This work develops a class of probabilistic algorithms for the numerical\nsolution of nonlinear, time-dependent partial differential equations (PDEs).\nCurrent state-of-the-art PDE solvers treat the space- and time-dimensions\nseparately, serially, and with black-box algorithms, which obscures the\ninteractions between spatial and temporal approximation errors and misguides\nthe quantification of the overall error. To fix this issue, we introduce a\nprobabilistic version of a technique called method of lines. The proposed\nalgorithm begins with a Gaussian process interpretation of finite difference\nmethods, which then interacts naturally with filtering-based probabilistic\nordinary differential equation (ODE) solvers because they share a common\nlanguage: Bayesian inference. Joint quantification of space- and\ntime-uncertainty becomes possible without losing the performance benefits of\nwell-tuned ODE solvers. Thereby, we extend the toolbox of probabilistic\nprograms for differential equation simulation to PDEs.\n