Uncertainty Quantification for Forward and Inverse Problems of PDEs via
  Latent Global Evolution
Tailin Wu Willie Neiswanger Hongtao Zheng Stefano Ermon Jure Leskovec
Deep learning-based surrogate models have demonstrated remarkable advantages
over classical solvers in terms of speed, often achieving speedups of 10 to
1000 times over traditional partial differential equation (PDE) solvers.
However, a significant challenge hindering their widespread adoption in both
scientific and industrial domains is the lack of understanding about their
prediction uncertainties, particularly in scenarios that involve critical
decision making. To address this limitation, we propose a method that
integrates efficient and precise uncertainty quantification into a deep
learning-based surrogate model. Our method, termed Latent Evolution of PDEs
with Uncertainty Quantification (LE-PDE-UQ), endows deep learning-based
surrogate models with robust and efficient uncertainty quantification
capabilities for both forward and inverse problems. LE-PDE-UQ leverages latent
vectors within a latent space to evolve both the system's state and its
corresponding uncertainty estimation. The latent vectors are decoded to provide
predictions for the system's state as well as estimates of its uncertainty. In
extensive experiments, we demonstrate the accurate uncertainty quantification
performance of our approach, surpassing that of strong baselines including deep
ensembles, Bayesian neural network layers, and dropout. Our method excels at
propagating uncertainty over extended auto-regressive rollouts, making it
suitable for scenarios involving long-term predictions. Our code is available
at: https://github.com/AI4Science-WestlakeU/le-pde-uq.