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An optimal Petrov-Galerkin framework for operator networks

arXiv 2503.04024 ↗First posted 6 Mar 2025math.NA cs.LG cs.NAcited by 0

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Abstract (arXiv)

The optimal Petrov-Galerkin formulation to solve partial differential equations (PDEs) recovers the best approximation in a specified finite-dimensional (trial) space with respect to a suitable norm. However, the recovery of this optimal solution is contingent on being able to construct the optimal weighting functions associated with the trial basis. While explicit constructions are available for simple one- and two-dimensional problems, such constructions for a general multidimensional problem remain elusive. In the present work, we revisit the optimal Petrov-Galerkin formulation through the lens of deep learning. We propose an operator network framework called Petrov-Galerkin Variationally Mimetic Operator Network (PG-VarMiON), which emulates the optimal Petrov-Galerkin weak form of the underlying PDE. The PG-VarMiON is trained in a supervised manner using a labeled dataset comprising the PDE data and the corresponding PDE solution, with the training loss depending on the choice of the optimal norm. The special architecture of the PG-VarMiON allows it to implicitly learn the optimal weighting functions, thus endowing the proposed operator network with the ability to generalize well beyond the training set. We derive approximation error estimates for PG-VarMiON, highlighting the contributions of various error sources, particularly the error in learning the true weighting functions. Several numerical results are presented for the advection-diffusion equation to demonstrate the efficacy of the proposed method. By embedding the Petrov-Galerkin structure into the network architecture, PG-VarMiON exhibits greater robustness and improved generalization compared to other popular deep operator frameworks, particularly when the training data is limited.

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References · 24

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12 more
  1. 2022Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problemscited by 57
  2. 2022Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Networkcited by 45
  3. 2022Scalable Uncertainty Quantification for Deep Operator Networks using Randomized Priorscited by 39
  4. 2019Deep learning observables in computational fluid dynamicscited by 38
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  6. 2022Transfer Learning Enhanced DeepONet for Long-Time Prediction of Evolution Equationscited by 28
  7. 2020Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networkscited by 26
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  12. 2024First Order System Least Squares Neural Networkscited by 4

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