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Tim Weiland

@timwei.land

410 followers 268 following 18 posts

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

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Tim Weiland @timwei.land · 30/06/2026

Last week, I released Latte.jl 🎉 It's a probabilistic programming framework for latent Gaussian models in Julia: spatial disease maps, time trends, hierarchical GLMMs, GPs on a mesh, and more. Recognizing the structure of these models gives you access to fast inference algorithms. `] add Latte`

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Tim Weiland @timwei.land · 17/03/2025

So here's the full pipeline in a nutshell: Construct a linear stochastic proxy to the PDE you want to solve -> discretize to get a GMRF -> Gauss-Newton + sparse linear algebra to get a posterior which is informed about your data and the PDE. 7/8

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.

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Tim Weiland @timwei.land · 17/03/2025

Turns out that these "physics-informed priors" indeed then converge much faster (in terms of the discretization resolution) to the true solution. 6/8

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.

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Tim Weiland @timwei.land · 17/03/2025

But wait a sec... With this approach, we express our prior belief through an SPDE. Then why should we use the Whittle-Matérn SPDE? Instead, why don't we construct a linear SPDE that more closely captures the dynamics we care about? 5/8

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.

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Tim Weiland @timwei.land · 17/03/2025

Still with me? So we get a FEM representation of the solution function, with stochastic weights given by a GMRF. Now we just need to "inform" these weights about a discretization of the PDE we want to solve. These computations are highly efficient, due to the magic of ✨sparse linear algebra✨. 4/8

A screenshot of equations that express the posterior of a Gaussian Markov Random Field under affine observations y = Au + b.

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Tim Weiland @timwei.land · 17/03/2025

Matérn GPs are solutions to the Whittle-Matérn stochastic PDE (SPDE). In 2011, Lindgren et al. used the finite element method (FEM) to discretize this SPDE. This results in a Gaussian Markov Random Field (GMRF), a Gaussian with a sparse precision matrix. 3/8

A screenshot of the equation for the Whittle-Matérn stochastic PDE.

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Tim Weiland @timwei.land · 17/03/2025

⚙️ Want to simulate physics under uncertainty, at FEM accuracy, without much computational overhead? Read on to learn about the exciting interplay of stochastic PDEs, Markov structures and sparse linear algebra that make it possible... 🧵 1/8

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.

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