Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs

• URL: http://arxiv.org/abs/2208.01565v1
• Date: Tue, 2 Aug 2022 16:10:27 GMT
• Title: Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs
• Authors: Emilia Magnani, Nicholas Kr\"amer, Runa Eschenhagen, Lorenzo Rosasco, Philipp Hennig
• Abstract summary: We provide a mathematically detailed Bayesian formulation of the ''shallow'' (linear) version of neural operators. We then extend this analytic treatment to general deep neural operators using approximate methods from Bayesian deep learning. As a result, our approach is able to identify cases, and provide structured uncertainty estimates, where the neural operator fails to predict well.
• Score: 34.179984253109346
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• Convergence analysis of wide shallow neural operators within the framework of Neural Tangent Kernel [4.313136216120379]
  We conduct the convergence analysis of gradient descent for the wide shallow neural operators and physics-informed shallow neural operators within the framework of Neural Tangent Kernel (NTK) Under the setting of over-parametrization, gradient descent can find the global minimum regardless of whether it is in continuous time or discrete time.
  arXiv  Detail & Related papers  (2024-12-07T05:47:28Z)
• Neural Operators for Predictor Feedback Control of Nonlinear Delay Systems [3.0248879829045388]
  We introduce a new perspective on predictor designs by recasting the predictor formulation as an operator learning problem. We prove the existence of an arbitrarily accurate neural operator approximation of the predictor operator. Under the approximated-predictor, we achieve semiglobal practical stability of the closed-loop nonlinear system.
  arXiv  Detail & Related papers  (2024-11-28T07:30:26Z)
• DimOL: Dimensional Awareness as A New 'Dimension' in Operator Learning [63.5925701087252]
  We introduce DimOL (Dimension-aware Operator Learning), drawing insights from dimensional analysis. To implement DimOL, we propose the ProdLayer, which can be seamlessly integrated into FNO-based and Transformer-based PDE solvers. Empirically, DimOL models achieve up to 48% performance gain within the PDE datasets.
  arXiv  Detail & Related papers  (2024-10-08T10:48:50Z)
• Linearization Turns Neural Operators into Function-Valued Gaussian Processes [23.85470417458593]
  We introduce LUNO, a novel framework for approximate Bayesian uncertainty quantification in trained neural operators. Our approach leverages model linearization to push (Gaussian) weight-space uncertainty forward to the neural operator's predictions. We show that this can be interpreted as a probabilistic version of the concept of currying from functional programming, yielding a function-valued (Gaussian) random process belief.
  arXiv  Detail & Related papers  (2024-06-07T16:43:54Z)
• Diffusion models as probabilistic neural operators for recovering unobserved states of dynamical systems [49.2319247825857]
  We show that diffusion-based generative models exhibit many properties favourable for neural operators. We propose to train a single model adaptable to multiple tasks, by alternating between the tasks during training.
  arXiv  Detail & Related papers  (2024-05-11T21:23:55Z)
• Operator Learning: Algorithms and Analysis [8.305111048568737]
  Operator learning refers to the application of ideas from machine learning to approximate operators mapping between Banach spaces of functions. This review focuses on neural operators, built on the success of deep neural networks in the approximation of functions defined on finite dimensional Euclidean spaces.
  arXiv  Detail & Related papers  (2024-02-24T04:40:27Z)
• Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems [3.2548794659022393]
  We explore using neural operators, or neural network representations of nonlinear maps between function spaces, to accelerate infinite-dimensional Bayesian inverse problems. We show that a trained neural operator with error correction can achieve a quadratic reduction of its approximation error. We demonstrate that posterior representations of two BIPs produced using trained neural operators are greatly and consistently enhanced by error correction.
  arXiv  Detail & Related papers  (2022-10-06T15:57:22Z)
• Neural Operator: Learning Maps Between Function Spaces [75.93843876663128]
  We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations.
  arXiv  Detail & Related papers  (2021-08-19T03:56:49Z)
• Fourier Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  We formulate a new neural operator by parameterizing the integral kernel directly in Fourier space. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. It is up to three orders of magnitude faster compared to traditional PDE solvers.
  arXiv  Detail & Related papers  (2020-10-18T00:34:21Z)
• Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach [144.21892195917758]
  We study estimation in a class of generalized Structural equation models (SEMs) We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
  arXiv  Detail & Related papers  (2020-07-02T17:55:47Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.