Neural Monte Carlo PDE Solvers
Presenter: Guandao Yang
Partial Differential Equation Solvers are Useful !
E.g. Laplace equation:
(Poisson eq) Image Editing
(Perez, Gangnet, and Blake, 2012)
(Biharmonic eq) Deformation
(Jacobson et. al, 2011)
(Navier-Stokes) Fluid Simulation
(Rioux-Lavoie et. al, 2022)
Solving PDEs - Finite-element method
Figure Credit: Keenan Crane
Solving PDEs - Finite-element method
Can we solve PDEs without discretization?
Figure Credit: Keenan Crane
Discretization can be difficult!
Can we solve PDEs without discretization?
Graphics: Monte Carlo Method
Learning: Neural fields / PINNs
(Shawney and Crane, 2020)
Neural network represent the mapping from spatial coordinate to the PDE solutions; train with losses to enforce PDE constraints.
Derive an integral solution for the PDE; estimate the integral by Monte Carlo method.
(Raissi et. al., 2019, Sitzmann et. al., 2020)
The “Bias and Variance Tradeoff” between MC and NF
Graphics: Monte Carlo Method
Learning: Neural fields / PINNs
Unbiased (accurate)
High variance (slow)
Low-variance (fast)
Biased (inaccurate)
Can we combine the advantages of these two methods?
Graphics: Monte Carlo Method
Learning: Neural fields / PINNs
Unbiased (accurate)
High-variance (slow)
Low-variance (fast)
Biased (inaccurate)
Hybrid Solver
Fast
Accurate
Monte Carlo Solver for Laplace
Figure Credit: Keenan Crane
Monte Carlo Solver for Laplace
Monte Carlo Solver for Laplace
…
…
Potentially Long Walk!!!
Ours: Hybrid Solver for Laplace
How do we obtain a Neural Field solution?
Supervise directly with noisy estimate of the MC solver
Hybrid is faster than Monte Carlo methods!
We are more accurate under the same compute!
Limitation: Hybrid solver is still Biased
Solution: Use network as Control Variates
Zilu
Li
Guandao
Yang
Xi
Deng
Bharath
Hariharan
Gordon
Wetzstein
Leonidas
Guibas
Solution: Use network as Control Variates
Achieve Lower Error with Equal Number of Samples
About Two Times Faster
Accurate Hybrid PDE Solvers can be fast and accurate!
NF as MC caching
NF as MC control variate
NF as MC sampling guidance?
SIGGRAPH Asia Conf 2023
(this talk)
On going…
Future direction?
Thanks for listening!
Happy to chat more during poster!
Zilu Li
Guandao Yang
Xi Deng
Chris
De Sa
Bharath
Hariharan
Steve
Marschner
Gordon
Wetzstein
Leonidas
Guibas
Neural Monte Carlo Method - Solving PDE without Discretization
Presenter: Guandao
Collaborators: Zilu Li, Xi Deng,
Bharath Hariharan, Chris De Sa, Steve Marschner, Leo Guibas, Gordon Wetzstein