Probabilistic ODE Solutions in Millions of Dimensions
Authors: Nicholas Krämer, Nathanael Bosch, Jonathan Schmidt, Philipp Hennig
ICML 2022 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | 5. Empirical Evaluation The remainder evaluates the efficiency of the proposals. Next to everything detailed above, our implementation uses the preconditioner suggested by Kr amer & Hennig (2020). Its complexity is neglibible in light of Propositions 3.3 and 4.3, because all preconditioners are diagonal matrices. The full code for the solver implementation and experiments is publicly available on Git Hub.2 |
| Researcher Affiliation | Academia | 1University of T ubingen, T ubingen, Germany 2Max Planck Institute for Intelligent Systems, T ubingen, Germany. |
| Pseudocode | No | The paper describes algorithmic steps in prose (e.g., Section 2: “ODE Filter Setup”, and Section 3: “Complexity” with “Proof” sections detailing steps) but does not provide a formal pseudocode block or algorithm listing. |
| Open Source Code | Yes | The full code for the solver implementation and experiments is publicly available on Git Hub.2 |
| Open Datasets | Yes | The Lorenz96 model describes a chaotic dynamical system for which the dimension can be chosen freely (Lorenz, 1996).The Pleiades system describes the motion of seven stars in a plane, with coordinates (xi, yi) and masses mi = i, i = 1, . . . , 7 (Hairer et al., 1993, Section II.10).The Fitz Hugh Nagumo partial differential equation (PDE) is (Ambrosio & Franc oise, 2009) |
| Dataset Splits | No | The paper describes simulation setups and parameters like “adaptive steps are chosen with tolerance levels τabs = 10 3, τrel = 10 1” and discusses “calibration”, but it does not specify explicit training, validation, or test dataset splits in the context of data partitioning for model training/evaluation. |
| Hardware Specification | No | A step with the Kronecker EK0 takes 1 second for a 16 million-dimensional ODE on a consumer-level CPU. GPUs improve the runtime for extremely high-dimensional problems (d 105). |
| Software Dependencies | No | The paper mentions software like “Differential Equations.jl”, “Sci Py (Virtanen et al., 2020)”, or “Matlab (Shampine & Reichelt, 1997)”, and “DOP853 (Sci Py)” in the context of comparisons, but it does not provide specific version numbers for the software dependencies used in their own implementation. |
| Experiment Setup | Yes | In our experiments, we chose a = 208 10 4, b = 5 10 3, k = 5 10 3, τ = 0.1. As initial values, we used random samples from the uniform distribution on (0, 1). We solve the PDE from t0 = 0 to tmax = 20 on a range of spatial domains x [0, W] [0, W] R2, with W {0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50}. To turn the PDE into a system of ODEs, we discretised the Laplacian with central, second-order finite differences schemes on a uniform grid. The mesh size of the grid determines the number of grid points, which controls the dimension of the ODE problem. |