Sampling from neural priors speeds solving partial differential equations
Sampling from Neural Operator Priors to Solve Partial Differential Equations
Machine Learning
Summary
Solving partial differential equations (PDEs) can be very hard, but it’s important for simulating physical systems like weather or fluid flow. The authors show how to use a kind of AI model called a neural operator as a starting point (a prior), combined with a special sampling technique called pseudo-bootstrap sequential Monte Carlo. They improve the process by adapting a method called DPM++ to guide the sampling better. Their approach works well across several example problems, especially when using only a few sampling steps.
What this means in practice
- •For engineering simulation teams: Accelerate solving PDE-based simulations with fewer computation steps using neural operator guided sampling.
- •For computational physics developers: Improve performance and accuracy in reconstructing physical fields from sparse data in inverse PDE problems.
Authors
Andrew Millard, Joshua Murphy, Zheng Zhao
Abstract
Guided sampling with sparse observations and PDE residuals has recently been combined with neural operator priors to solve partial differential equations in function space, their natural setting. We pair such a prior with pseudo-bootstrap sequential Monte Carlo (pBS SMC), a guided sampling framework that accommodates proposals without tractable densities, and introduce a new proposal by adapting the DPM++ algorithm to the guidance setting. Across five benchmark PDE systems this proposal attains the best error--time trade-off at every step count tested, and wrapping it in pBS gives the lowest error on most inverse-problem settings, with the largest gains when few sampling steps are used.