Tim Weiland
@timwei.land
PhD student @ ELLIS, IMPRS-IS.
Working on physics-informed ML and probabilistic numerics at Philipp Hennig's group in Tübingen.
https://timwei.land
Working on physics-informed ML and probabilistic numerics at Philipp Hennig's group in Tübingen.
https://timwei.land
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
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
March 17, 2025 at 12:25 PM
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
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
Turns out that these "physics-informed priors" indeed then converge much faster (in terms of the discretization resolution) to the true solution. 6/8
March 17, 2025 at 12:25 PM
Turns out that these "physics-informed priors" indeed then converge much faster (in terms of the discretization resolution) to the true solution. 6/8
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
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
March 17, 2025 at 12:25 PM
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
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
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
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
March 17, 2025 at 12:25 PM
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
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
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
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
March 17, 2025 at 12:25 PM
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
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
⚙️ 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
Read on to learn about the exciting interplay of stochastic PDEs, Markov structures and sparse linear algebra that make it possible... 🧵 1/8
March 17, 2025 at 12:25 PM
⚙️ 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
Read on to learn about the exciting interplay of stochastic PDEs, Markov structures and sparse linear algebra that make it possible... 🧵 1/8