35,364 research outputs found
Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs
Neural operators are a type of deep architecture that learns to solve (i.e.
learns the nonlinear solution operator of) partial differential equations
(PDEs). The current state of the art for these models does not provide explicit
uncertainty quantification. This is arguably even more of a problem for this
kind of tasks than elsewhere in machine learning, because the dynamical systems
typically described by PDEs often exhibit subtle, multiscale structure that
makes errors hard to spot by humans. In this work, we first provide a
mathematically detailed Bayesian formulation of the ''shallow'' (linear)
version of neural operators in the formalism of Gaussian processes. We then
extend this analytic treatment to general deep neural operators using
approximate methods from Bayesian deep learning. We extend previous results on
neural operators by providing them with uncertainty quantification. As a
result, our approach is able to identify cases, and provide structured
uncertainty estimates, where the neural operator fails to predict well
AReS and MaRS - Adversarial and MMD-Minimizing Regression for SDEs
Stochastic differential equations are an important modeling class in many
disciplines. Consequently, there exist many methods relying on various
discretization and numerical integration schemes. In this paper, we propose a
novel, probabilistic model for estimating the drift and diffusion given noisy
observations of the underlying stochastic system. Using state-of-the-art
adversarial and moment matching inference techniques, we avoid the
discretization schemes of classical approaches. This leads to significant
improvements in parameter accuracy and robustness given random initial guesses.
On four established benchmark systems, we compare the performance of our
algorithms to state-of-the-art solutions based on extended Kalman filtering and
Gaussian processes.Comment: Published at the Thirty-sixth International Conference on Machine
Learning (ICML 2019
- …