Neural functional a posteriori error estimates
Vladimir Fanaskov, Alexander Rudikov, Ivan Oseledets
TL;DR
This paper introduces a novel loss function for neural network training that incorporates a posteriori error estimates, enabling accurate error bounds and improved performance across various PDE-related tasks.
Contribution
It develops a systematic approach to embed a posteriori error estimates into neural network training for PDEs, enhancing accuracy and providing reliable error bounds.
Findings
Achieves better or comparable accuracy across multiple neural network architectures.
Enables cheap post-training error estimation with high-quality upper bounds.
Demonstrates effectiveness on elliptic PDE problems.
Abstract
We propose a new loss function for supervised and physics-informed training of neural networks and operators that incorporates a posteriori error estimate. More specifically, during the training stage, the neural network learns additional physical fields that lead to rigorous error majorants after a computationally cheap postprocessing stage. Theoretical results are based upon the theory of functional a posteriori error estimates, which allows for the systematic construction of such loss functions for a diverse class of practically relevant partial differential equations. From the numerical side, we demonstrate on a series of elliptic problems that for a variety of architectures and approaches (physics-informed neural networks, physics-informed neural operators, neural operators, and classical architectures in the regression and physics-informed settings), we can reach better or…
Peer Reviews
Decision·Submitted to ICLR 2024
1. By the theory from functional a posteriori error analysis, using Astral loss gives theoretical guarantee that the output of neural network is the exact solution in the sense that their distance is zero in some function space. 2. Astral loss can be computed explicitly for common PDE problems. The authors use elliptic equation as an example. 3. Some experiments are done to compare Astral loss with the commonly used residual loss using different models/equations.
1. Although the use of this type of loss in this setting might be new, this work does not prove any new theoretical results. 2. That being said, experiment is a very important component in this paper, however, I find the evaluation metric of the solution very interesting. More specifically, let $u$ be the output of neural networks and $u^*$ be the exact solution. The test error is usually computed using relative $L^2$ norm (See for example [1][2]), i.e. $$|| u - u^*||_2^2 / ||u^*||_2^2 = \int|u
- The paper is well-written and easy to follow. Figures are clear and informative. - The paper addresses a method for computing error bounds on solutions to PDEs, a challenge of high interest to the ML for physics community. - Experiments support the claims.
- It seems like this approach will have limited applicability. The first limitation is that the U function must be specified by the practitioner, but this is addressed in the paper, seems reasonable, and is a way to incorporate domain knowledge. The bigger issue is that in most scenarios, I imagine that predicting the error certificates is at least as difficult (and usually more difficult) than predicting the solution.
1. It is a novel idea to adopt the theoretical functional a posteriori error estimates for the learning objective. The functional a posteriori error estimate was initially established in the conventional finite element method analysis for PDE. 2. The entire framework has been clearly described (at least I can follow the main stream of the paper, although I am not the expert in this particular area). 3. The paper is well motivated with the goal to mitigate neural PDE solvers' inability guaran
Although this is a viable approach in a guaranteed way to produce reliable neural PDE solvers, the application could be limited, e.g., in the case for the problems where there is no theoretical posteriori estimates available.
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Taxonomy
TopicsNeural Networks and Applications