Tim Weiland

@timwei.land

406 Followers 266 Following

PhD student @ ELLIS, IMPRS-IS. Working on physics-informed ML and probabilistic numerics at Philipp Hennig's group in Tübingen. https://timwei.land

A diagram describing our method. You start with some PDE you want to solve. Then, you construct a linearized SPDE that closely resembles the dynamics of this PDE. Discretizing this SPDE (through FEM along space and some e.g. implicit Euler scheme along time) yields a GMRF prior with a sparse precision matrix. Then, you condition this prior on your (noisy) observations, e.g. the initial condition. Afterwards, you perform Gauss-Newton steps until convergence to integrate the (nonlinear) information about the PDE discretization. At convergence, you do Laplace to finally obtain a GMRF posterior over the solution of the PDE.
A plot comparing the two different priors we mentioned before for the task of solving a Burgers' equation. The x axis is the number of collocation points, and the y axis is the relative error in percentage. We observe that the advection-diffusion prior achieves a very low relative error already for a low number of collocation points, while the error drops off much more slowly for the Matérn-based prior.
A diagram comparing the effect of different priors for probabilistic numerical PDE solvers. For the task of solving a Burgers' equation, we compare a product of 1D Matérns with a prior constructed from an advection-diffusion SPDE. Both priors are conditioned only on the initial condition, and then we plot for two different time points how the dynamics evolve. We see that for the Matérn-based prior, the effect of the observations simply "dies out" over time. By contrast, the advection-diffusion prior actually propagates the observations through time in a way that much more closely resembles the Burgers' equation dynamics we want to approximate.
A screenshot of equations that express the posterior of a Gaussian Markov Random Field under affine observations y = Au + b.
A screenshot of the equation for the Whittle-Matérn stochastic PDE.
A plot comparing the performance of different methods to numerically solve PDEs. The x-axis depicts the compute time in seconds, the y-axis depicts the relative error to the ground-truth solution in percentage. The graph for the "standard" finite element method has the steepest downward slope and thus the best performance. Our methods based on Gaussian Markov Random Fields achieve the same accuracies as the finite element method, but at slight computational overheads, depicted by their graphs having a slightly flatter slope. For instance, the highest accuracy solves require ~8-9 seconds for FEM and ~25 seconds for our GMRF-based method.