Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations
This work addresses the issue of separate error handling in PDE solvers for researchers in computational mathematics and scientific computing, offering a novel integration approach.
The paper tackles the problem of numerical solution for nonlinear, time-dependent PDEs by developing a probabilistic algorithm that integrates spatial and temporal uncertainty quantification, extending probabilistic programming tools to PDE simulation.
This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately, serially, and with black-box algorithms, which obscures the interactions between spatial and temporal approximation errors and misguides the quantification of the overall error. To fix this issue, we introduce a probabilistic version of a technique called method of lines. The proposed algorithm begins with a Gaussian process interpretation of finite difference methods, which then interacts naturally with filtering-based probabilistic ordinary differential equation (ODE) solvers because they share a common language: Bayesian inference. Joint quantification of space- and time-uncertainty becomes possible without losing the performance benefits of well-tuned ODE solvers. Thereby, we extend the toolbox of probabilistic programs for differential equation simulation to PDEs.