Physics-constrained Unsupervised Learning of Partial Differential Equations using Meshes
This enables interactive PDE solving for applications like soft-body deformations, though it is incremental as it builds on existing graph neural network methods.
The paper tackles the problem of solving partial differential equations (PDEs) on irregular meshes without requiring large supervised datasets, by developing an unsupervised learning framework using graph neural networks with physics-based constraints, and shows it is computationally efficient and remains close to a classical numerical solver.
Enhancing neural networks with knowledge of physical equations has become an efficient way of solving various physics problems, from fluid flow to electromagnetism. Graph neural networks show promise in accurately representing irregularly meshed objects and learning their dynamics, but have so far required supervision through large datasets. In this work, we represent meshes naturally as graphs, process these using Graph Networks, and formulate our physics-based loss to provide an unsupervised learning framework for partial differential equations (PDE). We quantitatively compare our results to a classical numerical PDE solver, and show that our computationally efficient approach can be used as an interactive PDE solver that is adjusting boundary conditions in real-time and remains sufficiently close to the baseline solution. Our inherently differentiable framework will enable the application of PDE solvers in interactive settings, such as model-based control of soft-body deformations, or in gradient-based optimization methods that require a fully differentiable pipeline.