Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations

• URL: http://arxiv.org/abs/2110.11847v1
• Date: Fri, 22 Oct 2021 15:26:05 GMT
• Title: Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations
• Authors: Nicholas Kr\"amer, Jonathan Schmidt, and Philipp Hennig
• Abstract summary: Current state-of-the-art PDE solvers treat the space- and time-dimensions separately, serially, and with black-box algorithms. We introduce a probabilistic version of a technique called method of lines to fix this issue. Joint quantification of space- and time-uncertainty becomes possible without losing the performance benefits of well-tuned ODE solvers.
• Score: 20.86460521113266
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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