE solutions build either on independence assumptions or on Kronecker structure in the prior model. We evaluate the resulting efficiency on a range of problems, including the probabilistic numerical simulation of a differential equation with millions of dimensions.Preprint.

show abstract

A Probabilistic State Space Model for Joint Inference from Differential Equations and Data

Cited by 4 publications

References 0 publications

Manifold-constrained Gaussian process inference for time-varying parameters in dynamic systems

Manifold-constrained Gaussian process inference for time-varying parameters in dynamic systems

Linear-Time Probabilistic Solutions of Boundary Value Problems

Probabilistic ODE Solutions in Millions of Dimensions

Contact Info

Product

Resources

About