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.
Code (0)
등록된 구현이 없습니다.
Tasks
Gaussian ProcessesUncertainty QuantificationSimilar Papers 제목 키워드 기반
Linearization Turns Neural Operators into Function-Valued Gaussian Processes
Neural operators generalize neural networks to learn mappings between function spaces from data. They are commonly used to learn solution operators of parametric partial differential equations (PDEs) or propagators of ti…
Gaussian ProcessesUncertainty QuantificationBayesian Hidden Physics Models: Uncertainty Quantification for Discovery of Nonlinear Partial Differential Operators from Data
What do data tell us about physics-and what don't they tell us? There has been a surge of interest in using machine learning models to discover governing physical laws such as differential equations from data, but curren…
Uncertainty QuantificationFully Nonparametric Bayesian Additive Regression Trees
Bayesian Additive Regression Trees (BART) is a fully Bayesian approach to modeling with ensembles of trees. BART can uncover complex regression functions with high dimensional regressors in a fairly automatic way and pro…
regressionUncertainty QuantificationGoal-oriented Uncertainty Quantification for Inverse Problems via Variational Encoder-Decoder Networks
In this work, we describe a new approach that uses variational encoder-decoder (VED) networks for efficient goal-oriented uncertainty quantification for inverse problems. Contrary to standard inverse problems, these appr…
DecoderUncertainty QuantificationLight-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operators
Operator learning is a powerful paradigm for solving partial differential equations, with Fourier Neural Operators serving as a widely adopted foundation. However, FNOs face significant scalability challenges due to over…