Probabilistic ODE Solutions in Millions of Dimensions
This enables uncertainty quantification for large-scale scientific problems, such as discretized partial differential equations, but is incremental as it builds on existing probabilistic solver frameworks.
The paper tackles the challenge of solving high-dimensional ordinary differential equations (ODEs) probabilistically, which was previously infeasible due to computational bottlenecks, and demonstrates efficiency by simulating a differential equation with millions of dimensions.
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.