Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs
Authors: Emilia Magnani, Nicholas Krämer, Runa Eschenhagen, Lorenzo Rosasco, Philipp Hennig
TMLR 2025 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | In this section, we apply the theoretical framework from Section 3 to construct Bayesian neural operators that provide uncertainty estimates. We begin with the shallow case, leveraging the exact Gaussian process formulation of Section 3.1, and then proceed to the deep setting. By replicating experiments from Li et al. (2020b), we show that we can effectively detect wrong predictions. In Section 5.3, we evaluate our method on a benchmark for PDEs and compare its performance with other widely used uncertainty quantification methods. |
| Researcher Affiliation | Academia | Emilia Magnani EMAIL Tübingen AI Center, University of Tübingen Nicholas Krämer EMAIL Technical University of Denmark Runa Eschenhagen EMAIL University of Cambridge Lorenzo Rosasco EMAIL Ma LGa DIBRIS, University of Genova, Istituto Italiano di Tecnologia Philipp Hennig EMAIL Tübingen AI Center, University of Tübingen |
| Pseudocode | No | The paper describes methods and architectures in detail, such as the neural operator architecture (Figure 2) and the mathematical formulation of the Bayesian framework, but it does not include any structured pseudocode or algorithm blocks. |
| Open Source Code | No | The paper mentions using a third-party code for graph-based neural operators: "We first use their original code for graph-based neural operators1 using message-passing layers (Kipf & Welling, 2016; Gilmer et al., 2017)..." with a footnote pointing to a GitHub repository. However, it does not explicitly state that the authors' own implementation of the approximate Bayesian framework or the Laplace approximation is open-source or provide a link for it. |
| Open Datasets | Yes | We assess uncertainty quantification performance on a diverse set of 1d equations from APEBench (Koehler et al., 2024), including Burgers , hyper-diffusion, and Kuramoto Sivashinsky equations. |
| Dataset Splits | Yes | In particular, since the problem is relatively simple, we consider an extreme setting where we train on only two training functions and subsample only two points from a 16 16 grid for each. ... Figure 5 shows results on a dense 61 61 grid, analogous to the previous one, trained on 100 densely evaluated 16 16 grid solutions. ... We train for 100 epochs on 40 trajectories using mean squared error loss and Adam, with r 0.3 on grids of 256 points. ... Evaluation is performed on 100 input output pairs. |
| Hardware Specification | No | The paper does not provide any specific details about the hardware used to run the experiments, such as GPU models, CPU types, or memory specifications. |
| Software Dependencies | No | All components are implemented in jax (Bradbury et al., 2018). While JAX is mentioned as a software component, no specific version number is provided for it or any other libraries/frameworks used. |
| Experiment Setup | Yes | To recreate the results in Li et al. (2020b) we first use their original code for graph-based neural operators using message-passing layers (Kipf & Welling, 2016; Gilmer et al., 2017) with 64 hidden dimensions and Re LU activations. Training is performed via the Adam optimizer. ... Our graph neural operator implementation has four layers with 18 hidden features each. ... We train for 100 epochs on 40 trajectories using mean squared error loss and Adam, with r 0.3 on grids of 256 points. ... The prior precision is selected via grid search, optimizing for marginal negative log-likelihood (NLL). |