Fast and Accurate PDE Solvers via Neural Fields

演讲人: Yichen Chen MIT CSAIL
时间: 2023-04-17 16:00-2023-04-17 17:00
地点:C19-2 or 腾讯会议:203-746-249

Numerically solving partial differential equations (PDEs) often entails spatial discretizations. Traditional methods (e.g., finite difference, finite element, smoothed-particle hydrodynamics) adopt spatial discretizations, such as grids, meshes, and point clouds. While intuitive to model and understand, these discretizations suffer from (1) long runtime for large-scale problems with many spatial degrees of freedom (DOFs); (2) discretization errors, such as dissipation (fluid) and inaccurate contact (solid). In this talk, we will discuss our two recent works leveraging implicit neural representations (also known as neural fields) to tackle traditional representations’ excessive runtime and discretization errors.

The first work accelerates PDE solvers using reduced-order modeling (ROM). Whereas prior ROM approaches reduce the dimensionality of discretized vector fields, our continuous reduced-order modeling (CROM) approach builds a low-dimensional manifold of the continuous vector fields themselves, not their discretization. Compared to prior discretization-dependent ROM methods (e.g., POD, PCA, autoencoders), CROM features higher accuracy (49-79x), lower memory consumption (39-132x), dynamically adaptive resolutions, and applicability to any discretization. Experiments demonstrate 109x and 89x wall-clock speedups over unreduced models on CPUs and GPUs.

The second work encodes spatial information through neural network weights and computes PDE time-stepping with optimization time integrators. While slower to compute than traditional representations, our approach achieves higher accuracy under the same memory constraint and features adaptive allocation of DOFs without complex remeshing.


Peter Yichen Chen is a postdoctoral associate at MIT CSAIL, advised by Wojciech Matusik. He completed his CS PhD from Columbia, advised by Eitan Grinspun. Before that, he was a Sherwood-Prize-winning math undergrad from UCLA, working with Joey Teran. Peter's research centers around artificial intelligence (AI) and scientific computing. In particular, he focuses on building physics simulation frameworks with machine learning approaches.