Stable Implementation of Probabilistic ODE Solvers
Authors: Nicholas Krämer, Philipp Hennig
JMLR 2024 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | Section 4 benchmarks the probabilistic ODE solver in the suggested implementation against high-order Runge-Kutta methods. The test environments that are used throughout the paper are the logistic equation, Lotka-Volterra, the restricted three-body problem, the SIR model, the Pleiades problem, and a stiffversion of the van der Pol system. The resulting rapid convergence is shown to be competitive with high-order, state-of-the-art, classical methods. |
| Researcher Affiliation | Academia | Nicholas Kr amer EMAIL T ubingen AI Center, University of T ubingen Maria-von-Linden-Straße 6, T ubingen, Germany Philipp Hennig EMAIL T ubingen AI Center, University of T ubingen Maria-von-Linden-Straße 6, T ubingen, Germany |
| Pseudocode | Yes | Algorithm 1 Template for a probabilistic ODE solver step from tn to tn+1. Require: xn N(mn, Cn) Predict mean m n+1 := Anmn Equation (2) Predict covariance C n+1 = An Cn A n + Qn Equation (2) Linearise ℓ(x) δ[Hx b] at m n+1 Equation (6) or Equation (7) Compute observation mean z = Hm n+1 b (Tronarp et al., 2019, Eq. (12c)) Compute observation covariance S = HC n+1H (Tronarp et al., 2019, Eq. (12a)) Compute Kalman gain K = C n+1H S 1 (Tronarp et al., 2019, Eq. (12b)) Update mean mn+1 = m n+1 Kz (Tronarp et al., 2019, Eq. (12d)) Update covariance Cn+1 = C n+1 KSK (Tronarp et al., 2019, Eq. (12e)) |
| Open Source Code | Yes | The implementation tricks have been made available in Prob Num (Wenger et al., 2021) and have been adopted by related software implementations of probabilistic ODE solvers since.1 1. https://github.com/pnkraemer/tornadox, https://github.com/nathanaelbosch/Prob Num Diff Eq.jl |
| Open Datasets | Yes | The test environments that are used throughout the paper are the logistic equation, Lotka-Volterra, the restricted three-body problem, the SIR model, the Pleiades problem, and a stiffversion of the van der Pol system. We begin the experiments by numerically integrating the Lotka-Volterra predator-prey model (Lotka, 1978)... |
| Dataset Splits | No | The paper focuses on numerical solutions of Ordinary Differential Equations (ODEs) which are mathematical problems defined by an initial value and a differential equation. There is no concept of training, validation, or test data splits in this context, as the methods are applied to solve the entire ODE problem. |
| Hardware Specification | No | We only rely on standard, 64-bit floating point arithmetic. |
| Software Dependencies | No | In our Python code, we use a Taylor-mode AD implementation in JAX (Bradbury et al., 2020; Bettencourt et al., 2019). Sci Py (Virtanen et al., 2020) ... LSODA (Hindmarsh and Petzold, 2005) ... Prob Num (Wenger et al., 2021). The paper mentions various software components used but does not provide specific version numbers for any of them. |
| Experiment Setup | Yes | We solve this problem using EK1 with relative tolerance 10-6, absolute tolerance 10-3, order ν = 7, adaptive step-selection with a PI-controller (Gustafsson et al., 1988), and the time-varying diffusion model originally proposed by Schober et al. (2019) and extended to EK1 by Bosch et al. (2021). Both problems use an EK1 based on a once-integrated Wiener process prior, a time-varying diffusion model, and fixed steps of size h = 0.3 (Lotka Volterra), respectively h = 3.5 (SIR). |