New way to solve physics problems with less computation
Efficient Message Passing for Partial Differential Equation Priors
Machine LearningArtificial Intelligence
Summary
Solving physical problems often involves complicated equations called partial differential equations (PDEs) that describe how things like heat or fluids behave. The authors present a new method to solve these equations using a graphical model called a factor graph, which helps combine the rules from the equations with observed data. Their approach finds approximate solutions faster than some common methods and also provides a way to understand the uncertainty in the solutions. They tested their method on two example PDEs and found it gave accurate and reliable predictions more efficiently than traditional techniques.
What this means in practice
- •For computational engineers: Improve simulation accuracy and computational efficiency when modeling systems governed by PDEs using probabilistic message passing.
- •For environmental modelers: Incorporate physical laws and observed data to generate probabilistic forecasts of environmental processes like pollutant transport or population spread.
Authors
Anna Kazachkova, Leonhard Hennicke, Rainer Schlosser, Ralf Herbrich
Abstract
Prior information for real-world physical quantities is most elegantly expressed via partial differential equations (PDEs). In this paper, we propose a novel way to solve PDEs using probabilistic inference on a factor graph. In general, factor graphs provide a natural way to encode prior knowledge into a model as explicit factors; here, this knowledge is provided by a governing PDE, which narrows the solution space, while observed data further shape the posterior over the parameters. The approximate parameter posterior is inferred using message passing based on moment matching, without posterior sampling or global gradient-based optimization. We demonstrate our approach on the first-order advection and the second-order semi-linear Fisher-KPP equations, where it achieves predictive accuracy comparable to a standard baseline while providing structured predictive uncertainty. Moreover, the inferred posterior marginal means and uncertainty structure match more closely those obtained using Hamiltonian Monte Carlo than the evaluated mean-field variational inference baseline, while requiring up to 10x less training time in our experiments, with inference speed comparable to variational inference.